Research ArticleA New Gruumlnwald-Letnikov Derivative Derived from aSecond-Order Scheme

B A Jacobs12

1School of Computer Science and Applied Mathematics University of the Witwatersrand Johannesburg Private Bag 3Wits 2050 South Africa2DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) University of the WitwatersrandJohannesburg Private Bag 3 Wits 2050 South Africa

Correspondence should be addressed to B A Jacobs byronjacobswitsacza

Received 25 May 2015 Accepted 16 August 2015

Academic Editor Tiecheng Xia

Copyright copy 2015 B A Jacobs This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A novel derivation of a second-order accurate Grunwald-Letnikov-type approximation to the fractional derivative of a functionis presented This scheme is shown to be second-order accurate under certain modifications to account for poor accuracy inapproximating the asymptotic behavior near the lower limit of differentiation Some example functions are chosen and numericalresults are presented to illustrate the efficacy of this new method over some other popular choices for discretizing fractionalderivatives

1 Introduction

Fractional differential equations and fractional order deriva-tives have become increasingly popular in recent years Theuse of fractional order derivatives has become abundantin modeling techniques as well as computational methodsfor the numerical solution of these models Zeng et al[1] derive a generalized fractional diffusion equation byaddressing deficiencies in previously proposed models foranomalous diffusion An application of modelling plumesmoving at a fractional rate was investigated by Benson et al[2] which showed a fractional advection-dispersion model tobe appropriate as the mean squared displacement followed1199051120572 He [3] presented a more exact model for seepage flowby relaxing the continuity assumption and went on to solvethe arising equation using the variational iteration method(VIM) Metzler et al [4] provide a review of anomalousdiffusion equations presented as well as the formulation of ageneral diffusion case overcoming previous inconsistenciesMoving past anomalous diffusion Gafiychuk et al [5 6]studied a fractional reaction-diffusion system modelling anactivator-inhibitor dynamic The application of fractionalorder models is various and with this increased complexity

requires both analytic and numerical methods to find usefulsolutions

Meerschaert and Tadjeran [7] construct an implicitscheme for two-sided space-fractional partial differentialequations based on two first-order approximations to the left-and right-shifted Grunwald-Letnikov fractional derivativeScherer et al [8] also derive an implicit scheme for thefractional heat equation in the Caputo sense making useof the standard Grunwald-Letnikov scheme as well as therelationship between the Riemann-Liouville derivative andthe Caputo derivative There has been work conducted oncomparing numerical methods with analytical methods El-Sayed et al [9] for example compare a modified numericalmethod wherein the authors apply a transform to homoge-nize the initial conditions with the Adomian decompositionmethod Another approach taken byKaratay et al [10] derivesan unconditionally stable implicit scheme based on a first-order approximation to the Caputo derivative An interestingstudy is presented by Su et al [11] which combines the first-order shifted Grunwald-Letnikov scheme with characteristicmethods and a departure from the Riemann-Liouville andCaputo derivatives leadsChen et al [12] to derive a condition-ally stable explicit scheme and unconditionally stable implicit

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2015 Article ID 952057 9 pageshttpdxdoiorg1011552015952057

2 Abstract and Applied Analysis

scheme for the Caputo-Riesz fractional diffusion equationthat are superlinearly convergent

Tadjeran and Meerschaert [13] implement a second-order accurate Crank-Nicolson-type finite difference schemefor the space-fractional diffusion equation making use ofthe right-shifted Grunwald-Letnikov fractional derivativeDiethelm et al [14] propose a predictor-corrector-typeapproach for the numerical solution of both linear andnonlinear fractional differential equations including systemsof equations Mohebbi et al [15] study a high-order accu-rate numerical method for obtaining the solution to thesubdiffusion equation with a nonlinear source term Cui[16] goes on to solve the two-dimensional time-fractionaldiffusion equation using a high-order compact alternatingdirection implicit method Meerschaert et al [17] solve thetwo-dimensional space-fractional dispersion equation usinga right-shifted Grunwald-Letnikov discretization since theorder of the fractional derivative is above 1 but less than2 Lynch et al [18] implement a finite difference approachfor the space-fractional dispersion problem but implementdiscretization switching to maximize accuracy based on theorder of derivative Momani and Odibat [19] depart from thefinite-difference approach and instead compare two analyticmethods for the solution of linear fractional differentialequations namely the variational iteration method and theAdomian decomposition method These authors go on toexplore the application of these methods to nonlinear partialdifferential equations of fractional order in [20]

The present work derives a difference approximation in anovel way that produces a form similar to the known resultpresented by Oldham and Spanier in [21] This text cites anerror in the approximation of the fractional derivative of aconstant and linear function of order119873minus1 that is proportionalto the number of terms in the series approximation Anovel technique is coupled with this approximation to attainorder 119873minus2 accuracy Although the result obtained is knownto the best of the authorrsquos knowledge this derivation isnovel

Another important technique used in the numericalapplication of fractional differential equations is the ldquoshort-memoryrdquo principle [22]This technique truncates the approx-imation sum to avoid high computational costs It is shown inSection 4 that the presentmethod is highly accurate evenwithvery few terms in the approximation sum

The following section presents some preliminary resultsused in the remainder of the paper Section 3 goes on toderive the new method of discretizing a fractional derivativewith results on the performance of this technique reportedin Section 4 We make some concluding remarks and discussopportunities for further work in Section 5

2 Preliminaries

This section presents some preliminary results that are putto use in subsequent sections However in the interestof brevity we omit some useful properties but direct theinterested reader to the work by Li and Deng [23] Oftenfor numerical simulation of fractional order problems the

Grunwald-Letnikov derivative is used as ameans of discretiz-ing a fractional order derivative The Grunwald-Letnikovdiscretization has been used for numerical schemes forfractional partial differential equations [13 24 25] and isdefined as follows

Definition 1 The Grunwald-Letnikov derivative of order 120572 gt0 of a function 119906(119905) is

GL119863120572

0119905119906 (119905) = lim

Δ119905rarr0

1

Δ119905120572

119873

sum

119896=0

120596120572

119896119906 (119905 minus 119896Δ119905) (1)

where

120596120572

119896= (minus1)

119896(

120572

119896

) (2)

and119873Δ119905 = 119905

However the limit definition is only practically usefulfor finite-difference implementations hence the followingdefinition is used

Definition 2 TheGrunwald-Letnikov derivative of order 120572 gt0 of a function 119906(119905) isin 119862119898[0 119905] is [22]

119863120572

0119905119906 (119905) =

119898minus1

sum

119896=0

119906(119896)(0) 119905minus119886+119896

Γ (minus120572 + 119896 + 1)+

1

119898 minus 120572

sdot int

119905

0

(119905 minus 120591)119898minus120572minus1

119906(119898)

(120591) 119889120591

(3)

where119898 minus 1 le 120572 lt 119898 isin Z+

Definition 2 is used as a means of calculating theexact fractional derivative of a function The exact solutionobtained here is then used as a measure for accuracy whencompared with the discrete approximations It is prudent tomention that if 119906(119905) is sufficiently smooth 119906(119905) isin 119862

119898[0 119905]

then the Grunwald-Letnikov derivative is equivalent to thepopular Riemann-Liouville defined below

Definition 3 The Riemann-Liouville derivative of fractionalorder 120572 gt 0 of a function 119906(119905) is [22]

RL119863120572

0119905119906 (119905) =

1

Γ (119898 minus 120572)

119889119898

119889119905119898int

119905

0

(119905 minus 120591)119898minus120572minus1

119906 (120591) 119889120591 (4)

The right-shifted Grunwald-Letnikov approximationintroduced by Meerschaert and Tadjeran [25] as used in[13] leads to stable numerical schemes and is defined below

Definition 4 The right-shifted Grunwald-Letnikov approxi-mation of a function 119906(119905) for 1 lt 120572 le 2 is [25]

RSGL119863120572

0119905119906 (119905)

= lim119873rarrinfin

1

Δ119905120572

119873

sum

119896=0

(minus1)119896(

120572

119896

)119906 (119905 minus (119896 minus 1) Δ119905)

(5)

Abstract and Applied Analysis 3

Oldham and Spanier obtain a second-order Grunwald-Letnikov approximation simply by noting that the Riemannsumpresent in the derivativewould converge faster if a simplemodification was made

Definition 5 TheGrunwald-Letnikov derivative of order 120572 gt0 of a function 119906(119905) with a central difference modification is[21]

GL119863120572

0119905119906 (119905)

= limΔ119905rarr0

1

Δ119905120572

119873minus1

sum

119896=0

(minus1)119896(

120572

119896

)119906(119905 minus 119896Δ119905 +120572Δ119905

2)

(6)

where119873Δ119905 = 119905

The primary difference between Definition 5 and thepresent work is the derivation and the choice of119873 the upperlimit of the Riemann sum Although these discrepancies areslight we also address the issue of the asymptotic behaviorof the fractional derivative as 119905 approaches zero a difficultbehavior for a difference scheme to capture

3 Method

In much the same way the Grunwald-Letnikov deriva-tive is generalized from the first-order backward differencescheme as described by Podlubny in [22] we derive anew approximate fractional derivative in the form of adifference scheme which is derived from the second-ordercentral difference scheme Yuste and Acedo [24] state thatthe standard Grunwald-Letnikov derivative is accurate toO(Δ119905) while we derive a scheme based on the central-difference scheme we hope to achieve an approximationthat is O(Δ1199052) Podlubny [26] shows by example that theGrunwald-Letnikov derivative of a function 119891(119905) = 1 or119891(119905) = 119905

119898 is accurate to O(Δ119905) and hence conclude that theGrunwald-Letnikov derivative of any function that can bewritten as a power series will be accurate toO(Δ119905)The centraldifference approximation to the first-order derivative is givenby

1199061015840(119905) = limΔ119905rarr0

119906 (119905 + Δ119905) minus 119906 (119905 minus Δ119905)

2Δ119905+ O (Δ119905

2) (7)

or

1199061015840(119905) = limΔ119905rarr0

119906 (119905 + Δ1199052) minus 119906 (119905 minus Δ1199052)

Δ119905+ O(

Δ1199052

22) (8)

Similarly the second-order derivative can be approximated by

11990610158401015840(119905) = limΔ119905rarr0

119906 (119905 + Δ119905) minus 2119906 (119905) + 119906 (119905 minus Δ119905)

(Δ119905)2

+ O (Δ1199052)

(9)

Following this trend we may write

119906(119899)(119905)

= limΔ119905rarr0

1

(Δ119905)119899

119899

sum

119896=0

(minus1)119896(

119899

119896

)119906(119905 minus 119896Δ119905 +119899Δ119905

2)

(10)

Generalizing the order of the derivative to be a positive realvalue120572 and substituting into (10)we obtain a general formula

GL2119863120572

0119905119906 (119905) = 119906

(120572)(119905)

= limΔ119905rarr0

1

(Δ119905)120572

119873

sum

119896=0

(minus1)119896(

120572

119896

)119906(119905 minus 119896Δ119905 +120572Δ119905

2)

(11)

where the binomial coefficient is calculated by the use of theGamma function that generalizes the factorial operator

(

120572

119896

) =120572

119896 (120572 minus 119896)=

Γ (120572 + 1)

Γ (119896 + 1) Γ (120572 minus 119896 + 1) (12)

It is well known [26 27] that the fractional differentiationoperator is a nonlocal operator This effect is seen in thediscretization being affected by every point in the functionand not just a neighborhood of the point 119905This insight guidesour choice of 119873 as the upper limit of the summation Wechoose 119873 so that the last term in the series correspondswith the terminating point of the function 119906(0) and in termsof fractional differential equations and fractional partialdifferential equations this would correspondwith a boundaryor initial condition

119873 = lceil119905

Δ119905+120572

2rceil (13)

where lceilsdotrceil represents the ceiling function Due to the singu-larity of a fractional derivative GL119863

120572

0119905119906(119905) when 119905 = 0 it

is convenient to select 119906(119905) such that 119906(0) = 0 to alleviatethe asymptotic behavior However we do not want to restrictourselves only to functions that obey this property Hence wedefine

119892 (119905) = 119906 (119905) minus 119906 (0) (14)

so that 119892(0) = 0 and then

GL2119863120572

0119905119906 (119905) = GL2119863

120572

0119905119892 (119905) + GL2119863

120572

0119905119906 (0) (15)

since the fractional derivative is a linear operator We maynow exploit Property 22 in [23] which yields

L GL2119863120572

0119905119906 (0) = 119904

120572L 119906 (0) (16)

which then leads to

Lminus1119904120572L 119906 (0) =

119905minus120572119906 (0)

Γ (1 minus 120572) (17)

Finally (15) becomes

GL2119863120572

0119905119906 (119905) = GL2119863

120572

0119905119892 (119905) +

119905minus120572119906 (0)

Γ (1 minus 120572) (18)

While correcting for the asymptotic behavior at 119905 = 0 pro-duces sufficiently accurate results wemay go one step furtherand substitute the exact solution of the linear component of119906(119905) This is done to ensure that the accuracy of the scheme isat least O(119873minus2) Let

ℎ (119905) = 119906 (119905) minus 119906 (0) minus 1199051199061015840(0) (19)

then

GL2119863120572

0119905119906 (119905) = GL2119863

120572

0119905119892 (119905) +

119905minus120572119906 (0)

Γ (1 minus 120572)+1199051minus120572

1199061015840(0)

Γ (2 minus 120572) (20)

4 Abstract and Applied Analysis

31 Accuracy As described by Oldham and Spanier in [21]the error in the above scheme for functions 119906(119905) = 119862 119906(119905) = 119905and 119906(119905) = 119905

2 is given by 120572(120572+1)2119873 12057222119873 and 120572(120572minus1)(120572minus2)24119873

2 respectively Since we make the exact substitutionsfor the linear components we are able to increase the accuracyof the scheme to O(119873minus2) for any analytic function

Theorem 6 The error of GL2119863120572

0119905119906(119905) denoted by 120598(119906(119905))GL2 =

|119863120572

0119905119906(119905) minus GL2119863

120572

0119905119906(119905)| is at least O(119873minus2) accurate for the

function 119906(119905) = 119905119901 for all 119901 ge 2 given that the exact derivative

of order 120572 of 119905119901 is

119863120572

0119905119905119901=119905119901minus120572

Γ (119901 + 1)

Γ (119901 minus 120572 + 1) (21)

Proof Let 119901 = 2 Then (see [21] Chapter 8) (while Oldhamand Spanier cite the relative error it is easy to show thatthe more restrictive absolute error is also O(119873minus2) Moreinformation can be found in [22] Chapter 7)

120598 (119906 (119905))GL2 =120572 (120572 minus 1) (120572 minus 2)

241198732

(22)

Assume that the order of error is at least O(119873minus2) for 119901 = 119898that is

GL2119863120572

0119905119905119898=119905119898minus120572

Γ (119898 + 1)

Γ (119898 minus 120572 + 1)+ O (119873

minus2) (23)

Now let 119901 = 119898 + 1 then by the product rule for fractionalderivatives (see [21] Chapter 5) we have

GL2119863120572

0119905119905119898+1

= GL2119863120572

0119905(119905119898119905)

= 119905GL2119863120572

0119905119905119898+ 120572GL2119863

120572minus1

0119905119905119898

asymp 119905119905119898minus120572

Γ (119898 + 1)

Γ (119898 minus 120572 + 1)+ 120572

119905119898minus120572+1

Γ (119898 + 1)

Γ (119898 minus 120572)

+ O (119873minus2)

=119905119898minus120572+1

Γ (119898 + 1)

Γ (119898 minus 120572 + 1)(1 +

120572

119898 + 1 minus 120572)

+ O (119873minus2)

(24)

Then using the recursive property of the Gamma function119909Γ(119909) = Γ(119909 + 1) we have

GL2119863120572

0119905119905119898+1

=119905119898+1minus120572

Γ (119898 + 2)

Γ (119898 minus 120572 + 2)+ O (119873

minus2) (25)

Using Theorem 6 we can deduce that the accuracy ofGL2119863120572

0119905119906(119905) for any analytic function 119906(119905) isO(119873minus2) provided

that the linear and constant components of that function aredealt with accordingly

4 Results

This section presents some numerical results for differentfunctions Each example shows the absolute error over thedomain of the function the consistency of each derivativeas ℎ tends to zero the absolute error of the truncated seriesand the absolute error over a range of 120572 We comparethe aforementioned criteria for four different methods thestandard Grunwald-Letnikov approximation ( GL119863

