Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 256071 7 pageshttpdxdoiorg1011552013256071

Research ArticleA Numerical Method for Delayed Fractional-OrderDifferential Equations

Zhen Wang12

1 College of Information Science and Engineering Shandong University of Science and Technology Qingdao 266590 China2 State Key Laboratory for Novel Software Technology Nanjing University Nanjing 210093 China

Correspondence should be addressed to Zhen Wang wangzhensdgmailcom

Received 3 February 2013 Accepted 14 April 2013

Academic Editor Theodore E Simos

Copyright copy 2013 Zhen Wang 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 numerical method for nonlinear fractional-order differential equations with constant or time-varying delay is devised Theorder here is an arbitrary positive real number and the differential operator is with the Caputo definition The general Adams-Bashforth-Moulton method combined with the linear interpolation method is employed to approximate the delayed fractional-order differential equations Meanwhile the detailed error analysis for this algorithm is given In order to compare with the exactanalytical solution a numerical example is provided to illustrate the effectiveness of the proposed method

1 Introduction

Fractional calculus is an old mathematical problem and hasbeen always thought of as a pure mathematical problem fornearly three centuries [1ndash3] Though having a long historyit was not applied to physics and engineering for a longperiod of time However in the last few decades fractionalcalculus began to attract increasing attention of scientistsfrom an application point of view [4ndash6] In the fields ofcontinuous-time modeling many scholars have pointed outthat fractional derivatives are very helpful in describing linearviscoelasticity acoustics rheology polymeric chemistry andso forth Moreover fractional derivatives have proved to be avery suitable tool for the description of memory and heredi-tary properties of various materials and processes Nowadaysthe mathematical theories and practical applications of theseoperators are well established and their applicabilities toscience and engineering are being considered as attractivetopics Inmodeling dynamical processes equations describedwith fractional derivatives have appeared in various fieldssuch as physics chemistry and engineering [7 8] andtherefore fractional-order differential equations (FDEs) arebecoming the focus of many studies

The development of effective methods for numericallysolving FDEs has received increasing attention over the last

few years Several methods based on Caputo or Riemann-Liouville definition have been proposed and analyzed [9ndash24] For instance based on the predictor-corrector schemeDiethelm et al introduced Adams-Bashforth-Moulton algo-rithm [11ndash13] and meanwhile some error analysis and anextension of Richardson extrapolation were presented toimprove the numerical accuracyThese techniques elaboratedin [11ndash13] take advantage of the fact that the FDEs can bereduced to Volterra type integral equations And thereforeone can apply the numerical schemes for Volterra typeintegral equations to find the solution of FDEs one can referto [21 25ndash28] and the literature cited therein for more detailsThe technique presented in [11ndash13] has been further analyzedand extended for multiterm [15] and multiorder systems [17]Li and Tao [20] studied the error analyses of the fractionalAdams method for the fractional-order ordinary differentialequations for more general cases Deng [24] obtained a goodnumerical approximation by combining the short memoryprinciple with the predictor-corrector approach

However in real world systems delay is very oftenencountered in many practical systems such as automaticcontrol biology and hydraulic networks economics andlong transmission lines Delayed differential equations arecorrespondingly used to describe such dynamical systemsIn recent years delayed FDEs begin to arouse the attention

2 Journal of Applied Mathematics

Table 1 Error behavior at time 119905 = 119879 with analytical value with 120572 = 02 120591 = 01

119879 119864 ℎ = 110 ℎ = 120 ℎ = 140 ℎ = 180 ℎ = 1160 ℎ = 1320

5 119864119860

08037 02881 01074 00411 00160 00062119864119877

00402 00144 00054 00021 7980119890 minus 4 3090119890 minus 4

10 119864119860

1659 05964 02234 00861 00338 00134119864119877

00184 00066 00025 9562119890 minus 4 3757119890 minus 4 1494119890 minus 5

50 119864119860

8161 2953 1114 04322 01714 00690119864119877

00033 00012 4545119890 minus 4 17643119890 minus 4 69940119890 minus 5 28171119890 minus 5

of many researchers [29ndash31] To simulate these equations isan important technique in the research accordingly findingeffective numerical methods for the delayed FDEs is anecessary process However to the best of our knowledgethere are very few works devoted in the literature to suchwork so farThe aim of this paper is to generalize the Adams-Bashforth-Moulton algorithm introduced in [11ndash13] to thedelayed FDEs

The rest of this paper is organized as follows In Section 2the numerical scheme is devised based on the Adams-Bashforth-Moulton method and meanwhile the predictor-corrector scheme is developed in the case of constant andtime-varying delay respectively The detailed error analysisof the newly proposed algorithm is given in Section 3Numerical simulations are performed in Section 4 in orderto illustrate the effectiveness of the proposed scheme Finallysome concluding remarks are reported in Section 5

2 Formulation of the Numerical Method forDelayed FDEs

There are three commonly used definitions for fractionalderivatives [2] that is Riemann-Liouville Grunwald-Letnicov and Caputo definitions In pure mathematicsRiemann-Liouville derivative is more commonly used thanCaputo derivative However as we all know only the Caputodefinition has the same form as integer-order differentialequations in the initial conditions And therefore this paperis based on Caputo definition

21 Caputo Fractional Derivatives

Definition 1 (see [2]) The Caputo fractional derivative oforder 120572 of a continuous function 119891 119877

+rarr 119877 is defined

as follows

119863120572

119905119910 (119905) =

1

Γ (119898 minus 120572)int

119905

0

119891(119898)

(120591)

(119905 minus 120591)120572minus119898+1

119889120591 119898 minus 1 lt 120572 lt 119898

119889119898

119889119905119898119891 (119905) 120572 = 119898

(1)

where Γ is Γ-function and

Γ (119911) = int

infin

0

119890minus119905119905119911minus1

119889119905 Γ (119911 + 1) = 119911Γ (119911) (2)

22 The Procedure of Adams-Bashforth-Moulton AlgorithmThemain aim of this paper is to study a numerical scheme forthe approximate solution of delayed FDEs For this purposewe consider delayed FDEs described by

119863120572

119905119910 (119905) = 119891 (119905 119910 (119905) 119910 (119905 minus 120591)) 119905 ge 0 119898 minus 1 lt 120572 le 119898

(3)

119910 (119905) = 120593 (119905) 119905 le 0 (4)

where 120572 is the order of the differential equation 120593(119905) is theinitial value and 119898 is an integer

Before introducing the new algorithm for the delayedFDEs we first briefly recall the idea of the classical one-stepAdams-Bashforth-Moulton algorithm for the followingODE

119889119910 (119905)

119889119905= 119891 (119905 119910 (119905)) 119910 (0) = 119910

0 (5)

Let 119905 isin [0 119879] 119905119895

= 119895ℎ 119895 = 0 1 119873 and ℎ = 119879119873 bethe time step Assume that 119910

119895= 119910(119905119895) 119895 = 1 119899 then the

previous equation is equivalent to

119910 (119905119899+1

) = 119910 (119905119899) + int

119905119899+1

119905119899

119891 (119904 119910 (119904)) 119889119904 (6)

Since the integral in the right-hand side of (6) can beapproximated by trapezoidal quadrature formula

int

119905119899+1

119905119899

119891 (119904 119910 (119904)) 119889119904 asympℎ

2(119891 (119905119899 119910 (119905119899)) + 119891 (119905

119899+1 119910 (119905119899+1

)))

(7)

Thus we can get the approximation to 119910119899+1

as follows

119910119899+1

= 119910119899+

ℎ

2[119891 (119905119899 119910119899) + 119891 (119905

119899+1 119910119899+1

)] (8)

which is an implicit equation and cannot be solved for 119910119899+1

directly however one can adopt another numerical methodto approximate 119910

119899+1in the right-hand side preliminarily

The preliminary approximation 119910119901

119899+1is called predictor

and can be obtained by forward Euler method or any otherexplicit method here we take forward Euler method as anexample that is