120572

0119905) the

second-order Grunwald-Letnikov approximation ( GL2119863120572

0119905)

the second-order Grunwald-Letnikov approximation withconstant correction described in Section 3 (CCGL2119863

120572

0119905) and

finally the right-shifted Grunwald-Letnikov approximation( RSGL119863

120572

0119905)

41 Example 1 Let

119906 (119905) = 1199052 (26)

and let the order of derivative be 120572 = 075 with timestep Δ119905 = 0001 Since this choice of 119906(119905) satisfies 119906(0) =

0 the approximation does not suffer from poor accuracynear 119905 = 0 and hence the second-order Grunwald-Letnikov( GL2119863

120572

0119905) and the second-order Grunwald-Letnikov with

constant correction (CCGL2119863120572

0119905) approximations are similar

42 Example 2 Let

119906 (119905) = cos (119905) (27)

the order of derivative 120572 = 15 andΔ119905 = 0001 Asmentionedby Oldham and Spanier [21] this method is robust for all 120572this example shows highly accurate results for 120572 gt 1

43 Example 3 Let

119906 (119905) = cos (119905) + sin (119905) 1199052 (28)

the order of derivative 120572 = 09 and again Δ119905 = 0001This function suffers from the asymptotic behavior of thefractional derivative near 119905 = 0 and hence the constantcorrection method enjoys a vastly superior accuracy over theother methods

44 Example 4 Let

119906 (119905) = 119890minus119905 cos (119905) (29)

the order of derivative 120572 = 21 and Δ119905 = 0001

45 Example 5 Let

119906 (119905) = sin (119905) + cos (119905) (30)

the order of derivative 120572 = 075 and Δ119905 = 0001The above results indicate that the present method enjoys

a high accuracy The nature of the fractional derivative nearthe lower limit of differentiation results in a high gradientDifference schemes with fixed interval steps are unable to

Abstract and Applied Analysis 5

0 1 2 3 4 5 600000

00005

00010

00015

00020

t

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 1 Absolute error over time domain for four differentmethods

000 002 004 006 008 010000

002

004

006

008

h

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 2 Absolute error over varying values of Δ119905 for four differentmethods

accurately capture very steep gradients similar to those at theasymptote of the fractional derivativeThe use of the constantcorrection allows one to avoid the asymptotic behavior near119905 = 0 and the provable accuracy

Figures 1 5 9 13 and 17 show the increasing accuracyof the approximations as 119905 deviates from 0 as well as therobustness of our method to the asymptotic behavior Theconsistency of the methods is evident in Figures 2 6 10 14and 18These figures indicate that as ℎ tends to 0 the absoluteerror of the approximations also tends to 0 As mentionedin Section 1 the ldquoshort-memoryrdquo principle allows for thetruncation of the approximating series in order to alleviatecomputational cost our present method is highly accurateeven for small values of 119873 as is evident in Figures 3 7 1115 and 19 The final set of figures Figures 4 8 12 16 and 20indicate the robustness of our method to varying values of 120572It is important to note here the log scale of the graph

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 50 100 150 20000000

00005

00010

00015

00020

N

Abso

lute

erro

r

Figure 3Absolute error for increasing119873 for four differentmethods

00 05 10 15 20 25 30

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

120572

10minus2

10minus5

10minus8

10minus14

10minus11

Figure 4 Absolute error over varying values of 120572 for four differentmethods

5 Conclusion

This work has shown a novel derivation of a second-orderaccurate scheme for approximating the fractional orderderivative of a function To the best of the authorrsquos knowledgeperforming a linear transformation on the function to avoidasymptotic behavior near the origin is also novel Section 4provides several examples with varying parameters andbehaviors to illustrate the efficacy of the present method

Using a higher-order accurate approximation leads tothe potential for higher-order accurate numerical schemesfor fractional differential equations and fractional partialdifferential equations Tadjeran and Meerschaert [13] imple-ment the right-shifted Grunwald-Letnikov derivative forfractional derivatives of order 1 lt 120572 le 2 which lead toan unconditionally stable numerical scheme According toOldham and Spanier [21] the present method is robust for allvalues of alpha Potential extensions to this work include con-structing numerical schemes centered on this approximationand the fractional order derivative for nonlinear fractional

6 Abstract and Applied Analysis

0 1 2 3 4 5 600000

00005

00010

00015

t

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 5 Absolute error over time domain for four differentmethods

000 002 004 006 008 0100

1

2

3

4

5

6

7

h

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 6 Absolute error over varying values of Δ119905 for four differentmethods

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 50 100 150 200000

005

010

015

020

025

N

Abso

lute

erro

r

Figure 7 Absolute error for increasing 119873 for four different meth-ods

00 05 10 15 20 25 30

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

10minus7

10minus1

10minus4

120572

Figure 8 Absolute error over varying values of 120572 for four differentmethods

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 1 2 3 4 5 60000

0005

0010

0015

t

Figure 9 Absolute error over time domain for four differentmethods

000 002 004 006 008 01000

01

02

03

04

h

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 10 Absolute error over varying values ofΔ119905 for four differentmethods

Abstract and Applied Analysis 7

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 50 100 150 2000000

0002

0004

0006

0008

N

Abso

lute

erro

r

Figure 11 Absolute error for increasing 119873 for four differentmethods

00 05 10 15 20 25 30

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

120572

10minus1

10minus4

10minus7

Figure 12 Absolute error over varying values of 120572 for four differentmethods

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 1 2 3 4 5 600000

00001

00002

00003

00004

00005

t

Abso

lute

erro

r

Figure 13 Absolute error over time domain for four differentmethods

000 002 004 006 008 010000

005

010

015

020

h

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 14 Absolute error over varying values ofΔ119905 for four differentmethods

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 200 400 600 800 10000000

0001

0002

0003

0004

0005

N

Abso

lute

erro

r

Figure 15 Absolute error for increasing 119873 for four differentmethods

00 05 10 15 20 25 300000

0002

0004

0006

0008

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

120572

Figure 16 Absolute error over varying values of 120572 for four differentmethods

8 Abstract and Applied Analysis

0 1 2 3 4 5 600000

00005

00010

00015

t

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 17 Absolute error over time domain for four differentmethods

000 002 004 006 008 01000

01

02

03

04

05

06

h

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 18 Absolute error over varying values ofΔ119905 for four differentmethods

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 50 100 150 2000000

0005

0010

0015

N

Abso

lute

erro

r

Figure 19 Absolute error for increasing 119873 for four differentmethods

00 05 10 15 20 25 30

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

120572

10minus1

10minus4

10minus7

Figure 20 Absolute error over varying values of 120572 for four differentmethods

differential equations as well as time-marching schemes fortime or space fractional partial differential equations

The accuracy of the method has been proven to be atleast of O(119873minus2) which suggests that the accuracy should beof order 10minus6 for Δ119905 However in these examples the erroris observed to be in the region of 10minus8 Nevertheless themathematical proof of accuracy as well as the compellingresults presented substantiates the efforts toward this field

Disclaimer

The opinions expressed and conclusions arrived at herein arethose of the author and are not necessarily to be attributed tothe CoE-MaSS

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author would like to acknowledge and thank ProfessorCharis Harley for her helpful suggestions and useful insightThe DST-NRF Centre of Excellence in Mathematical andStatistical Sciences (CoE-MaSS) is acknowledged for fundingunder Grant no 94005 Thanks are due to the reviewers fortheir commentswhich helped to elevate the level of this work

References

[1] Q Zeng H Li and D Liu ldquoAnomalous fractional diffusionequation for transport phenomenardquo Communications in Non-linear Science and Numerical Simulation vol 4 no 2 pp 99ndash104 1999

[2] D A Benson S W Wheatcraft and M M MeerschaertldquoApplication of a fractional advection-dispersion equationrdquoWater Resources Research vol 36 no 6 pp 1403ndash1412 2000

Abstract and Applied Analysis 9

[3] J-H He ldquoApproximate analytical solution for seepage flowwithfractional derivatives in porous mediardquo Computer Methods inApplied Mechanics and Engineering vol 167 no 1-2 pp 57ndash681998

[4] R Metzler W G Glockle and T F Nonnenmacher ldquoFractionalmodel equation for anomalous diffusionrdquo Physica A StatisticalMechanics and its Applications vol 211 no 1 pp 13ndash24 1994

[5] V Gafiychuk B Datsko and V Meleshko ldquoMathematicalmodeling of time fractional reaction-diffusion systemsrdquo Journalof Computational and AppliedMathematics vol 220 no 1-2 pp215ndash225 2008

[6] B Y Datsko and V V Gafiychuk ldquoMathematical modeling offractional reaction-diffusion systems with different order timederivativesrdquo Journal of Mathematical Sciences vol 165 no 3 pp392ndash402 2010

[7] M M Meerschaert and C Tadjeran ldquoFinite difference approxi-mations for two-sided space-fractional partial differential equa-tionsrdquoApplied Numerical Mathematics vol 56 no 1 pp 80ndash902006

[8] R Scherer S L Kalla L Boyadjiev and B Al-Saqabi ldquoNumer-ical treatment of fractional heat equationsrdquo Applied NumericalMathematics vol 58 no 8 pp 1212ndash1223 2008

[9] A M El-Sayed I L El-Kalla and E A Ziada ldquoAnalytical andnumerical solutions of multi-term nonlinear fractional ordersdifferential equationsrdquo Applied Numerical Mathematics vol 60no 8 pp 788ndash797 2010

[10] I Karatay R S Bayramoglu and A Sahin ldquoImplicit differenceapproximation for the time fractional heat equation with thenonlocal conditionrdquo Applied Numerical Mathematics vol 61no 12 pp 1281ndash1288 2011

[11] L Su W Wang and H Wang ldquoA characteristic differencemethod for the transient fractional convection-diffusion equa-tionsrdquo Applied Numerical Mathematics vol 61 no 8 pp 946ndash960 2011

[12] M Chen W Deng and Y Wu ldquoSuperlinearly convergentalgorithms for the two-dimensional space-time caputo-rieszfractional diffusion equationrdquo Applied Numerical Mathematicsvol 70 pp 22ndash41 2013

[13] C Tadjeran and M M Meerschaert ldquoA second-order accuratenumerical method for the two-dimensional fractional diffusionequationrdquo Journal of Computational Physics vol 220 no 2 pp813ndash823 2007

[14] K Diethelm N J Ford and A D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 3ndash22 2002

[15] A Mohebbi M Abbaszadeh and M Dehghan ldquoA high-orderand unconditionally stable scheme for the modified anomalousfractional sub-diffusion equationwith a nonlinear source termrdquoJournal of Computational Physics vol 240 pp 36ndash48 2013

[16] M Cui ldquoCompact alternating direction implicit method fortwo-dimensional time fractional diffusion equationrdquo Journal ofComputational Physics vol 231 no 6 pp 2621ndash2633 2012

[17] M M Meerschaert H-P Scheffler and C Tadjeran ldquoFinitedifference methods for two-dimensional fractional dispersionequationrdquo Journal of Computational Physics vol 211 no 1 pp249ndash261 2006

[18] V E Lynch B A Carreras D del-Castillo-Negrete K MFerreira-Mejias and H R Hicks ldquoNumerical methods for thesolution of partial differential equations of fractional orderrdquoJournal of Computational Physics vol 192 no 2 pp 406ndash4212003

[19] S Momani and Z Odibat ldquoNumerical comparison of methodsfor solving linear differential equations of fractional orderrdquoChaos Solitons amp Fractals vol 31 no 5 pp 1248ndash1255 2007

[20] Z Odibat and S Momani ldquoNumerical methods for nonlinearpartial differential equations of fractional orderrdquo Applied Math-ematical Modelling vol 32 no 1 pp 28ndash39 2008

[21] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[22] I Podlubny Fractional Dfferential Equations vol 198 ofMathe-matics in Science and Engineering Academic Press 1999

[23] C Li andW Deng ldquoRemarks on fractional derivativesrdquoAppliedMathematics andComputation vol 187 no 2 pp 777ndash784 2007

[24] S B Yuste and L Acedo ldquoAn explicit finite difference methodand a new von Neumann-type stability analysis for fractionaldiffusion equationsrdquo SIAM Journal on Numerical Analysis vol42 no 5 pp 1862ndash1874 2005

[25] M M Meerschaert and C Tadjeran ldquoFinite difference approx-imations for fractional advection-dispersion flow equationsrdquoJournal of Computational and AppliedMathematics vol 172 no1 pp 65ndash77 2004

[26] I Podlubny ldquoGeometric and physical interpretation of frac-tional integration and fractional differentiationrdquo FractionalCalculus amp Applied Analysis vol 5 no 4 pp 367ndash386 2002

[27] A Compte and R Metzler ldquoThe generalized Cattaneo equationfor the description of anomalous transport processesrdquo Journal ofPhysics A Mathematical and General vol 30 no 21 pp 7277ndash7289 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

scheme for the Caputo-Riesz fractional diffusion equationthat are superlinearly convergent

Tadjeran and Meerschaert [13] implement a second-order accurate Crank-Nicolson-type finite difference schemefor the space-fractional diffusion equation making use ofthe right-shifted Grunwald-Letnikov fractional derivativeDiethelm et al [14] propose a predictor-corrector-typeapproach for the numerical solution of both linear andnonlinear fractional differential equations including systemsof equations Mohebbi et al [15] study a high-order accu-rate numerical method for obtaining the solution to thesubdiffusion equation with a nonlinear source term Cui[16] goes on to solve the two-dimensional time-fractionaldiffusion equation using a high-order compact alternatingdirection implicit method Meerschaert et al [17] solve thetwo-dimensional space-fractional dispersion equation usinga right-shifted Grunwald-Letnikov discretization since theorder of the fractional derivative is above 1 but less than2 Lynch et al [18] implement a finite difference approachfor the space-fractional dispersion problem but implementdiscretization switching to maximize accuracy based on theorder of derivative Momani and Odibat [19] depart from thefinite-difference approach and instead compare two analyticmethods for the solution of linear fractional differentialequations namely the variational iteration method and theAdomian decomposition method These authors go on toexplore the application of these methods to nonlinear partialdifferential equations of fractional order in [20]

The present work derives a difference approximation in anovel way that produces a form similar to the known resultpresented by Oldham and Spanier in [21] This text cites anerror in the approximation of the fractional derivative of aconstant and linear function of order119873minus1 that is proportionalto the number of terms in the series approximation Anovel technique is coupled with this approximation to attainorder 119873minus2 accuracy Although the result obtained is knownto the best of the authorrsquos knowledge this derivation isnovel

Another important technique used in the numericalapplication of fractional differential equations is the ldquoshort-memoryrdquo principle [22]This technique truncates the approx-imation sum to avoid high computational costs It is shown inSection 4 that the presentmethod is highly accurate evenwithvery few terms in the approximation sum

The following section presents some preliminary resultsused in the remainder of the paper Section 3 goes on toderive the new method of discretizing a fractional derivativewith results on the performance of this technique reportedin Section 4 We make some concluding remarks and discussopportunities for further work in Section 5

2 Preliminaries

This section presents some preliminary results that are putto use in subsequent sections However in the interestof brevity we omit some useful properties but direct theinterested reader to the work by Li and Deng [23] Oftenfor numerical simulation of fractional order problems the

Grunwald-Letnikov derivative is used as ameans of discretiz-ing a fractional order derivative The Grunwald-Letnikovdiscretization has been used for numerical schemes forfractional partial differential equations [13 24 25] and isdefined as follows

Definition 1 The Grunwald-Letnikov derivative of order 120572 gt0 of a function 119906(119905) is

GL119863120572

0119905119906 (119905) = lim

Δ119905rarr0

1

Δ119905120572

119873

sum

119896=0

120596120572

119896119906 (119905 minus 119896Δ119905) (1)

where

120596120572

119896= (minus1)

119896(

120572

119896

) (2)

and119873Δ119905 = 119905

However the limit definition is only practically usefulfor finite-difference implementations hence the followingdefinition is used

Definition 2 TheGrunwald-Letnikov derivative of order 120572 gt0 of a function 119906(119905) isin 119862119898[0 119905] is [22]

119863120572

0119905119906 (119905) =

119898minus1

sum

119896=0

119906(119896)(0) 119905minus119886+119896

Γ (minus120572 + 119896 + 1)+

1

119898 minus 120572

sdot int

119905

0

(119905 minus 120591)119898minus120572minus1

119906(119898)

(120591) 119889120591

(3)

where119898 minus 1 le 120572 lt 119898 isin Z+

Definition 2 is used as a means of calculating theexact fractional derivative of a function The exact solutionobtained here is then used as a measure for accuracy whencompared with the discrete approximations It is prudent tomention that if 119906(119905) is sufficiently smooth 119906(119905) isin 119862