119910119901

119899+1= 119910119899+ ℎ119891 (119905

119899 119910119899) (9)

Then the so-called one-step Adams-Bashforth-Moultonmethod is

119910119899+1

= 119910119899+

ℎ

2[119891 (119905119899 119910119899) + 119891 (119905

119899+1 119910119901

119899+1)] (10)

Journal of Applied Mathematics 3

Table 2 Error behavior at time 119905 = 119879 with analytical value with 120572 = 09 120591 = 01

119879 119864 ℎ = 110 ℎ = 120 ℎ = 140 ℎ = 180 ℎ = 1160 ℎ = 1320

5 119864119860

00534 00163 00038 minus75742119890 minus 4 minus00026 minus00033119864119877

00027 81293119890 minus 4 19072119890 minus 4 minus37871119890 minus 5 minus12853119890 minus 4 minus16663119890 minus 4

10 119864119860

00865 00247 00061 minus51223119890 minus 5 minus00022 minus00031119864119877

96095119890 minus 4 27404119890 minus 4 67297119890 minus 5 minus56910119890 minus 7 minus24824119890 minus 5 minus34191119890 minus 5

50 119864119860

03610 00955 00248 00051 minus53786119890 minus 4 minus00023119864119877

14736119890 minus 4 38962119890 minus 5 10103119890 minus 5 20997119890 minus 6 minus71954119890 minus 7 minus92911119890 minus 7

Table 3 Error behavior at time 119905 = 119879 with analytical value with 120572 = 02 120591 = 001 exp(minus119905)

119879 119864 ℎ = 110 ℎ = 120 ℎ = 140 ℎ = 180 ℎ = 1160 ℎ = 1320

5 119864119860

00425 00218 00133 00097 00081 00074119864119877

00021 00011 66680119890 minus 4 48592119890 minus 4 40585119890 minus 4 36960119890 minus 4

10 119864119860

00411 00217 00138 00104 00089 00082119864119877

45679119890 minus 4 24130119890 minus 4 15319119890 minus 4 11553119890 minus 4 98859119890 minus 5 91315119890 minus 5

50 119864119860

minus00351 00184 00116 00087 00074 00052119864119877

14317119890 minus 5 75036119890 minus 6 47294119890 minus 6 35467119890 minus 6 30242119890 minus 6 24322119890 minus 6

in which (9) is called the predictor and (10) is called thecorrector It is well known that thismethod is convergent thatis

max119895=01119873

10038161003816100381610038161003816119910 (119905119895) minus 119910119895

10038161003816100381610038161003816= 119874 (ℎ

2) (11)

23 The Representation of Algorithm for Delayed FDEs Hav-ing introduced the Adams-Bashforth-Moulton algorithmnow the key problem is to establish approximation to thedelayed term 119910(119905 minus 120591) which consists of the following twocases

Case I (when 120591 is constant) It can be seen that for anypositive constant 120591 119905

119895minus 120591 may not be a grid point 119905

119899for any

119899 Suppose that (119898 minus 120575)ℎ = 120591 and 0 le 120575 lt 1 When 120575 = 0119910(119905119899minus 120591) can be approximated by

119910 (119905119899minus 120591) asymp

119910119899minus119898

if 119899 gt 119898

120593119899 if 119899 le 119898

(12)

and when 0 lt 120575 lt 1 119910(119905119899minus 120591) cannot be calculated directly

For (119898 minus 1)ℎ lt 120591 lt 119898ℎ let V119899+1

be the approximation to119910(119905119899+1

minus120591) we interpolate it by the two nearest points that is

V119899+1

= 120575119910119899minus119898+2

+ (1 minus 120575) 119910119899minus119898+1 (13)

It can be seen from (13) that if119898 gt 1 the numerical equationis explicit and thus can be computed directly However when119898 = 1 and 120575 = 0 that is 120591 lt ℎ then the first term in the right-hand side of the previous equation is 120575119910

119899+1 It still needs to

predict in this case that is

V119899+1

= 120575119910119901

119899+1+ (1 minus 120575) 119910119899 (14)

Case II (when 120591 is time varying) If 120591 is time varying thatis 120591 = 120591(119905) then the approximation in this case is morecomplicated Let V

119899+1be the approximation to 119910(119905

119899+1minus 120591)

and the linear interpolation of 119910119895 at point 119905 = 119905

119899+1minus 120591(119905119899+1

)

is employed to approximate the delay term Let 120591(119905119899+1

) =

(119898119899+1

minus 120575119899+1

)ℎ where 119898119899+1

is a positive integer and 120575119899+1

isin

[0 1) then

V119899+1

= 120575119899+1

119910119899minus119898119899+1+2

+ (1 minus 120575119899+1

) 119910119899minus119898119899+1+1

(15)

It should be noted that when 120591 is constant for given ℎ and 120591we can judge whether 119898 = 1 or 119898 gt 1 holds at the beginningof the programme However when 120591 is time varying 119898 isalso time varying that is at one moment it is equal to 1 andat another moment it may greater than 1 If 119898

119899+1= 1 the

first term in the right-hand side of (15) still needs to predictotherwise when119898

119899+1gt 1 it needs not So in each step of the

computing it needs to test whether119898119899= 1 or not firstly then

computation of the first term in the right-hand side of (15)depends on whether it requires predicting or not in differentsteps

Now we begin the numerical algorithm for the delayedFDEs (3)-(4)

From the theory introduced in [10ndash14] we know that thedelayed initial value problem (3)-(4) is equivalent to Volterraintegral equation as follows

119910 (119905) =

119898minus1

sum

119896=0

120593 (119905)119905119896

119896+

1

Γ (120572)int

119905

0

119891 (119904 119910 (119904) 119910 (119904 minus 120591))

(119905 minus 119904)1minus120572

119889119904 (16)

Just as the method used in [13] the remaining problem isto compute the integral term in (16) Firstly the producttrapezoidal quadrature formula is applied to replace theintegrals term where nodes 119905

119895(119895 = 0 1 119899 + 1) are taken

with respect to the weight function (119905119899+1

minus sdot)120572minus1 That is we

employ the approximation

int

119905119899+1

0

(119905119896+1

minus 119911)120572minus1

119892 (119911) 119889119911 asymp int

119905119899+1

0

(119905119899+1

minus 119911)120572minus1

119892119899+1 (119911) 119889119911

(17)

4 Journal of Applied Mathematics

Table 4 Error behavior at time 119905 = 119879 with analytical value with 120572 = 09 120591 = 001 exp(minus119905)

119879 119864 ℎ = 110 ℎ = 120 ℎ = 140 ℎ = 180 ℎ = 1160 ℎ = 1320

5 119864119860

00031 00016 00012 00011 00011 00010119864119877

15375119890 minus 4 81279119890 minus 5 61399119890 minus 5 55480119890 minus 5 53282119890 minus 5 52136119890 minus 5

10 119864119860

00022 91466119890 minus 4 58474119890 minus 4 49919119890 minus 4 47680119890 minus 4 47115119890 minus 4

119864119877

24189119890 minus 5 10163119890 minus 5 64971119890 minus 6 55465119890 minus 6 52300119890 minus 6 52350119890 minus 6

50 119864119860

00016 49082119890 minus 4 19954119890 minus 4 12510119890 minus 4 10616119890 minus 4 70303119890 minus 4