119898[0 119905]

then the Grunwald-Letnikov derivative is equivalent to thepopular Riemann-Liouville defined below

Definition 3 The Riemann-Liouville derivative of fractionalorder 120572 gt 0 of a function 119906(119905) is [22]

RL119863120572

0119905119906 (119905) =

1

Γ (119898 minus 120572)

119889119898

119889119905119898int

119905

0

(119905 minus 120591)119898minus120572minus1

119906 (120591) 119889120591 (4)

The right-shifted Grunwald-Letnikov approximationintroduced by Meerschaert and Tadjeran [25] as used in[13] leads to stable numerical schemes and is defined below

Definition 4 The right-shifted Grunwald-Letnikov approxi-mation of a function 119906(119905) for 1 lt 120572 le 2 is [25]

RSGL119863120572

0119905119906 (119905)

= lim119873rarrinfin

1

Δ119905120572

119873

sum

119896=0

(minus1)119896(

120572

119896

)119906 (119905 minus (119896 minus 1) Δ119905)

(5)

Abstract and Applied Analysis 3

Oldham and Spanier obtain a second-order Grunwald-Letnikov approximation simply by noting that the Riemannsumpresent in the derivativewould converge faster if a simplemodification was made

Definition 5 TheGrunwald-Letnikov derivative of order 120572 gt0 of a function 119906(119905) with a central difference modification is[21]

GL119863120572

0119905119906 (119905)

= limΔ119905rarr0

1

Δ119905120572

119873minus1

sum

119896=0

(minus1)119896(

120572

119896

)119906(119905 minus 119896Δ119905 +120572Δ119905

2)

(6)

where119873Δ119905 = 119905

The primary difference between Definition 5 and thepresent work is the derivation and the choice of119873 the upperlimit of the Riemann sum Although these discrepancies areslight we also address the issue of the asymptotic behaviorof the fractional derivative as 119905 approaches zero a difficultbehavior for a difference scheme to capture

3 Method

In much the same way the Grunwald-Letnikov deriva-tive is generalized from the first-order backward differencescheme as described by Podlubny in [22] we derive anew approximate fractional derivative in the form of adifference scheme which is derived from the second-ordercentral difference scheme Yuste and Acedo [24] state thatthe standard Grunwald-Letnikov derivative is accurate toO(Δ119905) while we derive a scheme based on the central-difference scheme we hope to achieve an approximationthat is O(Δ1199052) Podlubny [26] shows by example that theGrunwald-Letnikov derivative of a function 119891(119905) = 1 or119891(119905) = 119905

119898 is accurate to O(Δ119905) and hence conclude that theGrunwald-Letnikov derivative of any function that can bewritten as a power series will be accurate toO(Δ119905)The centraldifference approximation to the first-order derivative is givenby

1199061015840(119905) = limΔ119905rarr0

119906 (119905 + Δ119905) minus 119906 (119905 minus Δ119905)

2Δ119905+ O (Δ119905

2) (7)

or

1199061015840(119905) = limΔ119905rarr0

119906 (119905 + Δ1199052) minus 119906 (119905 minus Δ1199052)

Δ119905+ O(

Δ1199052

22) (8)

Similarly the second-order derivative can be approximated by

11990610158401015840(119905) = limΔ119905rarr0

119906 (119905 + Δ119905) minus 2119906 (119905) + 119906 (119905 minus Δ119905)

(Δ119905)2

+ O (Δ1199052)

(9)

Following this trend we may write

119906(119899)(119905)

= limΔ119905rarr0

1

(Δ119905)119899

119899

sum

119896=0

(minus1)119896(

119899

119896

)119906(119905 minus 119896Δ119905 +119899Δ119905

2)

(10)

Generalizing the order of the derivative to be a positive realvalue120572 and substituting into (10)we obtain a general formula

GL2119863120572

0119905119906 (119905) = 119906

(120572)(119905)

= limΔ119905rarr0

1

(Δ119905)120572

119873

sum

119896=0

(minus1)119896(

120572

119896

)119906(119905 minus 119896Δ119905 +120572Δ119905

2)

(11)

where the binomial coefficient is calculated by the use of theGamma function that generalizes the factorial operator

(

120572

119896

) =120572

119896 (120572 minus 119896)=

Γ (120572 + 1)

Γ (119896 + 1) Γ (120572 minus 119896 + 1) (12)

It is well known [26 27] that the fractional differentiationoperator is a nonlocal operator This effect is seen in thediscretization being affected by every point in the functionand not just a neighborhood of the point 119905This insight guidesour choice of 119873 as the upper limit of the summation Wechoose 119873 so that the last term in the series correspondswith the terminating point of the function 119906(0) and in termsof fractional differential equations and fractional partialdifferential equations this would correspondwith a boundaryor initial condition

119873 = lceil119905

Δ119905+120572

2rceil (13)

where lceilsdotrceil represents the ceiling function Due to the singu-larity of a fractional derivative GL119863

120572

0119905119906(119905) when 119905 = 0 it

is convenient to select 119906(119905) such that 119906(0) = 0 to alleviatethe asymptotic behavior However we do not want to restrictourselves only to functions that obey this property Hence wedefine

119892 (119905) = 119906 (119905) minus 119906 (0) (14)

so that 119892(0) = 0 and then

GL2119863120572

0119905119906 (119905) = GL2119863

120572

0119905119892 (119905) + GL2119863

120572

0119905119906 (0) (15)

since the fractional derivative is a linear operator We maynow exploit Property 22 in [23] which yields

L GL2119863120572

0119905119906 (0) = 119904

120572L 119906 (0) (16)

which then leads to

Lminus1119904120572L 119906 (0) =

119905minus120572119906 (0)

Γ (1 minus 120572) (17)

Finally (15) becomes

GL2119863120572

0119905119906 (119905) = GL2119863

120572

0119905119892 (119905) +

119905minus120572119906 (0)

Γ (1 minus 120572) (18)

While correcting for the asymptotic behavior at 119905 = 0 pro-duces sufficiently accurate results wemay go one step furtherand substitute the exact solution of the linear component of119906(119905) This is done to ensure that the accuracy of the scheme isat least O(119873minus2) Let

ℎ (119905) = 119906 (119905) minus 119906 (0) minus 1199051199061015840(0) (19)

then

GL2119863120572

0119905119906 (119905) = GL2119863

120572

0119905119892 (119905) +

119905minus120572119906 (0)

Γ (1 minus 120572)+1199051minus120572

1199061015840(0)

Γ (2 minus 120572) (20)

4 Abstract and Applied Analysis

31 Accuracy As described by Oldham and Spanier in [21]the error in the above scheme for functions 119906(119905) = 119862 119906(119905) = 119905and 119906(119905) = 119905

2 is given by 120572(120572+1)2119873 12057222119873 and 120572(120572minus1)(120572minus2)24119873

2 respectively Since we make the exact substitutionsfor the linear components we are able to increase the accuracyof the scheme to O(119873minus2) for any analytic function

Theorem 6 The error of GL2119863120572

0119905119906(119905) denoted by 120598(119906(119905))GL2 =

|119863120572

0119905119906(119905) minus GL2119863

120572

0119905119906(119905)| is at least O(119873minus2) accurate for the

function 119906(119905) = 119905119901 for all 119901 ge 2 given that the exact derivative

of order 120572 of 119905119901 is

119863120572

0119905119905119901=119905119901minus120572

Γ (119901 + 1)

Γ (119901 minus 120572 + 1) (21)

Proof Let 119901 = 2 Then (see [21] Chapter 8) (while Oldhamand Spanier cite the relative error it is easy to show thatthe more restrictive absolute error is also O(119873minus2) Moreinformation can be found in [22] Chapter 7)

120598 (119906 (119905))GL2 =120572 (120572 minus 1) (120572 minus 2)

241198732

(22)

Assume that the order of error is at least O(119873minus2) for 119901 = 119898that is

GL2119863120572

0119905119905119898=119905119898minus120572

Γ (119898 + 1)

Γ (119898 minus 120572 + 1)+ O (119873

minus2) (23)

Now let 119901 = 119898 + 1 then by the product rule for fractionalderivatives (see [21] Chapter 5) we have

GL2119863120572

0119905119905119898+1

= GL2119863120572

0119905(119905119898119905)

= 119905GL2119863120572

0119905119905119898+ 120572GL2119863

120572minus1

0119905119905119898

asymp 119905119905119898minus120572

Γ (119898 + 1)

Γ (119898 minus 120572 + 1)+ 120572

119905119898minus120572+1

Γ (119898 + 1)

Γ (119898 minus 120572)

+ O (119873minus2)

=119905119898minus120572+1

Γ (119898 + 1)

Γ (119898 minus 120572 + 1)(1 +

120572

119898 + 1 minus 120572)

+ O (119873minus2)

(24)

Then using the recursive property of the Gamma function119909Γ(119909) = Γ(119909 + 1) we have

GL2119863120572

0119905119905119898+1

=119905119898+1minus120572

Γ (119898 + 2)

Γ (119898 minus 120572 + 2)+ O (119873

minus2) (25)

Using Theorem 6 we can deduce that the accuracy ofGL2119863120572

0119905119906(119905) for any analytic function 119906(119905) isO(119873minus2) provided

that the linear and constant components of that function aredealt with accordingly

4 Results

This section presents some numerical results for differentfunctions Each example shows the absolute error over thedomain of the function the consistency of each derivativeas ℎ tends to zero the absolute error of the truncated seriesand the absolute error over a range of 120572 We comparethe aforementioned criteria for four different methods thestandard Grunwald-Letnikov approximation ( GL119863

120572

0119905) the

second-order Grunwald-Letnikov approximation ( GL2119863120572

0119905)

the second-order Grunwald-Letnikov approximation withconstant correction described in Section 3 (CCGL2119863

120572

0119905) and

finally the right-shifted Grunwald-Letnikov approximation( RSGL119863

120572

0119905)

41 Example 1 Let

119906 (119905) = 1199052 (26)

and let the order of derivative be 120572 = 075 with timestep Δ119905 = 0001 Since this choice of 119906(119905) satisfies 119906(0) =

0 the approximation does not suffer from poor accuracynear 119905 = 0 and hence the second-order Grunwald-Letnikov( GL2119863

120572

0119905) and the second-order Grunwald-Letnikov with

constant correction (CCGL2119863120572

0119905) approximations are similar

42 Example 2 Let

119906 (119905) = cos (119905) (27)

the order of derivative 120572 = 15 andΔ119905 = 0001 Asmentionedby Oldham and Spanier [21] this method is robust for all 120572this example shows highly accurate results for 120572 gt 1

43 Example 3 Let

119906 (119905) = cos (119905) + sin (119905) 1199052 (28)

the order of derivative 120572 = 09 and again Δ119905 = 0001This function suffers from the asymptotic behavior of thefractional derivative near 119905 = 0 and hence the constantcorrection method enjoys a vastly superior accuracy over theother methods

44 Example 4 Let

119906 (119905) = 119890minus119905 cos (119905) (29)

the order of derivative 120572 = 21 and Δ119905 = 0001

45 Example 5 Let

119906 (119905) = sin (119905) + cos (119905) (30)

the order of derivative 120572 = 075 and Δ119905 = 0001The above results indicate that the present method enjoys

a high accuracy The nature of the fractional derivative nearthe lower limit of differentiation results in a high gradientDifference schemes with fixed interval steps are unable to

Abstract and Applied Analysis 5

0 1 2 3 4 5 600000

00005

00010

00015

00020

t

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 1 Absolute error over time domain for four differentmethods

000 002 004 006 008 010000

002

004

006

008

h

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 2 Absolute error over varying values of Δ119905 for four differentmethods

accurately capture very steep gradients similar to those at theasymptote of the fractional derivativeThe use of the constantcorrection allows one to avoid the asymptotic behavior near119905 = 0 and the provable accuracy

Figures 1 5 9 13 and 17 show the increasing accuracyof the approximations as 119905 deviates from 0 as well as therobustness of our method to the asymptotic behavior Theconsistency of the methods is evident in Figures 2 6 10 14and 18These figures indicate that as ℎ tends to 0 the absoluteerror of the approximations also tends to 0 As mentionedin Section 1 the ldquoshort-memoryrdquo principle allows for thetruncation of the approximating series in order to alleviatecomputational cost our present method is highly accurateeven for small values of 119873 as is evident in Figures 3 7 1115 and 19 The final set of figures Figures 4 8 12 16 and 20indicate the robustness of our method to varying values of 120572It is important to note here the log scale of the graph

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 50 100 150 20000000

00005

00010

00015

00020

N

Abso

lute

erro

r

Figure 3Absolute error for increasing119873 for four differentmethods

00 05 10 15 20 25 30

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

120572

10minus2

10minus5

10minus8

10minus14

10minus11

Figure 4 Absolute error over varying values of 120572 for four differentmethods

5 Conclusion

This work has shown a novel derivation of a second-orderaccurate scheme for approximating the fractional orderderivative of a function To the best of the authorrsquos knowledgeperforming a linear transformation on the function to avoidasymptotic behavior near the origin is also novel Section 4provides several examples with varying parameters andbehaviors to illustrate the efficacy of the present method

Using a higher-order accurate approximation leads tothe potential for higher-order accurate numerical schemesfor fractional differential equations and fractional partialdifferential equations Tadjeran and Meerschaert [13] imple-ment the right-shifted Grunwald-Letnikov derivative forfractional derivatives of order 1 lt 120572 le 2 which lead toan unconditionally stable numerical scheme According toOldham and Spanier [21] the present method is robust for allvalues of alpha Potential extensions to this work include con-structing numerical schemes centered on this approximationand the fractional order derivative for nonlinear fractional

6 Abstract and Applied Analysis

0 1 2 3 4 5 600000

00005

00010

00015

t

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 5 Absolute error over time domain for four differentmethods

000 002 004 006 008 0100

1

2

3

4

5

6

7

h

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 6 Absolute error over varying values of Δ119905 for four differentmethods

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 50 100 150 200000

005

010

015

020

025

N

Abso

lute

erro

r

Figure 7 Absolute error for increasing 119873 for four different meth-ods

00 05 10 15 20 25 30

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

10minus7

10minus1

10minus4

120572

Figure 8 Absolute error over varying values of 120572 for four differentmethods

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 1 2 3 4 5 60000

0005

0010

0015

t

Figure 9 Absolute error over time domain for four differentmethods

000 002 004 006 008 01000

01

02

03

04

h

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 10 Absolute error over varying values ofΔ119905 for four differentmethods

Abstract and Applied Analysis 7

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 50 100 150 2000000

0002

0004

0006

0008

N

Abso

lute

erro

r

Figure 11 Absolute error for increasing 119873 for four differentmethods

00 05 10 15 20 25 30

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

120572

10minus1

10minus4

10minus7

Figure 12 Absolute error over varying values of 120572 for four differentmethods

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 1 2 3 4 5 600000

00001

00002

00003

00004

00005

t

Abso

lute

erro

r

Figure 13 Absolute error over time domain for four differentmethods

000 002 004 006 008 010000

005

010

015

020

h

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 14 Absolute error over varying values ofΔ119905 for four differentmethods

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 200 400 600 800 10000000

0001

0002

0003

0004

0005

N

Abso

lute

erro

r

Figure 15 Absolute error for increasing 119873 for four differentmethods

00 05 10 15 20 25 300000

0002

0004

0006

0008

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

120572

Figure 16 Absolute error over varying values of 120572 for four differentmethods

8 Abstract and Applied Analysis

0 1 2 3 4 5 600000

00005

00010

00015

t

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 17 Absolute error over time domain for four differentmethods

000 002 004 006 008 01000

01

02

03

04

05

06

h

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 18 Absolute error over varying values ofΔ119905 for four differentmethods

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 50 100 150 2000000

0005

0010

0015

N

Abso

lute

erro

r

Figure 19 Absolute error for increasing 119873 for four differentmethods

00 05 10 15 20 25 30

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

120572

10minus1

10minus4

10minus7

Figure 20 Absolute error over varying values of 120572 for four differentmethods

differential equations as well as time-marching schemes fortime or space fractional partial differential equations

The accuracy of the method has been proven to be atleast of O(119873minus2) which suggests that the accuracy should beof order 10minus6 for Δ119905 However in these examples the erroris observed to be in the region of 10minus8 Nevertheless themathematical proof of accuracy as well as the compellingresults presented substantiates the efforts toward this field

Disclaimer

The opinions expressed and conclusions arrived at herein arethose of the author and are not necessarily to be attributed tothe CoE-MaSS

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author would like to acknowledge and thank ProfessorCharis Harley for her helpful suggestions and useful insightThe DST-NRF Centre of Excellence in Mathematical andStatistical Sciences (CoE-MaSS) is acknowledged for fundingunder Grant no 94005 Thanks are due to the reviewers fortheir commentswhich helped to elevate the level of this work

References

[1] Q Zeng H Li and D Liu ldquoAnomalous fractional diffusionequation for transport phenomenardquo Communications in Non-linear Science and Numerical Simulation vol 4 no 2 pp 99ndash104 1999

[2] D A Benson S W Wheatcraft and M M MeerschaertldquoApplication of a fractional advection-dispersion equationrdquoWater Resources Research vol 36 no 6 pp 1403ndash1412 2000

Abstract and Applied Analysis 9

[3] J-H He ldquoApproximate analytical solution for seepage flowwithfractional derivatives in porous mediardquo Computer Methods inApplied Mechanics and Engineering vol 167 no 1-2 pp 57ndash681998

[4] R Metzler W G Glockle and T F Nonnenmacher ldquoFractionalmodel equation for anomalous diffusionrdquo Physica A StatisticalMechanics and its Applications vol 211 no 1 pp 13ndash24 1994

[5] V Gafiychuk B Datsko and V Meleshko ldquoMathematicalmodeling of time fractional reaction-diffusion systemsrdquo Journalof Computational and AppliedMathematics vol 220 no 1-2 pp215ndash225 2008

[6] B Y Datsko and V V Gafiychuk ldquoMathematical modeling offractional reaction-diffusion systems with different order timederivativesrdquo Journal of Mathematical Sciences vol 165 no 3 pp392ndash402 2010

[7] M M Meerschaert and C Tadjeran ldquoFinite difference approxi-mations for two-sided space-fractional partial differential equa-tionsrdquoApplied Numerical Mathematics vol 56 no 1 pp 80ndash902006

[8] R Scherer S L Kalla L Boyadjiev and B Al-Saqabi ldquoNumer-ical treatment of fractional heat equationsrdquo Applied NumericalMathematics vol 58 no 8 pp 1212ndash1223 2008

[9] A M El-Sayed I L El-Kalla and E A Ziada ldquoAnalytical andnumerical solutions of multi-term nonlinear fractional ordersdifferential equationsrdquo Applied Numerical Mathematics vol 60no 8 pp 788ndash797 2010

[10] I Karatay R S Bayramoglu and A Sahin ldquoImplicit differenceapproximation for the time fractional heat equation with thenonlocal conditionrdquo Applied Numerical Mathematics vol 61no 12 pp 1281ndash1288 2011

[11] L Su W Wang and H Wang ldquoA characteristic differencemethod for the transient fractional convection-diffusion equa-tionsrdquo Applied Numerical Mathematics vol 61 no 8 pp 946ndash960 2011

[12] M Chen W Deng and Y Wu ldquoSuperlinearly convergentalgorithms for the two-dimensional space-time caputo-rieszfractional diffusion equationrdquo Applied Numerical Mathematicsvol 70 pp 22ndash41 2013

[13] C Tadjeran and M M Meerschaert ldquoA second-order accuratenumerical method for the two-dimensional fractional diffusionequationrdquo Journal of Computational Physics vol 220 no 2 pp813ndash823 2007

[14] K Diethelm N J Ford and A D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 3ndash22 2002

[15] A Mohebbi M Abbaszadeh and M Dehghan ldquoA high-orderand unconditionally stable scheme for the modified anomalousfractional sub-diffusion equationwith a nonlinear source termrdquoJournal of Computational Physics vol 240 pp 36ndash48 2013

[16] M Cui ldquoCompact alternating direction implicit method fortwo-dimensional time fractional diffusion equationrdquo Journal ofComputational Physics vol 231 no 6 pp 2621ndash2633 2012

[17] M M Meerschaert H-P Scheffler and C Tadjeran ldquoFinitedifference methods for two-dimensional fractional dispersionequationrdquo Journal of Computational Physics vol 211 no 1 pp249ndash261 2006

[18] V E Lynch B A Carreras D del-Castillo-Negrete K MFerreira-Mejias and H R Hicks ldquoNumerical methods for thesolution of partial differential equations of fractional orderrdquoJournal of Computational Physics vol 192 no 2 pp 406ndash4212003

[19] S Momani and Z Odibat ldquoNumerical comparison of methodsfor solving linear differential equations of fractional orderrdquoChaos Solitons amp Fractals vol 31 no 5 pp 1248ndash1255 2007

[20] Z Odibat and S Momani ldquoNumerical methods for nonlinearpartial differential equations of fractional orderrdquo Applied Math-ematical Modelling vol 32 no 1 pp 28ndash39 2008

[21] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[22] I Podlubny Fractional Dfferential Equations vol 198 ofMathe-matics in Science and Engineering Academic Press 1999

[23] C Li andW Deng ldquoRemarks on fractional derivativesrdquoAppliedMathematics andComputation vol 187 no 2 pp 777ndash784 2007

[24] S B Yuste and L Acedo ldquoAn explicit finite difference methodand a new von Neumann-type stability analysis for fractionaldiffusion equationsrdquo SIAM Journal on Numerical Analysis vol42 no 5 pp 1862ndash1874 2005

[25] M M Meerschaert and C Tadjeran ldquoFinite difference approx-imations for fractional advection-dispersion flow equationsrdquoJournal of Computational and AppliedMathematics vol 172 no1 pp 65ndash77 2004

[26] I Podlubny ldquoGeometric and physical interpretation of frac-tional integration and fractional differentiationrdquo FractionalCalculus amp Applied Analysis vol 5 no 4 pp 367ndash386 2002

[27] A Compte and R Metzler ldquoThe generalized Cattaneo equationfor the description of anomalous transport processesrdquo Journal ofPhysics A Mathematical and General vol 30 no 21 pp 7277ndash7289 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

Oldham and Spanier obtain a second-order Grunwald-Letnikov approximation simply by noting that the Riemannsumpresent in the derivativewould converge faster if a simplemodification was made

Definition 5 TheGrunwald-Letnikov derivative of order 120572 gt0 of a function 119906(119905) with a central difference modification is[21]

GL119863120572

0119905119906 (119905)

= limΔ119905rarr0

1

Δ119905120572

119873minus1

sum

119896=0

(minus1)119896(

120572

119896

)119906(119905 minus 119896Δ119905 +120572Δ119905

2)

(6)

where119873Δ119905 = 119905

The primary difference between Definition 5 and thepresent work is the derivation and the choice of119873 the upperlimit of the Riemann sum Although these discrepancies areslight we also address the issue of the asymptotic behaviorof the fractional derivative as 119905 approaches zero a difficultbehavior for a difference scheme to capture

3 Method

In much the same way the Grunwald-Letnikov deriva-tive is generalized from the first-order backward differencescheme as described by Podlubny in [22] we derive anew approximate fractional derivative in the form of adifference scheme which is derived from the second-ordercentral difference scheme Yuste and Acedo [24] state thatthe standard Grunwald-Letnikov derivative is accurate toO(Δ119905) while we derive a scheme based on the central-difference scheme we hope to achieve an approximationthat is O(Δ1199052) Podlubny [26] shows by example that theGrunwald-Letnikov derivative of a function 119891(119905) = 1 or119891(119905) = 119905

119898 is accurate to O(Δ119905) and hence conclude that theGrunwald-Letnikov derivative of any function that can bewritten as a power series will be accurate toO(Δ119905)The centraldifference approximation to the first-order derivative is givenby

1199061015840(119905) = limΔ119905rarr0

119906 (119905 + Δ119905) minus 119906 (119905 minus Δ119905)

2Δ119905+ O (Δ119905

2) (7)

or

1199061015840(119905) = limΔ119905rarr0

119906 (119905 + Δ1199052) minus 119906 (119905 minus Δ1199052)

Δ119905+ O(

Δ1199052

22) (8)

Similarly the second-order derivative can be approximated by

11990610158401015840(119905) = limΔ119905rarr0

119906 (119905 + Δ119905) minus 2119906 (119905) + 119906 (119905 minus Δ119905)

(Δ119905)2

+ O (Δ1199052)

(9)

Following this trend we may write

119906(119899)(119905)

= limΔ119905rarr0

1

(Δ119905)119899

119899

sum

119896=0

(minus1)119896(

119899

119896

)119906(119905 minus 119896Δ119905 +119899Δ119905

2)

(10)

Generalizing the order of the derivative to be a positive realvalue120572 and substituting into (10)we obtain a general formula

GL2119863120572

0119905119906 (119905) = 119906

(120572)(119905)

= limΔ119905rarr0

1

(Δ119905)120572

119873

sum

119896=0

(minus1)119896(

120572

119896

)119906(119905 minus 119896Δ119905 +120572Δ119905

2)

(11)

where the binomial coefficient is calculated by the use of theGamma function that generalizes the factorial operator

(

120572

119896

) =120572

119896 (120572 minus 119896)=

Γ (120572 + 1)

Γ (119896 + 1) Γ (120572 minus 119896 + 1) (12)

It is well known [26 27] that the fractional differentiationoperator is a nonlocal operator This effect is seen in thediscretization being affected by every point in the functionand not just a neighborhood of the point 119905This insight guidesour choice of 119873 as the upper limit of the summation Wechoose 119873 so that the last term in the series correspondswith the terminating point of the function 119906(0) and in termsof fractional differential equations and fractional partialdifferential equations this would correspondwith a boundaryor initial condition

119873 = lceil119905

Δ119905+120572

2rceil (13)

where lceilsdotrceil represents the ceiling function Due to the singu-larity of a fractional derivative GL119863

120572

0119905119906(119905) when 119905 = 0 it

is convenient to select 119906(119905) such that 119906(0) = 0 to alleviatethe asymptotic behavior However we do not want to restrictourselves only to functions that obey this property Hence wedefine

119892 (119905) = 119906 (119905) minus 119906 (0) (14)

so that 119892(0) = 0 and then

GL2119863120572

0119905119906 (119905) = GL2119863

120572

0119905119892 (119905) + GL2119863

120572

0119905119906 (0) (15)

since the fractional derivative is a linear operator We maynow exploit Property 22 in [23] which yields

L GL2119863120572

0119905119906 (0) = 119904

120572L 119906 (0) (16)

which then leads to

Lminus1119904120572L 119906 (0) =

119905minus120572119906 (0)

Γ (1 minus 120572) (17)

Finally (15) becomes

GL2119863120572

0119905119906 (119905) = GL2119863

120572

0119905119892 (119905) +

119905minus120572119906 (0)

Γ (1 minus 120572) (18)

While correcting for the asymptotic behavior at 119905 = 0 pro-duces sufficiently accurate results wemay go one step furtherand substitute the exact solution of the linear component of119906(119905) This is done to ensure that the accuracy of the scheme isat least O(119873minus2) Let

ℎ (119905) = 119906 (119905) minus 119906 (0) minus 1199051199061015840(0) (19)

then

GL2119863120572

0119905119906 (119905) = GL2119863

120572

0119905119892 (119905) +

119905minus120572119906 (0)

Γ (1 minus 120572)+1199051minus120572

1199061015840(0)

Γ (2 minus 120572) (20)

4 Abstract and Applied Analysis

31 Accuracy As described by Oldham and Spanier in [21]the error in the above scheme for functions 119906(119905) = 119862 119906(119905) = 119905and 119906(119905) = 119905

2 is given by 120572(120572+1)2119873 12057222119873 and 120572(120572minus1)(120572minus2)24119873

2 respectively Since we make the exact substitutionsfor the linear components we are able to increase the accuracyof the scheme to O(119873minus2) for any analytic function

Theorem 6 The error of GL2119863120572

0119905119906(119905) denoted by 120598(119906(119905))GL2 =

|119863120572

0119905119906(119905) minus GL2119863

120572

0119905119906(119905)| is at least O(119873minus2) accurate for the

function 119906(119905) = 119905119901 for all 119901 ge 2 given that the exact derivative

of order 120572 of 119905119901 is

119863120572

0119905119905119901=119905119901minus120572

Γ (119901 + 1)

Γ (119901 minus 120572 + 1) (21)

Proof Let 119901 = 2 Then (see [21] Chapter 8) (while Oldhamand Spanier cite the relative error it is easy to show thatthe more restrictive absolute error is also O(119873minus2) Moreinformation can be found in [22] Chapter 7)

120598 (119906 (119905))GL2 =120572 (120572 minus 1) (120572 minus 2)

241198732

(22)

Assume that the order of error is at least O(119873minus2) for 119901 = 119898that is

GL2119863120572

0119905119905119898=119905119898minus120572

Γ (119898 + 1)

Γ (119898 minus 120572 + 1)+ O (119873

minus2) (23)

Now let 119901 = 119898 + 1 then by the product rule for fractionalderivatives (see [21] Chapter 5) we have

GL2119863120572

0119905119905119898+1

= GL2119863120572

0119905(119905119898119905)

= 119905GL2119863120572

0119905119905119898+ 120572GL2119863

120572minus1

0119905119905119898

asymp 119905119905119898minus120572

Γ (119898 + 1)

Γ (119898 minus 120572 + 1)+ 120572

119905119898minus120572+1

Γ (119898 + 1)

Γ (119898 minus 120572)

+ O (119873minus2)

=119905119898minus120572+1

Γ (119898 + 1)

Γ (119898 minus 120572 + 1)(1 +

120572

119898 + 1 minus 120572)

+ O (119873minus2)

(24)

Then using the recursive property of the Gamma function119909Γ(119909) = Γ(119909 + 1) we have

GL2119863120572

0119905119905119898+1

=119905119898+1minus120572

Γ (119898 + 2)

Γ (119898 minus 120572 + 2)+ O (119873

minus2) (25)

Using Theorem 6 we can deduce that the accuracy ofGL2119863120572

0119905119906(119905) for any analytic function 119906(119905) isO(119873minus2) provided

that the linear and constant components of that function aredealt with accordingly

4 Results

This section presents some numerical results for differentfunctions Each example shows the absolute error over thedomain of the function the consistency of each derivativeas ℎ tends to zero the absolute error of the truncated seriesand the absolute error over a range of 120572 We comparethe aforementioned criteria for four different methods thestandard Grunwald-Letnikov approximation ( GL119863

120572

0119905) the

second-order Grunwald-Letnikov approximation ( GL2119863120572

0119905)

the second-order Grunwald-Letnikov approximation withconstant correction described in Section 3 (CCGL2119863

120572

0119905) and

finally the right-shifted Grunwald-Letnikov approximation( RSGL119863

120572

0119905)

41 Example 1 Let

119906 (119905) = 1199052 (26)

and let the order of derivative be 120572 = 075 with timestep Δ119905 = 0001 Since this choice of 119906(119905) satisfies 119906(0) =

0 the approximation does not suffer from poor accuracynear 119905 = 0 and hence the second-order Grunwald-Letnikov( GL2119863

120572

0119905) and the second-order Grunwald-Letnikov with

constant correction (CCGL2119863120572

0119905) approximations are similar

42 Example 2 Let

119906 (119905) = cos (119905) (27)

the order of derivative 120572 = 15 andΔ119905 = 0001 Asmentionedby Oldham and Spanier [21] this method is robust for all 120572this example shows highly accurate results for 120572 gt 1

43 Example 3 Let

119906 (119905) = cos (119905) + sin (119905) 1199052 (28)

the order of derivative 120572 = 09 and again Δ119905 = 0001This function suffers from the asymptotic behavior of thefractional derivative near 119905 = 0 and hence the constantcorrection method enjoys a vastly superior accuracy over theother methods

44 Example 4 Let

119906 (119905) = 119890minus119905 cos (119905) (29)

the order of derivative 120572 = 21 and Δ119905 = 0001

45 Example 5 Let

119906 (119905) = sin (119905) + cos (119905) (30)

the order of derivative 120572 = 075 and Δ119905 = 0001The above results indicate that the present method enjoys

a high accuracy The nature of the fractional derivative nearthe lower limit of differentiation results in a high gradientDifference schemes with fixed interval steps are unable to

Abstract and Applied Analysis 5

0 1 2 3 4 5 600000

00005

00010

00015

00020

t

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 1 Absolute error over time domain for four differentmethods