119864119877

66311119890 minus 7 20033119890 minus 7 81447119890 minus 8 51062119890 minus 8 43330119890 minus 8 27081119890 minus 8

where 119892119899+1

(sdot) is the piecewise linear interpolation for 119892(sdot)

with nodes and knots chosen at 119905119895 119895 = 0 1 119899 + 1 Using

the standard technique of quadrature theory it can be foundthat we can write the integrals on the right-hand side of (17)as

int

119905119899+1

0

(119905119899+1

minus 119911)120572minus1

119892119899+1 (119911) 119889119911 =

119899+1

sum

119895=0

119886119895119899+1

119892 (119905119895) (18)

where119886119895119899+1

=ℎ120572

120572 (120572 + 1)

times

119899120572+1

minus(119899 minus 120572) (119899 + 1)120572 119895 = 0

(119899 minus 119895+2)120572+1

+(119899minus119895)120572+1

minus2(119899minus119895+1)120572+1

1 le 119895 le 119899

1 119895 = 119899 + 1

(19)

Consequently the numerical scheme for FDEs (3)-(4) can bedepicted as

119910119899+1

=

119898minus1

sum

119896=0

120593119896

119905119896

119899+1

119896+

ℎ120572

Γ (120572 + 2)119891 (119905119899+1

119910119901

119899+1 V119899+1

)

+1

Γ (120572)

119899

sum

119895=0

119886119895119899+1

119891 (119905119895 119910119895 V119895)

V119899+1

= 120575119910119899minus119898+2

+ (1 minus 120575) 119910119899minus119898+1 if 119898 gt 1

120575119910119901

119899+1+ (1 minus 120575) 119910119899 if 119898 = 1

119910119901

119899+1=

119898minus1

sum

119896=0

120593119896

119905119896

119899+1

119896+

1

Γ (120572)

119899

sum

119895=0

119887119895119899+1

119891 (119905119895 119910119895 V119895)

(20)

where

119887119895119899+1

=ℎ120572

120572((119899 minus 119895 + 1)

120572minus (119899 minus 119895)

120572) (21)

3 Error Analysis ofthe Newly-Proposed Algorithm

Before giving the main results we firstly introduce thefollowing two lemmas which will be used in the proof of thetheorem

Lemma 2 (see [13]) (i) Let 119911 isin 1198621[0 119879] then

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119911 (119905) 119889119905 minus

119899

sum

119895=0

119887119895119899+1

119911 (119905119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

120572

10038171003817100381710038171003817119911101584010038171003817100381710038171003817infin

119905120572

119899+1ℎ

(22)

(ii) Let 119911(119905) isin 1198622[0 119879] then

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119911 (119905) 119889119905 minus

119899+1

sum

119895=0

119886119895119899+1

119911 (119905119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862119879119903

120572

100381710038171003817100381710038171199111015840101584010038171003817100381710038171003817infin

119905120572

119899+1ℎ2

(23)

In this paper we further assume that 119891(sdot) in (3) satisfiesthe following Lipschitz conditionswith respect to its variablesas follows

1003816100381610038161003816119891 (119905 119910

1 119906) minus 119891 (119905 119910

2 119906)

1003816100381610038161003816le 1198711

10038161003816100381610038161199101minus 1199102

1003816100381610038161003816

1003816100381610038161003816119891 (119905 119910 119906

1) minus 119891 (119905 119910 119906

2)1003816100381610038161003816le 1198712

10038161003816100381610038161199061minus 1199062

1003816100381610038161003816

(24)

where 1198711 1198712are positive constants

Theorem 3 Suppose the solution 119910(119905) isin 1198622[0 119879] of the initial

value problem satisfies the following two conditions

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119863120572

119905119910 (119905) 119889119905 minus

119899

sum

119895=0

119887119895119896+1

119863120572

119905119910 (119905119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 1198621199051205741

119899+1ℎ1205751

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119863120572

119905119910 (119905) 119889119905 minus

119899

sum

119895=0

119886119895119896+1

119863120572

119905119910 (119905119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 1198621199051205742