000 002 004 006 008 010000

002

004

006

008

h

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 2 Absolute error over varying values of Δ119905 for four differentmethods

accurately capture very steep gradients similar to those at theasymptote of the fractional derivativeThe use of the constantcorrection allows one to avoid the asymptotic behavior near119905 = 0 and the provable accuracy

Figures 1 5 9 13 and 17 show the increasing accuracyof the approximations as 119905 deviates from 0 as well as therobustness of our method to the asymptotic behavior Theconsistency of the methods is evident in Figures 2 6 10 14and 18These figures indicate that as ℎ tends to 0 the absoluteerror of the approximations also tends to 0 As mentionedin Section 1 the ldquoshort-memoryrdquo principle allows for thetruncation of the approximating series in order to alleviatecomputational cost our present method is highly accurateeven for small values of 119873 as is evident in Figures 3 7 1115 and 19 The final set of figures Figures 4 8 12 16 and 20indicate the robustness of our method to varying values of 120572It is important to note here the log scale of the graph

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 50 100 150 20000000

00005

00010

00015

00020

N

Abso

lute

erro

r

Figure 3Absolute error for increasing119873 for four differentmethods

00 05 10 15 20 25 30

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

120572

10minus2

10minus5

10minus8

10minus14

10minus11

Figure 4 Absolute error over varying values of 120572 for four differentmethods

5 Conclusion

This work has shown a novel derivation of a second-orderaccurate scheme for approximating the fractional orderderivative of a function To the best of the authorrsquos knowledgeperforming a linear transformation on the function to avoidasymptotic behavior near the origin is also novel Section 4provides several examples with varying parameters andbehaviors to illustrate the efficacy of the present method

Using a higher-order accurate approximation leads tothe potential for higher-order accurate numerical schemesfor fractional differential equations and fractional partialdifferential equations Tadjeran and Meerschaert [13] imple-ment the right-shifted Grunwald-Letnikov derivative forfractional derivatives of order 1 lt 120572 le 2 which lead toan unconditionally stable numerical scheme According toOldham and Spanier [21] the present method is robust for allvalues of alpha Potential extensions to this work include con-structing numerical schemes centered on this approximationand the fractional order derivative for nonlinear fractional

6 Abstract and Applied Analysis

0 1 2 3 4 5 600000

00005

00010

00015

t

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 5 Absolute error over time domain for four differentmethods

000 002 004 006 008 0100

1

2

3

4

5

6

7

h

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 6 Absolute error over varying values of Δ119905 for four differentmethods

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 50 100 150 200000

005

010

015

020

025

N

Abso

lute

erro

r

Figure 7 Absolute error for increasing 119873 for four different meth-ods

00 05 10 15 20 25 30

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

10minus7

10minus1

10minus4

120572

Figure 8 Absolute error over varying values of 120572 for four differentmethods

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 1 2 3 4 5 60000

0005

0010

0015

t

Figure 9 Absolute error over time domain for four differentmethods

000 002 004 006 008 01000

01

02

03

04

h

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 10 Absolute error over varying values ofΔ119905 for four differentmethods

Abstract and Applied Analysis 7

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 50 100 150 2000000

0002

0004

0006

0008

N

Abso

lute

erro

r

Figure 11 Absolute error for increasing 119873 for four differentmethods

00 05 10 15 20 25 30

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

120572

10minus1

10minus4

10minus7

Figure 12 Absolute error over varying values of 120572 for four differentmethods

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 1 2 3 4 5 600000

00001

00002

00003

00004

00005

t

Abso

lute

erro

r

Figure 13 Absolute error over time domain for four differentmethods

000 002 004 006 008 010000

005

010

015

020

h

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 14 Absolute error over varying values ofΔ119905 for four differentmethods

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 200 400 600 800 10000000

0001

0002

0003

0004

0005

N

Abso

lute

erro

r

Figure 15 Absolute error for increasing 119873 for four differentmethods

00 05 10 15 20 25 300000

0002

0004

0006

0008

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

120572

Figure 16 Absolute error over varying values of 120572 for four differentmethods

8 Abstract and Applied Analysis

0 1 2 3 4 5 600000

00005

00010

00015

t

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 17 Absolute error over time domain for four differentmethods

000 002 004 006 008 01000

01

02

03

04

05

06

h

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 18 Absolute error over varying values ofΔ119905 for four differentmethods

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 50 100 150 2000000

0005

0010

0015

N

Abso

lute

erro

r

Figure 19 Absolute error for increasing 119873 for four differentmethods

00 05 10 15 20 25 30

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

120572

10minus1

10minus4

10minus7

Figure 20 Absolute error over varying values of 120572 for four differentmethods

differential equations as well as time-marching schemes fortime or space fractional partial differential equations

The accuracy of the method has been proven to be atleast of O(119873minus2) which suggests that the accuracy should beof order 10minus6 for Δ119905 However in these examples the erroris observed to be in the region of 10minus8 Nevertheless themathematical proof of accuracy as well as the compellingresults presented substantiates the efforts toward this field

Disclaimer

The opinions expressed and conclusions arrived at herein arethose of the author and are not necessarily to be attributed tothe CoE-MaSS

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author would like to acknowledge and thank ProfessorCharis Harley for her helpful suggestions and useful insightThe DST-NRF Centre of Excellence in Mathematical andStatistical Sciences (CoE-MaSS) is acknowledged for fundingunder Grant no 94005 Thanks are due to the reviewers fortheir commentswhich helped to elevate the level of this work

References

[1] Q Zeng H Li and D Liu ldquoAnomalous fractional diffusionequation for transport phenomenardquo Communications in Non-linear Science and Numerical Simulation vol 4 no 2 pp 99ndash104 1999

[2] D A Benson S W Wheatcraft and M M MeerschaertldquoApplication of a fractional advection-dispersion equationrdquoWater Resources Research vol 36 no 6 pp 1403ndash1412 2000

Abstract and Applied Analysis 9

[3] J-H He ldquoApproximate analytical solution for seepage flowwithfractional derivatives in porous mediardquo Computer Methods inApplied Mechanics and Engineering vol 167 no 1-2 pp 57ndash681998

[4] R Metzler W G Glockle and T F Nonnenmacher ldquoFractionalmodel equation for anomalous diffusionrdquo Physica A StatisticalMechanics and its Applications vol 211 no 1 pp 13ndash24 1994

[5] V Gafiychuk B Datsko and V Meleshko ldquoMathematicalmodeling of time fractional reaction-diffusion systemsrdquo Journalof Computational and AppliedMathematics vol 220 no 1-2 pp215ndash225 2008

[6] B Y Datsko and V V Gafiychuk ldquoMathematical modeling offractional reaction-diffusion systems with different order timederivativesrdquo Journal of Mathematical Sciences vol 165 no 3 pp392ndash402 2010

[7] M M Meerschaert and C Tadjeran ldquoFinite difference approxi-mations for two-sided space-fractional partial differential equa-tionsrdquoApplied Numerical Mathematics vol 56 no 1 pp 80ndash902006

[8] R Scherer S L Kalla L Boyadjiev and B Al-Saqabi ldquoNumer-ical treatment of fractional heat equationsrdquo Applied NumericalMathematics vol 58 no 8 pp 1212ndash1223 2008

[9] A M El-Sayed I L El-Kalla and E A Ziada ldquoAnalytical andnumerical solutions of multi-term nonlinear fractional ordersdifferential equationsrdquo Applied Numerical Mathematics vol 60no 8 pp 788ndash797 2010

[10] I Karatay R S Bayramoglu and A Sahin ldquoImplicit differenceapproximation for the time fractional heat equation with thenonlocal conditionrdquo Applied Numerical Mathematics vol 61no 12 pp 1281ndash1288 2011

[11] L Su W Wang and H Wang ldquoA characteristic differencemethod for the transient fractional convection-diffusion equa-tionsrdquo Applied Numerical Mathematics vol 61 no 8 pp 946ndash960 2011

[12] M Chen W Deng and Y Wu ldquoSuperlinearly convergentalgorithms for the two-dimensional space-time caputo-rieszfractional diffusion equationrdquo Applied Numerical Mathematicsvol 70 pp 22ndash41 2013

[13] C Tadjeran and M M Meerschaert ldquoA second-order accuratenumerical method for the two-dimensional fractional diffusionequationrdquo Journal of Computational Physics vol 220 no 2 pp813ndash823 2007

[14] K Diethelm N J Ford and A D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 3ndash22 2002

[15] A Mohebbi M Abbaszadeh and M Dehghan ldquoA high-orderand unconditionally stable scheme for the modified anomalousfractional sub-diffusion equationwith a nonlinear source termrdquoJournal of Computational Physics vol 240 pp 36ndash48 2013

[16] M Cui ldquoCompact alternating direction implicit method fortwo-dimensional time fractional diffusion equationrdquo Journal ofComputational Physics vol 231 no 6 pp 2621ndash2633 2012

[17] M M Meerschaert H-P Scheffler and C Tadjeran ldquoFinitedifference methods for two-dimensional fractional dispersionequationrdquo Journal of Computational Physics vol 211 no 1 pp249ndash261 2006

[18] V E Lynch B A Carreras D del-Castillo-Negrete K MFerreira-Mejias and H R Hicks ldquoNumerical methods for thesolution of partial differential equations of fractional orderrdquoJournal of Computational Physics vol 192 no 2 pp 406ndash4212003

[19] S Momani and Z Odibat ldquoNumerical comparison of methodsfor solving linear differential equations of fractional orderrdquoChaos Solitons amp Fractals vol 31 no 5 pp 1248ndash1255 2007

[20] Z Odibat and S Momani ldquoNumerical methods for nonlinearpartial differential equations of fractional orderrdquo Applied Math-ematical Modelling vol 32 no 1 pp 28ndash39 2008

[21] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[22] I Podlubny Fractional Dfferential Equations vol 198 ofMathe-matics in Science and Engineering Academic Press 1999

[23] C Li andW Deng ldquoRemarks on fractional derivativesrdquoAppliedMathematics andComputation vol 187 no 2 pp 777ndash784 2007

[24] S B Yuste and L Acedo ldquoAn explicit finite difference methodand a new von Neumann-type stability analysis for fractionaldiffusion equationsrdquo SIAM Journal on Numerical Analysis vol42 no 5 pp 1862ndash1874 2005

[25] M M Meerschaert and C Tadjeran ldquoFinite difference approx-imations for fractional advection-dispersion flow equationsrdquoJournal of Computational and AppliedMathematics vol 172 no1 pp 65ndash77 2004

[26] I Podlubny ldquoGeometric and physical interpretation of frac-tional integration and fractional differentiationrdquo FractionalCalculus amp Applied Analysis vol 5 no 4 pp 367ndash386 2002

[27] A Compte and R Metzler ldquoThe generalized Cattaneo equationfor the description of anomalous transport processesrdquo Journal ofPhysics A Mathematical and General vol 30 no 21 pp 7277ndash7289 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

31 Accuracy As described by Oldham and Spanier in [21]the error in the above scheme for functions 119906(119905) = 119862 119906(119905) = 119905and 119906(119905) = 119905

2 is given by 120572(120572+1)2119873 12057222119873 and 120572(120572minus1)(120572minus2)24119873

2 respectively Since we make the exact substitutionsfor the linear components we are able to increase the accuracyof the scheme to O(119873minus2) for any analytic function

Theorem 6 The error of GL2119863120572

0119905119906(119905) denoted by 120598(119906(119905))GL2 =

|119863120572

0119905119906(119905) minus GL2119863

120572

0119905119906(119905)| is at least O(119873minus2) accurate for the

function 119906(119905) = 119905119901 for all 119901 ge 2 given that the exact derivative

of order 120572 of 119905119901 is

119863120572

0119905119905119901=119905119901minus120572

Γ (119901 + 1)

Γ (119901 minus 120572 + 1) (21)

Proof Let 119901 = 2 Then (see [21] Chapter 8) (while Oldhamand Spanier cite the relative error it is easy to show thatthe more restrictive absolute error is also O(119873minus2) Moreinformation can be found in [22] Chapter 7)

120598 (119906 (119905))GL2 =120572 (120572 minus 1) (120572 minus 2)

241198732

(22)

Assume that the order of error is at least O(119873minus2) for 119901 = 119898that is

GL2119863120572

0119905119905119898=119905119898minus120572

Γ (119898 + 1)

Γ (119898 minus 120572 + 1)+ O (119873

minus2) (23)

Now let 119901 = 119898 + 1 then by the product rule for fractionalderivatives (see [21] Chapter 5) we have

GL2119863120572

0119905119905119898+1

= GL2119863120572

0119905(119905119898119905)

= 119905GL2119863120572

0119905119905119898+ 120572GL2119863

120572minus1

0119905119905119898

asymp 119905119905119898minus120572

Γ (119898 + 1)

Γ (119898 minus 120572 + 1)+ 120572

119905119898minus120572+1

Γ (119898 + 1)

Γ (119898 minus 120572)

+ O (119873minus2)

=119905119898minus120572+1

Γ (119898 + 1)

Γ (119898 minus 120572 + 1)(1 +

120572

119898 + 1 minus 120572)

+ O (119873minus2)

(24)

Then using the recursive property of the Gamma function119909Γ(119909) = Γ(119909 + 1) we have

GL2119863120572

0119905119905119898+1

=119905119898+1minus120572

Γ (119898 + 2)

Γ (119898 minus 120572 + 2)+ O (119873

minus2) (25)

Using Theorem 6 we can deduce that the accuracy ofGL2119863120572

0119905119906(119905) for any analytic function 119906(119905) isO(119873minus2) provided

that the linear and constant components of that function aredealt with accordingly

4 Results

This section presents some numerical results for differentfunctions Each example shows the absolute error over thedomain of the function the consistency of each derivativeas ℎ tends to zero the absolute error of the truncated seriesand the absolute error over a range of 120572 We comparethe aforementioned criteria for four different methods thestandard Grunwald-Letnikov approximation ( GL119863

120572

0119905) the

second-order Grunwald-Letnikov approximation ( GL2119863120572

0119905)

the second-order Grunwald-Letnikov approximation withconstant correction described in Section 3 (CCGL2119863

120572

0119905) and

finally the right-shifted Grunwald-Letnikov approximation( RSGL119863

120572

0119905)

41 Example 1 Let

119906 (119905) = 1199052 (26)

and let the order of derivative be 120572 = 075 with timestep Δ119905 = 0001 Since this choice of 119906(119905) satisfies 119906(0) =

0 the approximation does not suffer from poor accuracynear 119905 = 0 and hence the second-order Grunwald-Letnikov( GL2119863

120572

0119905) and the second-order Grunwald-Letnikov with

constant correction (CCGL2119863120572

0119905) approximations are similar

42 Example 2 Let

119906 (119905) = cos (119905) (27)

the order of derivative 120572 = 15 andΔ119905 = 0001 Asmentionedby Oldham and Spanier [21] this method is robust for all 120572this example shows highly accurate results for 120572 gt 1

43 Example 3 Let

119906 (119905) = cos (119905) + sin (119905) 1199052 (28)

the order of derivative 120572 = 09 and again Δ119905 = 0001This function suffers from the asymptotic behavior of thefractional derivative near 119905 = 0 and hence the constantcorrection method enjoys a vastly superior accuracy over theother methods

44 Example 4 Let

119906 (119905) = 119890minus119905 cos (119905) (29)

the order of derivative 120572 = 21 and Δ119905 = 0001

45 Example 5 Let

119906 (119905) = sin (119905) + cos (119905) (30)

the order of derivative 120572 = 075 and Δ119905 = 0001The above results indicate that the present method enjoys

a high accuracy The nature of the fractional derivative nearthe lower limit of differentiation results in a high gradientDifference schemes with fixed interval steps are unable to

Abstract and Applied Analysis 5

0 1 2 3 4 5 600000

00005

00010

00015

00020

t

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 1 Absolute error over time domain for four differentmethods

000 002 004 006 008 010000

002

004

006

008

h

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 2 Absolute error over varying values of Δ119905 for four differentmethods

accurately capture very steep gradients similar to those at theasymptote of the fractional derivativeThe use of the constantcorrection allows one to avoid the asymptotic behavior near119905 = 0 and the provable accuracy

Figures 1 5 9 13 and 17 show the increasing accuracyof the approximations as 119905 deviates from 0 as well as therobustness of our method to the asymptotic behavior Theconsistency of the methods is evident in Figures 2 6 10 14and 18These figures indicate that as ℎ tends to 0 the absoluteerror of the approximations also tends to 0 As mentionedin Section 1 the ldquoshort-memoryrdquo principle allows for thetruncation of the approximating series in order to alleviatecomputational cost our present method is highly accurateeven for small values of 119873 as is evident in Figures 3 7 1115 and 19 The final set of figures Figures 4 8 12 16 and 20indicate the robustness of our method to varying values of 120572It is important to note here the log scale of the graph

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 50 100 150 20000000

00005

00010

00015

00020

N

Abso

lute

erro

r

Figure 3Absolute error for increasing119873 for four differentmethods

00 05 10 15 20 25 30

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

120572

10minus2

10minus5

10minus8

10minus14

10minus11

Figure 4 Absolute error over varying values of 120572 for four differentmethods

5 Conclusion

This work has shown a novel derivation of a second-orderaccurate scheme for approximating the fractional orderderivative of a function To the best of the authorrsquos knowledgeperforming a linear transformation on the function to avoidasymptotic behavior near the origin is also novel Section 4provides several examples with varying parameters andbehaviors to illustrate the efficacy of the present method

Using a higher-order accurate approximation leads tothe potential for higher-order accurate numerical schemesfor fractional differential equations and fractional partialdifferential equations Tadjeran and Meerschaert [13] imple-ment the right-shifted Grunwald-Letnikov derivative forfractional derivatives of order 1 lt 120572 le 2 which lead toan unconditionally stable numerical scheme According toOldham and Spanier [21] the present method is robust for allvalues of alpha Potential extensions to this work include con-structing numerical schemes centered on this approximationand the fractional order derivative for nonlinear fractional

6 Abstract and Applied Analysis

0 1 2 3 4 5 600000

00005

00010

00015

t

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 5 Absolute error over time domain for four differentmethods

000 002 004 006 008 0100

1

2

3

4

5

6

7

h

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 6 Absolute error over varying values of Δ119905 for four differentmethods

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 50 100 150 200000

005

010

015

020

025

N

Abso

lute

erro

r

Figure 7 Absolute error for increasing 119873 for four different meth-ods

00 05 10 15 20 25 30

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

10minus7

10minus1

10minus4

120572

Figure 8 Absolute error over varying values of 120572 for four differentmethods

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 1 2 3 4 5 60000

0005

0010

0015

t

Figure 9 Absolute error over time domain for four differentmethods

000 002 004 006 008 01000

01

02

03

04

h

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 10 Absolute error over varying values ofΔ119905 for four differentmethods

Abstract and Applied Analysis 7

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 50 100 150 2000000

0002

0004

0006

0008

N

Abso

lute

erro

r

Figure 11 Absolute error for increasing 119873 for four differentmethods

00 05 10 15 20 25 30

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

120572

10minus1

10minus4

10minus7

Figure 12 Absolute error over varying values of 120572 for four differentmethods

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 1 2 3 4 5 600000

00001

00002

00003

00004

00005

t

Abso

lute

erro

r

Figure 13 Absolute error over time domain for four differentmethods

000 002 004 006 008 010000

005

010

015

020

h

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 14 Absolute error over varying values ofΔ119905 for four differentmethods

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 200 400 600 800 10000000

0001

0002

0003

0004

0005

N

Abso

lute

erro

r

Figure 15 Absolute error for increasing 119873 for four differentmethods

00 05 10 15 20 25 300000

0002

0004

0006

0008

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

120572

Figure 16 Absolute error over varying values of 120572 for four differentmethods

8 Abstract and Applied Analysis

0 1 2 3 4 5 600000

00005

00010

00015

t

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 17 Absolute error over time domain for four differentmethods

000 002 004 006 008 01000

01

02

03

04

05

06

h

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 18 Absolute error over varying values ofΔ119905 for four differentmethods

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 50 100 150 2000000

0005

0010

0015

N

Abso

lute

erro

r

Figure 19 Absolute error for increasing 119873 for four differentmethods

00 05 10 15 20 25 30

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

120572

10minus1

10minus4

10minus7

Figure 20 Absolute error over varying values of 120572 for four differentmethods

differential equations as well as time-marching schemes fortime or space fractional partial differential equations

The accuracy of the method has been proven to be atleast of O(119873minus2) which suggests that the accuracy should beof order 10minus6 for Δ119905 However in these examples the erroris observed to be in the region of 10minus8 Nevertheless themathematical proof of accuracy as well as the compellingresults presented substantiates the efforts toward this field

Disclaimer

The opinions expressed and conclusions arrived at herein arethose of the author and are not necessarily to be attributed tothe CoE-MaSS

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author would like to acknowledge and thank ProfessorCharis Harley for her helpful suggestions and useful insightThe DST-NRF Centre of Excellence in Mathematical andStatistical Sciences (CoE-MaSS) is acknowledged for fundingunder Grant no 94005 Thanks are due to the reviewers fortheir commentswhich helped to elevate the level of this work

References

[1] Q Zeng H Li and D Liu ldquoAnomalous fractional diffusionequation for transport phenomenardquo Communications in Non-linear Science and Numerical Simulation vol 4 no 2 pp 99ndash104 1999

[2] D A Benson S W Wheatcraft and M M MeerschaertldquoApplication of a fractional advection-dispersion equationrdquoWater Resources Research vol 36 no 6 pp 1403ndash1412 2000

Abstract and Applied Analysis 9

[3] J-H He ldquoApproximate analytical solution for seepage flowwithfractional derivatives in porous mediardquo Computer Methods inApplied Mechanics and Engineering vol 167 no 1-2 pp 57ndash681998

[4] R Metzler W G Glockle and T F Nonnenmacher ldquoFractionalmodel equation for anomalous diffusionrdquo Physica A StatisticalMechanics and its Applications vol 211 no 1 pp 13ndash24 1994

[5] V Gafiychuk B Datsko and V Meleshko ldquoMathematicalmodeling of time fractional reaction-diffusion systemsrdquo Journalof Computational and AppliedMathematics vol 220 no 1-2 pp215ndash225 2008

[6] B Y Datsko and V V Gafiychuk ldquoMathematical modeling offractional reaction-diffusion systems with different order timederivativesrdquo Journal of Mathematical Sciences vol 165 no 3 pp392ndash402 2010

[7] M M Meerschaert and C Tadjeran ldquoFinite difference approxi-mations for two-sided space-fractional partial differential equa-tionsrdquoApplied Numerical Mathematics vol 56 no 1 pp 80ndash902006

[8] R Scherer S L Kalla L Boyadjiev and B Al-Saqabi ldquoNumer-ical treatment of fractional heat equationsrdquo Applied NumericalMathematics vol 58 no 8 pp 1212ndash1223 2008

[9] A M El-Sayed I L El-Kalla and E A Ziada ldquoAnalytical andnumerical solutions of multi-term nonlinear fractional ordersdifferential equationsrdquo Applied Numerical Mathematics vol 60no 8 pp 788ndash797 2010

[10] I Karatay R S Bayramoglu and A Sahin ldquoImplicit differenceapproximation for the time fractional heat equation with thenonlocal conditionrdquo Applied Numerical Mathematics vol 61no 12 pp 1281ndash1288 2011

[11] L Su W Wang and H Wang ldquoA characteristic differencemethod for the transient fractional convection-diffusion equa-tionsrdquo Applied Numerical Mathematics vol 61 no 8 pp 946ndash960 2011

[12] M Chen W Deng and Y Wu ldquoSuperlinearly convergentalgorithms for the two-dimensional space-time caputo-rieszfractional diffusion equationrdquo Applied Numerical Mathematicsvol 70 pp 22ndash41 2013

[13] C Tadjeran and M M Meerschaert ldquoA second-order accuratenumerical method for the two-dimensional fractional diffusionequationrdquo Journal of Computational Physics vol 220 no 2 pp813ndash823 2007

[14] K Diethelm N J Ford and A D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 3ndash22 2002

[15] A Mohebbi M Abbaszadeh and M Dehghan ldquoA high-orderand unconditionally stable scheme for the modified anomalousfractional sub-diffusion equationwith a nonlinear source termrdquoJournal of Computational Physics vol 240 pp 36ndash48 2013

[16] M Cui ldquoCompact alternating direction implicit method fortwo-dimensional time fractional diffusion equationrdquo Journal ofComputational Physics vol 231 no 6 pp 2621ndash2633 2012

[17] M M Meerschaert H-P Scheffler and C Tadjeran ldquoFinitedifference methods for two-dimensional fractional dispersionequationrdquo Journal of Computational Physics vol 211 no 1 pp249ndash261 2006

[18] V E Lynch B A Carreras D del-Castillo-Negrete K MFerreira-Mejias and H R Hicks ldquoNumerical methods for thesolution of partial differential equations of fractional orderrdquoJournal of Computational Physics vol 192 no 2 pp 406ndash4212003

[19] S Momani and Z Odibat ldquoNumerical comparison of methodsfor solving linear differential equations of fractional orderrdquoChaos Solitons amp Fractals vol 31 no 5 pp 1248ndash1255 2007

[20] Z Odibat and S Momani ldquoNumerical methods for nonlinearpartial differential equations of fractional orderrdquo Applied Math-ematical Modelling vol 32 no 1 pp 28ndash39 2008

[21] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[22] I Podlubny Fractional Dfferential Equations vol 198 ofMathe-matics in Science and Engineering Academic Press 1999

[23] C Li andW Deng ldquoRemarks on fractional derivativesrdquoAppliedMathematics andComputation vol 187 no 2 pp 777ndash784 2007

[24] S B Yuste and L Acedo ldquoAn explicit finite difference methodand a new von Neumann-type stability analysis for fractionaldiffusion equationsrdquo SIAM Journal on Numerical Analysis vol42 no 5 pp 1862ndash1874 2005

[25] M M Meerschaert and C Tadjeran ldquoFinite difference approx-imations for fractional advection-dispersion flow equationsrdquoJournal of Computational and AppliedMathematics vol 172 no1 pp 65ndash77 2004

[26] I Podlubny ldquoGeometric and physical interpretation of frac-tional integration and fractional differentiationrdquo FractionalCalculus amp Applied Analysis vol 5 no 4 pp 367ndash386 2002

[27] A Compte and R Metzler ldquoThe generalized Cattaneo equationfor the description of anomalous transport processesrdquo Journal ofPhysics A Mathematical and General vol 30 no 21 pp 7277ndash7289 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

0 1 2 3 4 5 600000

00005

00010

00015

00020

t

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 1 Absolute error over time domain for four differentmethods

000 002 004 006 008 010000

002

004

006

008

h

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 2 Absolute error over varying values of Δ119905 for four differentmethods

accurately capture very steep gradients similar to those at theasymptote of the fractional derivativeThe use of the constantcorrection allows one to avoid the asymptotic behavior near119905 = 0 and the provable accuracy

Figures 1 5 9 13 and 17 show the increasing accuracyof the approximations as 119905 deviates from 0 as well as therobustness of our method to the asymptotic behavior Theconsistency of the methods is evident in Figures 2 6 10 14and 18These figures indicate that as ℎ tends to 0 the absoluteerror of the approximations also tends to 0 As mentionedin Section 1 the ldquoshort-memoryrdquo principle allows for thetruncation of the approximating series in order to alleviatecomputational cost our present method is highly accurateeven for small values of 119873 as is evident in Figures 3 7 1115 and 19 The final set of figures Figures 4 8 12 16 and 20indicate the robustness of our method to varying values of 120572It is important to note here the log scale of the graph

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 50 100 150 20000000

00005

00010

00015

00020

N

Abso

lute

erro

r

Figure 3Absolute error for increasing119873 for four differentmethods

00 05 10 15 20 25 30

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

120572

10minus2

10minus5

10minus8

10minus14

10minus11

Figure 4 Absolute error over varying values of 120572 for four differentmethods

5 Conclusion

This work has shown a novel derivation of a second-orderaccurate scheme for approximating the fractional orderderivative of a function To the best of the authorrsquos knowledgeperforming a linear transformation on the function to avoidasymptotic behavior near the origin is also novel Section 4provides several examples with varying parameters andbehaviors to illustrate the efficacy of the present method

Using a higher-order accurate approximation leads tothe potential for higher-order accurate numerical schemesfor fractional differential equations and fractional partialdifferential equations Tadjeran and Meerschaert [13] imple-ment the right-shifted Grunwald-Letnikov derivative forfractional derivatives of order 1 lt 120572 le 2 which lead toan unconditionally stable numerical scheme According toOldham and Spanier [21] the present method is robust for allvalues of alpha Potential extensions to this work include con-structing numerical schemes centered on this approximationand the fractional order derivative for nonlinear fractional

6 Abstract and Applied Analysis

0 1 2 3 4 5 600000

00005

00010

00015

t

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 5 Absolute error over time domain for four differentmethods

000 002 004 006 008 0100

1

2

3

4

5

6

7

h

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 6 Absolute error over varying values of Δ119905 for four differentmethods

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 50 100 150 200000

005

010

015

020

025

N

Abso

lute

erro

r

Figure 7 Absolute error for increasing 119873 for four different meth-ods

00 05 10 15 20 25 30

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

10minus7

10minus1

10minus4

120572

Figure 8 Absolute error over varying values of 120572 for four differentmethods

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 1 2 3 4 5 60000

0005

0010

0015

t

Figure 9 Absolute error over time domain for four differentmethods

000 002 004 006 008 01000

01

02

03

04

h

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 10 Absolute error over varying values ofΔ119905 for four differentmethods

Abstract and Applied Analysis 7

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 50 100 150 2000000

0002

0004

0006

0008

N

Abso

lute

erro

r

Figure 11 Absolute error for increasing 119873 for four differentmethods

00 05 10 15 20 25 30

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

120572

10minus1

10minus4

10minus7

Figure 12 Absolute error over varying values of 120572 for four differentmethods

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 1 2 3 4 5 600000

00001

00002

00003

00004

00005

t

Abso

lute

erro

r

Figure 13 Absolute error over time domain for four differentmethods

000 002 004 006 008 010000

005

010

015

020

h

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 14 Absolute error over varying values ofΔ119905 for four differentmethods

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 200 400 600 800 10000000

0001

0002

0003

0004

0005

N

Abso

lute

erro

r

Figure 15 Absolute error for increasing 119873 for four differentmethods

00 05 10 15 20 25 300000

0002

0004

0006

0008

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

120572

Figure 16 Absolute error over varying values of 120572 for four differentmethods

8 Abstract and Applied Analysis

0 1 2 3 4 5 600000

00005

00010

00015

t

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 17 Absolute error over time domain for four differentmethods

000 002 004 006 008 01000

01

02

03

04

05

06

h

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 18 Absolute error over varying values ofΔ119905 for four differentmethods

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 50 100 150 2000000

0005

0010

0015

N

Abso

lute

erro

r

Figure 19 Absolute error for increasing 119873 for four differentmethods

00 05 10 15 20 25 30

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

120572

10minus1

10minus4

10minus7

Figure 20 Absolute error over varying values of 120572 for four differentmethods

differential equations as well as time-marching schemes fortime or space fractional partial differential equations

The accuracy of the method has been proven to be atleast of O(119873minus2) which suggests that the accuracy should beof order 10minus6 for Δ119905 However in these examples the erroris observed to be in the region of 10minus8 Nevertheless themathematical proof of accuracy as well as the compellingresults presented substantiates the efforts toward this field

Disclaimer

The opinions expressed and conclusions arrived at herein arethose of the author and are not necessarily to be attributed tothe CoE-MaSS

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author would like to acknowledge and thank ProfessorCharis Harley for her helpful suggestions and useful insightThe DST-NRF Centre of Excellence in Mathematical andStatistical Sciences (CoE-MaSS) is acknowledged for fundingunder Grant no 94005 Thanks are due to the reviewers fortheir commentswhich helped to elevate the level of this work

References

[1] Q Zeng H Li and D Liu ldquoAnomalous fractional diffusionequation for transport phenomenardquo Communications in Non-linear Science and Numerical Simulation vol 4 no 2 pp 99ndash104 1999

[2] D A Benson S W Wheatcraft and M M MeerschaertldquoApplication of a fractional advection-dispersion equationrdquoWater Resources Research vol 36 no 6 pp 1403ndash1412 2000

Abstract and Applied Analysis 9

[3] J-H He ldquoApproximate analytical solution for seepage flowwithfractional derivatives in porous mediardquo Computer Methods inApplied Mechanics and Engineering vol 167 no 1-2 pp 57ndash681998

[4] R Metzler W G Glockle and T F Nonnenmacher ldquoFractionalmodel equation for anomalous diffusionrdquo Physica A StatisticalMechanics and its Applications vol 211 no 1 pp 13ndash24 1994