119899+1ℎ1205752

(25)

with some 1205741 1205742

ge 0 and 1205751 1205752

gt 0 then for some suitable119879 gt 0 we have

max0le119869le119873

10038161003816100381610038161003816119910 (119905119895) minus 119910119895

10038161003816100381610038161003816le 119862ℎ119902 (26)

where 119902 = min1205751+ 120572 120575

2 119873 = [119879ℎ] and 119862 is a positive

constant

Journal of Applied Mathematics 5

Proof We will prove the result by virtue of mathematicalinduction Suppose that the conclusion is true for 119895 =

0 1 119899 thenwe have to prove that the inequality also holdsfor 119895 = 119899 + 1 To do this we first consider the error of thepredictor 119910

119901

119899+1

From the assumptions (24) we have

10038161003816100381610038161003816119891 (119905119895 119910 (119905119895) 119910 (119905

119895minus 120591)) minus 119891 (119905

119895 119910119895 V119895)10038161003816100381610038161003816

le10038161003816100381610038161003816119891 (119905119895 119910 (119905119895) 119910 (119905

119895minus 120591)) minus 119891 (119905

119895 119910119895 119910 (119905119895minus 120591))

10038161003816100381610038161003816

+10038161003816100381610038161003816119891 (119905119895 119910119895 119910 (119905119895minus 120591)) minus 119891 (119905

119895 119910119895 V119895)10038161003816100381610038161003816

le 1198711ℎ119902+ 1198712ℎ119902= (1198711+ 1198712) ℎ119902

(27)

So

10038161003816100381610038161003816119910 (119905119899+1

) minus 119910119901

119899+1

10038161003816100381610038161003816

=1

Γ (120572)

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119891 (119905 119910 (119905) 119910 (119905 minus 120591)) 119889119905

minus

119899

sum

119895=0

119887119895119896+1

119891 (119905119895 119910119895 V119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

Γ (120572)

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119863120572

119905119910 (119905) 119889119905 minus

119899

sum

119895=0

119887119895119899+1

119863120572

119905119910 (119905119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

+1

Γ (120572)

times

119899

sum

119895=0

119887119895119899+1

10038161003816100381610038161003816119891 (119905119895 119910 (119905119895) 119910 (119905

119895minus 120591))

minus119891 (119905119895 119910119895 119910 (119905119895minus 120591))

10038161003816100381610038161003816

+10038161003816100381610038161003816119891 (119905119895 119910119895 119910 (119905119895minus 120591)) minus 119891 (119905

119895 119910119895 V119895)10038161003816100381610038161003816

le1198621199051205741

119899+1

Γ (120572)ℎ1205751

+1

Γ (120572)

119899

sum

119895=0

119887119895119899+1

(1198711+ 1198712) 119862ℎ119902

(28)

and from

119899

sum

119895=0

119887119895119899+1

=

119899

sum

119895=0

int

(119895+1)ℎ

119895ℎ

(119905119899+1

minus 119905)120572minus1

119889119905 =1

120572119905120572

119899+1le

1

120572119879120572

(29)

we have

10038161003816100381610038161003816119910 (119905119899+1

) minus 119910119901

119899+1

10038161003816100381610038161003816le

1198621198791205741

Γ (120572)ℎ1205751

+119862119879120572

Γ (120572 + 1)ℎ119902 (30)

Since119899

sum

119895=0

119886119895119899+1

le

119899

sum

119895=0

ℎ120572

120572 (120572 + 1)

times [(119896 minus 119895 + 2)120572+1

minus (119896 minus 119895 + 1)120572+1

]

minus [(119896 minus 119895 + 1)120572+1

minus (119896 minus 119895)120572+1

]

= [int

119905119899+1

0

(119905119899+2

minus 119905)120572119889119905 minus int

119905119899+1

0

(119905119899+1

minus 119905)120572119889119905]