[5] V Gafiychuk B Datsko and V Meleshko ldquoMathematicalmodeling of time fractional reaction-diffusion systemsrdquo Journalof Computational and AppliedMathematics vol 220 no 1-2 pp215ndash225 2008

[6] B Y Datsko and V V Gafiychuk ldquoMathematical modeling offractional reaction-diffusion systems with different order timederivativesrdquo Journal of Mathematical Sciences vol 165 no 3 pp392ndash402 2010

[7] M M Meerschaert and C Tadjeran ldquoFinite difference approxi-mations for two-sided space-fractional partial differential equa-tionsrdquoApplied Numerical Mathematics vol 56 no 1 pp 80ndash902006

[8] R Scherer S L Kalla L Boyadjiev and B Al-Saqabi ldquoNumer-ical treatment of fractional heat equationsrdquo Applied NumericalMathematics vol 58 no 8 pp 1212ndash1223 2008

[9] A M El-Sayed I L El-Kalla and E A Ziada ldquoAnalytical andnumerical solutions of multi-term nonlinear fractional ordersdifferential equationsrdquo Applied Numerical Mathematics vol 60no 8 pp 788ndash797 2010

[10] I Karatay R S Bayramoglu and A Sahin ldquoImplicit differenceapproximation for the time fractional heat equation with thenonlocal conditionrdquo Applied Numerical Mathematics vol 61no 12 pp 1281ndash1288 2011

[11] L Su W Wang and H Wang ldquoA characteristic differencemethod for the transient fractional convection-diffusion equa-tionsrdquo Applied Numerical Mathematics vol 61 no 8 pp 946ndash960 2011

[12] M Chen W Deng and Y Wu ldquoSuperlinearly convergentalgorithms for the two-dimensional space-time caputo-rieszfractional diffusion equationrdquo Applied Numerical Mathematicsvol 70 pp 22ndash41 2013

[13] C Tadjeran and M M Meerschaert ldquoA second-order accuratenumerical method for the two-dimensional fractional diffusionequationrdquo Journal of Computational Physics vol 220 no 2 pp813ndash823 2007

[14] K Diethelm N J Ford and A D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 3ndash22 2002

[15] A Mohebbi M Abbaszadeh and M Dehghan ldquoA high-orderand unconditionally stable scheme for the modified anomalousfractional sub-diffusion equationwith a nonlinear source termrdquoJournal of Computational Physics vol 240 pp 36ndash48 2013

[16] M Cui ldquoCompact alternating direction implicit method fortwo-dimensional time fractional diffusion equationrdquo Journal ofComputational Physics vol 231 no 6 pp 2621ndash2633 2012

[17] M M Meerschaert H-P Scheffler and C Tadjeran ldquoFinitedifference methods for two-dimensional fractional dispersionequationrdquo Journal of Computational Physics vol 211 no 1 pp249ndash261 2006

[18] V E Lynch B A Carreras D del-Castillo-Negrete K MFerreira-Mejias and H R Hicks ldquoNumerical methods for thesolution of partial differential equations of fractional orderrdquoJournal of Computational Physics vol 192 no 2 pp 406ndash4212003

[19] S Momani and Z Odibat ldquoNumerical comparison of methodsfor solving linear differential equations of fractional orderrdquoChaos Solitons amp Fractals vol 31 no 5 pp 1248ndash1255 2007

[20] Z Odibat and S Momani ldquoNumerical methods for nonlinearpartial differential equations of fractional orderrdquo Applied Math-ematical Modelling vol 32 no 1 pp 28ndash39 2008

[21] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[22] I Podlubny Fractional Dfferential Equations vol 198 ofMathe-matics in Science and Engineering Academic Press 1999

[23] C Li andW Deng ldquoRemarks on fractional derivativesrdquoAppliedMathematics andComputation vol 187 no 2 pp 777ndash784 2007

[24] S B Yuste and L Acedo ldquoAn explicit finite difference methodand a new von Neumann-type stability analysis for fractionaldiffusion equationsrdquo SIAM Journal on Numerical Analysis vol42 no 5 pp 1862ndash1874 2005

[25] M M Meerschaert and C Tadjeran ldquoFinite difference approx-imations for fractional advection-dispersion flow equationsrdquoJournal of Computational and AppliedMathematics vol 172 no1 pp 65ndash77 2004

[26] I Podlubny ldquoGeometric and physical interpretation of frac-tional integration and fractional differentiationrdquo FractionalCalculus amp Applied Analysis vol 5 no 4 pp 367ndash386 2002

[27] A Compte and R Metzler ldquoThe generalized Cattaneo equationfor the description of anomalous transport processesrdquo Journal ofPhysics A Mathematical and General vol 30 no 21 pp 7277ndash7289 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

0 1 2 3 4 5 600000

00005

00010

00015

t

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 5 Absolute error over time domain for four differentmethods

000 002 004 006 008 0100

1

2

3

4

5

6

7

h

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 6 Absolute error over varying values of Δ119905 for four differentmethods

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 50 100 150 200000

005

010

015

020

025

N

Abso

lute

erro

r

Figure 7 Absolute error for increasing 119873 for four different meth-ods

00 05 10 15 20 25 30

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

10minus7

10minus1

10minus4

120572

Figure 8 Absolute error over varying values of 120572 for four differentmethods

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 1 2 3 4 5 60000

0005

0010

0015

t

Figure 9 Absolute error over time domain for four differentmethods

000 002 004 006 008 01000

01

02

03

04

h

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 10 Absolute error over varying values ofΔ119905 for four differentmethods

Abstract and Applied Analysis 7

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 50 100 150 2000000

0002

0004

0006

0008

N

Abso

lute

erro

r

Figure 11 Absolute error for increasing 119873 for four differentmethods

00 05 10 15 20 25 30

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

120572

10minus1

10minus4

10minus7

Figure 12 Absolute error over varying values of 120572 for four differentmethods

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 1 2 3 4 5 600000

00001

00002

00003

00004

00005

t

Abso

lute

erro

r

Figure 13 Absolute error over time domain for four differentmethods

000 002 004 006 008 010000

005

010

015

020

h

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 14 Absolute error over varying values ofΔ119905 for four differentmethods

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 200 400 600 800 10000000

0001

0002

0003

0004

0005

N

Abso

lute

erro

r

Figure 15 Absolute error for increasing 119873 for four differentmethods

00 05 10 15 20 25 300000

0002

0004

0006

0008

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

120572

Figure 16 Absolute error over varying values of 120572 for four differentmethods

8 Abstract and Applied Analysis

0 1 2 3 4 5 600000

00005

00010

00015

t

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 17 Absolute error over time domain for four differentmethods

000 002 004 006 008 01000

01

02

03

04

05

06

h

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 18 Absolute error over varying values ofΔ119905 for four differentmethods

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 50 100 150 2000000

0005

0010

0015

N

Abso

lute

erro

r

Figure 19 Absolute error for increasing 119873 for four differentmethods

00 05 10 15 20 25 30

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

120572

10minus1

10minus4

10minus7

Figure 20 Absolute error over varying values of 120572 for four differentmethods

differential equations as well as time-marching schemes fortime or space fractional partial differential equations

The accuracy of the method has been proven to be atleast of O(119873minus2) which suggests that the accuracy should beof order 10minus6 for Δ119905 However in these examples the erroris observed to be in the region of 10minus8 Nevertheless themathematical proof of accuracy as well as the compellingresults presented substantiates the efforts toward this field

Disclaimer

The opinions expressed and conclusions arrived at herein arethose of the author and are not necessarily to be attributed tothe CoE-MaSS

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author would like to acknowledge and thank ProfessorCharis Harley for her helpful suggestions and useful insightThe DST-NRF Centre of Excellence in Mathematical andStatistical Sciences (CoE-MaSS) is acknowledged for fundingunder Grant no 94005 Thanks are due to the reviewers fortheir commentswhich helped to elevate the level of this work

References

[1] Q Zeng H Li and D Liu ldquoAnomalous fractional diffusionequation for transport phenomenardquo Communications in Non-linear Science and Numerical Simulation vol 4 no 2 pp 99ndash104 1999

[2] D A Benson S W Wheatcraft and M M MeerschaertldquoApplication of a fractional advection-dispersion equationrdquoWater Resources Research vol 36 no 6 pp 1403ndash1412 2000

Abstract and Applied Analysis 9

[3] J-H He ldquoApproximate analytical solution for seepage flowwithfractional derivatives in porous mediardquo Computer Methods inApplied Mechanics and Engineering vol 167 no 1-2 pp 57ndash681998

[4] R Metzler W G Glockle and T F Nonnenmacher ldquoFractionalmodel equation for anomalous diffusionrdquo Physica A StatisticalMechanics and its Applications vol 211 no 1 pp 13ndash24 1994

[5] V Gafiychuk B Datsko and V Meleshko ldquoMathematicalmodeling of time fractional reaction-diffusion systemsrdquo Journalof Computational and AppliedMathematics vol 220 no 1-2 pp215ndash225 2008

[6] B Y Datsko and V V Gafiychuk ldquoMathematical modeling offractional reaction-diffusion systems with different order timederivativesrdquo Journal of Mathematical Sciences vol 165 no 3 pp392ndash402 2010

[7] M M Meerschaert and C Tadjeran ldquoFinite difference approxi-mations for two-sided space-fractional partial differential equa-tionsrdquoApplied Numerical Mathematics vol 56 no 1 pp 80ndash902006

[8] R Scherer S L Kalla L Boyadjiev and B Al-Saqabi ldquoNumer-ical treatment of fractional heat equationsrdquo Applied NumericalMathematics vol 58 no 8 pp 1212ndash1223 2008

[9] A M El-Sayed I L El-Kalla and E A Ziada ldquoAnalytical andnumerical solutions of multi-term nonlinear fractional ordersdifferential equationsrdquo Applied Numerical Mathematics vol 60no 8 pp 788ndash797 2010

[10] I Karatay R S Bayramoglu and A Sahin ldquoImplicit differenceapproximation for the time fractional heat equation with thenonlocal conditionrdquo Applied Numerical Mathematics vol 61no 12 pp 1281ndash1288 2011

[11] L Su W Wang and H Wang ldquoA characteristic differencemethod for the transient fractional convection-diffusion equa-tionsrdquo Applied Numerical Mathematics vol 61 no 8 pp 946ndash960 2011

[12] M Chen W Deng and Y Wu ldquoSuperlinearly convergentalgorithms for the two-dimensional space-time caputo-rieszfractional diffusion equationrdquo Applied Numerical Mathematicsvol 70 pp 22ndash41 2013

[13] C Tadjeran and M M Meerschaert ldquoA second-order accuratenumerical method for the two-dimensional fractional diffusionequationrdquo Journal of Computational Physics vol 220 no 2 pp813ndash823 2007

[14] K Diethelm N J Ford and A D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 3ndash22 2002

[15] A Mohebbi M Abbaszadeh and M Dehghan ldquoA high-orderand unconditionally stable scheme for the modified anomalousfractional sub-diffusion equationwith a nonlinear source termrdquoJournal of Computational Physics vol 240 pp 36ndash48 2013

[16] M Cui ldquoCompact alternating direction implicit method fortwo-dimensional time fractional diffusion equationrdquo Journal ofComputational Physics vol 231 no 6 pp 2621ndash2633 2012

[17] M M Meerschaert H-P Scheffler and C Tadjeran ldquoFinitedifference methods for two-dimensional fractional dispersionequationrdquo Journal of Computational Physics vol 211 no 1 pp249ndash261 2006

[18] V E Lynch B A Carreras D del-Castillo-Negrete K MFerreira-Mejias and H R Hicks ldquoNumerical methods for thesolution of partial differential equations of fractional orderrdquoJournal of Computational Physics vol 192 no 2 pp 406ndash4212003

[19] S Momani and Z Odibat ldquoNumerical comparison of methodsfor solving linear differential equations of fractional orderrdquoChaos Solitons amp Fractals vol 31 no 5 pp 1248ndash1255 2007

[20] Z Odibat and S Momani ldquoNumerical methods for nonlinearpartial differential equations of fractional orderrdquo Applied Math-ematical Modelling vol 32 no 1 pp 28ndash39 2008

[21] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[22] I Podlubny Fractional Dfferential Equations vol 198 ofMathe-matics in Science and Engineering Academic Press 1999

[23] C Li andW Deng ldquoRemarks on fractional derivativesrdquoAppliedMathematics andComputation vol 187 no 2 pp 777ndash784 2007

[24] S B Yuste and L Acedo ldquoAn explicit finite difference methodand a new von Neumann-type stability analysis for fractionaldiffusion equationsrdquo SIAM Journal on Numerical Analysis vol42 no 5 pp 1862ndash1874 2005

[25] M M Meerschaert and C Tadjeran ldquoFinite difference approx-imations for fractional advection-dispersion flow equationsrdquoJournal of Computational and AppliedMathematics vol 172 no1 pp 65ndash77 2004

[26] I Podlubny ldquoGeometric and physical interpretation of frac-tional integration and fractional differentiationrdquo FractionalCalculus amp Applied Analysis vol 5 no 4 pp 367ndash386 2002

[27] A Compte and R Metzler ldquoThe generalized Cattaneo equationfor the description of anomalous transport processesrdquo Journal ofPhysics A Mathematical and General vol 30 no 21 pp 7277ndash7289 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 50 100 150 2000000

0002

0004

0006

0008

N

Abso

lute

erro

r

Figure 11 Absolute error for increasing 119873 for four differentmethods

00 05 10 15 20 25 30

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

120572

10minus1

10minus4

10minus7

Figure 12 Absolute error over varying values of 120572 for four differentmethods

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 1 2 3 4 5 600000

00001

00002

00003

00004

00005

t

Abso

lute

erro

r

Figure 13 Absolute error over time domain for four differentmethods

000 002 004 006 008 010000

005

010

015

020

h

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 14 Absolute error over varying values ofΔ119905 for four differentmethods

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 200 400 600 800 10000000

0001

0002

0003

0004

0005

N

Abso

lute

erro

r

Figure 15 Absolute error for increasing 119873 for four differentmethods

00 05 10 15 20 25 300000

0002

0004

0006

0008

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

120572

Figure 16 Absolute error over varying values of 120572 for four differentmethods

8 Abstract and Applied Analysis

0 1 2 3 4 5 600000

00005

00010

00015

t

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 17 Absolute error over time domain for four differentmethods

000 002 004 006 008 01000

01

02

03

04

05

06

h

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 18 Absolute error over varying values ofΔ119905 for four differentmethods

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 50 100 150 2000000

0005

0010

0015

N

Abso

lute

erro

r

Figure 19 Absolute error for increasing 119873 for four differentmethods

00 05 10 15 20 25 30

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

120572

10minus1

10minus4

10minus7

Figure 20 Absolute error over varying values of 120572 for four differentmethods

differential equations as well as time-marching schemes fortime or space fractional partial differential equations

The accuracy of the method has been proven to be atleast of O(119873minus2) which suggests that the accuracy should beof order 10minus6 for Δ119905 However in these examples the erroris observed to be in the region of 10minus8 Nevertheless themathematical proof of accuracy as well as the compellingresults presented substantiates the efforts toward this field

Disclaimer

The opinions expressed and conclusions arrived at herein arethose of the author and are not necessarily to be attributed tothe CoE-MaSS

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author would like to acknowledge and thank ProfessorCharis Harley for her helpful suggestions and useful insightThe DST-NRF Centre of Excellence in Mathematical andStatistical Sciences (CoE-MaSS) is acknowledged for fundingunder Grant no 94005 Thanks are due to the reviewers fortheir commentswhich helped to elevate the level of this work

References

[1] Q Zeng H Li and D Liu ldquoAnomalous fractional diffusionequation for transport phenomenardquo Communications in Non-linear Science and Numerical Simulation vol 4 no 2 pp 99ndash104 1999

[2] D A Benson S W Wheatcraft and M M MeerschaertldquoApplication of a fractional advection-dispersion equationrdquoWater Resources Research vol 36 no 6 pp 1403ndash1412 2000

Abstract and Applied Analysis 9

[3] J-H He ldquoApproximate analytical solution for seepage flowwithfractional derivatives in porous mediardquo Computer Methods inApplied Mechanics and Engineering vol 167 no 1-2 pp 57ndash681998

[4] R Metzler W G Glockle and T F Nonnenmacher ldquoFractionalmodel equation for anomalous diffusionrdquo Physica A StatisticalMechanics and its Applications vol 211 no 1 pp 13ndash24 1994

[5] V Gafiychuk B Datsko and V Meleshko ldquoMathematicalmodeling of time fractional reaction-diffusion systemsrdquo Journalof Computational and AppliedMathematics vol 220 no 1-2 pp215ndash225 2008

[6] B Y Datsko and V V Gafiychuk ldquoMathematical modeling offractional reaction-diffusion systems with different order timederivativesrdquo Journal of Mathematical Sciences vol 165 no 3 pp392ndash402 2010

[7] M M Meerschaert and C Tadjeran ldquoFinite difference approxi-mations for two-sided space-fractional partial differential equa-tionsrdquoApplied Numerical Mathematics vol 56 no 1 pp 80ndash902006

[8] R Scherer S L Kalla L Boyadjiev and B Al-Saqabi ldquoNumer-ical treatment of fractional heat equationsrdquo Applied NumericalMathematics vol 58 no 8 pp 1212ndash1223 2008

[9] A M El-Sayed I L El-Kalla and E A Ziada ldquoAnalytical andnumerical solutions of multi-term nonlinear fractional ordersdifferential equationsrdquo Applied Numerical Mathematics vol 60no 8 pp 788ndash797 2010

[10] I Karatay R S Bayramoglu and A Sahin ldquoImplicit differenceapproximation for the time fractional heat equation with thenonlocal conditionrdquo Applied Numerical Mathematics vol 61no 12 pp 1281ndash1288 2011

[11] L Su W Wang and H Wang ldquoA characteristic differencemethod for the transient fractional convection-diffusion equa-tionsrdquo Applied Numerical Mathematics vol 61 no 8 pp 946ndash960 2011

[12] M Chen W Deng and Y Wu ldquoSuperlinearly convergentalgorithms for the two-dimensional space-time caputo-rieszfractional diffusion equationrdquo Applied Numerical Mathematicsvol 70 pp 22ndash41 2013

[13] C Tadjeran and M M Meerschaert ldquoA second-order accuratenumerical method for the two-dimensional fractional diffusionequationrdquo Journal of Computational Physics vol 220 no 2 pp813ndash823 2007

[14] K Diethelm N J Ford and A D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 3ndash22 2002

[15] A Mohebbi M Abbaszadeh and M Dehghan ldquoA high-orderand unconditionally stable scheme for the modified anomalousfractional sub-diffusion equationwith a nonlinear source termrdquoJournal of Computational Physics vol 240 pp 36ndash48 2013

[16] M Cui ldquoCompact alternating direction implicit method fortwo-dimensional time fractional diffusion equationrdquo Journal ofComputational Physics vol 231 no 6 pp 2621ndash2633 2012

[17] M M Meerschaert H-P Scheffler and C Tadjeran ldquoFinitedifference methods for two-dimensional fractional dispersionequationrdquo Journal of Computational Physics vol 211 no 1 pp249ndash261 2006

[18] V E Lynch B A Carreras D del-Castillo-Negrete K MFerreira-Mejias and H R Hicks ldquoNumerical methods for thesolution of partial differential equations of fractional orderrdquoJournal of Computational Physics vol 192 no 2 pp 406ndash4212003

[19] S Momani and Z Odibat ldquoNumerical comparison of methodsfor solving linear differential equations of fractional orderrdquoChaos Solitons amp Fractals vol 31 no 5 pp 1248ndash1255 2007

[20] Z Odibat and S Momani ldquoNumerical methods for nonlinearpartial differential equations of fractional orderrdquo Applied Math-ematical Modelling vol 32 no 1 pp 28ndash39 2008

[21] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[22] I Podlubny Fractional Dfferential Equations vol 198 ofMathe-matics in Science and Engineering Academic Press 1999

[23] C Li andW Deng ldquoRemarks on fractional derivativesrdquoAppliedMathematics andComputation vol 187 no 2 pp 777ndash784 2007

[24] S B Yuste and L Acedo ldquoAn explicit finite difference methodand a new von Neumann-type stability analysis for fractionaldiffusion equationsrdquo SIAM Journal on Numerical Analysis vol42 no 5 pp 1862ndash1874 2005

[25] M M Meerschaert and C Tadjeran ldquoFinite difference approx-imations for fractional advection-dispersion flow equationsrdquoJournal of Computational and AppliedMathematics vol 172 no1 pp 65ndash77 2004

[26] I Podlubny ldquoGeometric and physical interpretation of frac-tional integration and fractional differentiationrdquo FractionalCalculus amp Applied Analysis vol 5 no 4 pp 367ndash386 2002

[27] A Compte and R Metzler ldquoThe generalized Cattaneo equationfor the description of anomalous transport processesrdquo Journal ofPhysics A Mathematical and General vol 30 no 21 pp 7277ndash7289 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

0 1 2 3 4 5 600000

00005

00010

00015

t

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 17 Absolute error over time domain for four differentmethods

000 002 004 006 008 01000

01

02

03

04

05

06

h

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

Figure 18 Absolute error over varying values ofΔ119905 for four differentmethods

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

0 50 100 150 2000000

0005

0010

0015

N

Abso

lute

erro

r

Figure 19 Absolute error for increasing 119873 for four differentmethods

00 05 10 15 20 25 30

Abso

lute

erro

r

GLD2nd-order GLD

2nd-order GLD with CCRS GLD

120572

10minus1

10minus4

10minus7

Figure 20 Absolute error over varying values of 120572 for four differentmethods

differential equations as well as time-marching schemes fortime or space fractional partial differential equations

The accuracy of the method has been proven to be atleast of O(119873minus2) which suggests that the accuracy should beof order 10minus6 for Δ119905 However in these examples the erroris observed to be in the region of 10minus8 Nevertheless themathematical proof of accuracy as well as the compellingresults presented substantiates the efforts toward this field

Disclaimer

The opinions expressed and conclusions arrived at herein arethose of the author and are not necessarily to be attributed tothe CoE-MaSS

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author would like to acknowledge and thank ProfessorCharis Harley for her helpful suggestions and useful insightThe DST-NRF Centre of Excellence in Mathematical andStatistical Sciences (CoE-MaSS) is acknowledged for fundingunder Grant no 94005 Thanks are due to the reviewers fortheir commentswhich helped to elevate the level of this work

References

[1] Q Zeng H Li and D Liu ldquoAnomalous fractional diffusionequation for transport phenomenardquo Communications in Non-linear Science and Numerical Simulation vol 4 no 2 pp 99ndash104 1999

[2] D A Benson S W Wheatcraft and M M MeerschaertldquoApplication of a fractional advection-dispersion equationrdquoWater Resources Research vol 36 no 6 pp 1403ndash1412 2000

Abstract and Applied Analysis 9

[3] J-H He ldquoApproximate analytical solution for seepage flowwithfractional derivatives in porous mediardquo Computer Methods inApplied Mechanics and Engineering vol 167 no 1-2 pp 57ndash681998

[4] R Metzler W G Glockle and T F Nonnenmacher ldquoFractionalmodel equation for anomalous diffusionrdquo Physica A StatisticalMechanics and its Applications vol 211 no 1 pp 13ndash24 1994

[5] V Gafiychuk B Datsko and V Meleshko ldquoMathematicalmodeling of time fractional reaction-diffusion systemsrdquo Journalof Computational and AppliedMathematics vol 220 no 1-2 pp215ndash225 2008

[6] B Y Datsko and V V Gafiychuk ldquoMathematical modeling offractional reaction-diffusion systems with different order timederivativesrdquo Journal of Mathematical Sciences vol 165 no 3 pp392ndash402 2010

[7] M M Meerschaert and C Tadjeran ldquoFinite difference approxi-mations for two-sided space-fractional partial differential equa-tionsrdquoApplied Numerical Mathematics vol 56 no 1 pp 80ndash902006

[8] R Scherer S L Kalla L Boyadjiev and B Al-Saqabi ldquoNumer-ical treatment of fractional heat equationsrdquo Applied NumericalMathematics vol 58 no 8 pp 1212ndash1223 2008

[9] A M El-Sayed I L El-Kalla and E A Ziada ldquoAnalytical andnumerical solutions of multi-term nonlinear fractional ordersdifferential equationsrdquo Applied Numerical Mathematics vol 60no 8 pp 788ndash797 2010

[10] I Karatay R S Bayramoglu and A Sahin ldquoImplicit differenceapproximation for the time fractional heat equation with thenonlocal conditionrdquo Applied Numerical Mathematics vol 61no 12 pp 1281ndash1288 2011

[11] L Su W Wang and H Wang ldquoA characteristic differencemethod for the transient fractional convection-diffusion equa-tionsrdquo Applied Numerical Mathematics vol 61 no 8 pp 946ndash960 2011

[12] M Chen W Deng and Y Wu ldquoSuperlinearly convergentalgorithms for the two-dimensional space-time caputo-rieszfractional diffusion equationrdquo Applied Numerical Mathematicsvol 70 pp 22ndash41 2013

[13] C Tadjeran and M M Meerschaert ldquoA second-order accuratenumerical method for the two-dimensional fractional diffusionequationrdquo Journal of Computational Physics vol 220 no 2 pp813ndash823 2007

[14] K Diethelm N J Ford and A D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 3ndash22 2002

[15] A Mohebbi M Abbaszadeh and M Dehghan ldquoA high-orderand unconditionally stable scheme for the modified anomalousfractional sub-diffusion equationwith a nonlinear source termrdquoJournal of Computational Physics vol 240 pp 36ndash48 2013

[16] M Cui ldquoCompact alternating direction implicit method fortwo-dimensional time fractional diffusion equationrdquo Journal ofComputational Physics vol 231 no 6 pp 2621ndash2633 2012

[17] M M Meerschaert H-P Scheffler and C Tadjeran ldquoFinitedifference methods for two-dimensional fractional dispersionequationrdquo Journal of Computational Physics vol 211 no 1 pp249ndash261 2006

[18] V E Lynch B A Carreras D del-Castillo-Negrete K MFerreira-Mejias and H R Hicks ldquoNumerical methods for thesolution of partial differential equations of fractional orderrdquoJournal of Computational Physics vol 192 no 2 pp 406ndash4212003

[19] S Momani and Z Odibat ldquoNumerical comparison of methodsfor solving linear differential equations of fractional orderrdquoChaos Solitons amp Fractals vol 31 no 5 pp 1248ndash1255 2007

[20] Z Odibat and S Momani ldquoNumerical methods for nonlinearpartial differential equations of fractional orderrdquo Applied Math-ematical Modelling vol 32 no 1 pp 28ndash39 2008

[21] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[22] I Podlubny Fractional Dfferential Equations vol 198 ofMathe-matics in Science and Engineering Academic Press 1999

[23] C Li andW Deng ldquoRemarks on fractional derivativesrdquoAppliedMathematics andComputation vol 187 no 2 pp 777ndash784 2007

[24] S B Yuste and L Acedo ldquoAn explicit finite difference methodand a new von Neumann-type stability analysis for fractionaldiffusion equationsrdquo SIAM Journal on Numerical Analysis vol42 no 5 pp 1862ndash1874 2005

[25] M M Meerschaert and C Tadjeran ldquoFinite difference approx-imations for fractional advection-dispersion flow equationsrdquoJournal of Computational and AppliedMathematics vol 172 no1 pp 65ndash77 2004

[26] I Podlubny ldquoGeometric and physical interpretation of frac-tional integration and fractional differentiationrdquo FractionalCalculus amp Applied Analysis vol 5 no 4 pp 367ndash386 2002

[27] A Compte and R Metzler ldquoThe generalized Cattaneo equationfor the description of anomalous transport processesrdquo Journal ofPhysics A Mathematical and General vol 30 no 21 pp 7277ndash7289 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

[3] J-H He ldquoApproximate analytical solution for seepage flowwithfractional derivatives in porous mediardquo Computer Methods inApplied Mechanics and Engineering vol 167 no 1-2 pp 57ndash681998

[4] R Metzler W G Glockle and T F Nonnenmacher ldquoFractionalmodel equation for anomalous diffusionrdquo Physica A StatisticalMechanics and its Applications vol 211 no 1 pp 13ndash24 1994

[5] V Gafiychuk B Datsko and V Meleshko ldquoMathematicalmodeling of time fractional reaction-diffusion systemsrdquo Journalof Computational and AppliedMathematics vol 220 no 1-2 pp215ndash225 2008

[6] B Y Datsko and V V Gafiychuk ldquoMathematical modeling offractional reaction-diffusion systems with different order timederivativesrdquo Journal of Mathematical Sciences vol 165 no 3 pp392ndash402 2010

[7] M M Meerschaert and C Tadjeran ldquoFinite difference approxi-mations for two-sided space-fractional partial differential equa-tionsrdquoApplied Numerical Mathematics vol 56 no 1 pp 80ndash902006

[8] R Scherer S L Kalla L Boyadjiev and B Al-Saqabi ldquoNumer-ical treatment of fractional heat equationsrdquo Applied NumericalMathematics vol 58 no 8 pp 1212ndash1223 2008

[9] A M El-Sayed I L El-Kalla and E A Ziada ldquoAnalytical andnumerical solutions of multi-term nonlinear fractional ordersdifferential equationsrdquo Applied Numerical Mathematics vol 60no 8 pp 788ndash797 2010

[10] I Karatay R S Bayramoglu and A Sahin ldquoImplicit differenceapproximation for the time fractional heat equation with thenonlocal conditionrdquo Applied Numerical Mathematics vol 61no 12 pp 1281ndash1288 2011

[11] L Su W Wang and H Wang ldquoA characteristic differencemethod for the transient fractional convection-diffusion equa-tionsrdquo Applied Numerical Mathematics vol 61 no 8 pp 946ndash960 2011

[12] M Chen W Deng and Y Wu ldquoSuperlinearly convergentalgorithms for the two-dimensional space-time caputo-rieszfractional diffusion equationrdquo Applied Numerical Mathematicsvol 70 pp 22ndash41 2013

[13] C Tadjeran and M M Meerschaert ldquoA second-order accuratenumerical method for the two-dimensional fractional diffusionequationrdquo Journal of Computational Physics vol 220 no 2 pp813ndash823 2007

[14] K Diethelm N J Ford and A D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 3ndash22 2002

[15] A Mohebbi M Abbaszadeh and M Dehghan ldquoA high-orderand unconditionally stable scheme for the modified anomalousfractional sub-diffusion equationwith a nonlinear source termrdquoJournal of Computational Physics vol 240 pp 36ndash48 2013

[16] M Cui ldquoCompact alternating direction implicit method fortwo-dimensional time fractional diffusion equationrdquo Journal ofComputational Physics vol 231 no 6 pp 2621ndash2633 2012

[17] M M Meerschaert H-P Scheffler and C Tadjeran ldquoFinitedifference methods for two-dimensional fractional dispersionequationrdquo Journal of Computational Physics vol 211 no 1 pp249ndash261 2006

[18] V E Lynch B A Carreras D del-Castillo-Negrete K MFerreira-Mejias and H R Hicks ldquoNumerical methods for thesolution of partial differential equations of fractional orderrdquoJournal of Computational Physics vol 192 no 2 pp 406ndash4212003

[19] S Momani and Z Odibat ldquoNumerical comparison of methodsfor solving linear differential equations of fractional orderrdquoChaos Solitons amp Fractals vol 31 no 5 pp 1248ndash1255 2007

[20] Z Odibat and S Momani ldquoNumerical methods for nonlinearpartial differential equations of fractional orderrdquo Applied Math-ematical Modelling vol 32 no 1 pp 28ndash39 2008

[21] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[22] I Podlubny Fractional Dfferential Equations vol 198 ofMathe-matics in Science and Engineering Academic Press 1999

[23] C Li andW Deng ldquoRemarks on fractional derivativesrdquoAppliedMathematics andComputation vol 187 no 2 pp 777ndash784 2007

[24] S B Yuste and L Acedo ldquoAn explicit finite difference methodand a new von Neumann-type stability analysis for fractionaldiffusion equationsrdquo SIAM Journal on Numerical Analysis vol42 no 5 pp 1862ndash1874 2005

[25] M M Meerschaert and C Tadjeran ldquoFinite difference approx-imations for fractional advection-dispersion flow equationsrdquoJournal of Computational and AppliedMathematics vol 172 no1 pp 65ndash77 2004

[26] I Podlubny ldquoGeometric and physical interpretation of frac-tional integration and fractional differentiationrdquo FractionalCalculus amp Applied Analysis vol 5 no 4 pp 367ndash386 2002

[27] A Compte and R Metzler ldquoThe generalized Cattaneo equationfor the description of anomalous transport processesrdquo Journal ofPhysics A Mathematical and General vol 30 no 21 pp 7277ndash7289 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of