=1

120572int

119905119899+1

0

[(119905119899+1

minus 119905)120572]1015840

(119905) 119889119905 =1

120572119905120572

119899+1le 119879120572

(31)

we have1003816100381610038161003816119910 (119905119899+1

) minus 119910119899+1

1003816100381610038161003816

=1

Γ (120572)

10038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119891 (119905 119910 (119905)

119910 (119905 minus 120591)) 119889119905 minus

119899

sum

119895=0

119886119895119899+1

119891 (119905119895 119910119895 V119895)

minus 119886119899+1119899+1

119891 (119905119899+1

119910119901

119899+1 V119899+1

)

10038161003816100381610038161003816100381610038161003816

le1

Γ (120572)

10038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119891 (119905 119910 (119905) 119910 (119905 minus 120591))

minus

119899+1

sum

119895=0

119886119895119899+1

119891 (119905119895 119910 (119905119895) 119910 (119905

119895minus 120591))

10038161003816100381610038161003816100381610038161003816

+

119899

sum

119895=0

119886119895119899+1

10038161003816100381610038161003816119891 (119905119895 119910 (119905119895) 119910 (119905

119895minus120591)) minus 119891 (119905

119895 119910119895 V119895)10038161003816100381610038161003816

+ 119886119899+1119899+1

10038161003816100381610038161003816119891 (119905119899+1

119910 (119905119899+1

) 119910 (119905119899+1

minus 120591))

minus 119891 (119905119899+1

119910119901

119899+1 V119899+1

)10038161003816100381610038161003816

le1

Γ (120572)

[

[

1198621199051205742

119899+1ℎ1205752

+ 119862ℎ119902

119899

sum

119895=0

119886119895119899+1

+ 119886119899+1119899+1

119871(ℎ119902+

1198621198791205741

Γ (120572)ℎ1205751

+119862119879120572

Γ (120572 + 1)ℎ119902)]

]

le1

Γ (120572)[1198621199051205742

119899+1ℎ1205752

+119871ℎ119902+ℎ

120572 (120572 + 1)

+1198621198711198791205741

Γ (120572 + 2)+

119862119879120572ℎ120572+119902

120572Γ (120572 + 2)] = 119862ℎ

119902

(32)

which completes the proof

When 1205751

= 1 1205752

= 2 applying Theorem 3 it is notdifficult to derive the following result on the error bound

6 Journal of Applied Mathematics

Theorem 4 Suppose that 120572 gt 0 and 119891(119905 119910(119905) 119911(119905)) isin

1198621015840[0 119879] then

max0le119895le119873

10038161003816100381610038161003816119910 (119905119895) minus 119910119895

10038161003816100381610038161003816le 119862ℎ119902 (33)

where 119902 = min 1 + 120572 2 and 119862 is a positive constant

Remark 5 We can see from Cases I and II that if the solution119910(119905) is not smooth then the truncation error of V

119899+1 which

approximates to 119910(119905 minus 120591) will not be 119874(ℎ2) Therefore by the

error analysis elaborated in Section 3 we cannot yield thesame conclusion as in Theorem 4 We will consider such aproblem in our future work

4 A Numerical Example

In this section the following delayed FDE is considered

119863120572

119905119910 (119905) =

2

Γ (3 minus 120572)1199052minus120572

minus1

Γ (2 minus 120572)1199051minus120572

+ 2120591119905 minus 1205912

minus 120591 minus 119910 (119905) + 119910 (119905 minus 120591) 120572 isin (0 1)

119910 (119905) = 0 119905 le 0

(34)

Notice that the exact solution to this equation is

119910 (119905) = 1199052minus 119905 (35)

In accordance with delay 120591 being constant or time varyingthe numerical results are displayed in Tables 1 2 3 and4 respectively where 119864

119860denotes the absolute numerical

error and 119864119877denotes the relative numerical error From the

numerical results we can see that the computing errors are ingeneral acceptable for engineering

5 Conclusions

In this paper a numerical algorithm is formulated basedon Adams-Bashforth-Moulton method for fractional-orderdifferential equations with time delay The error analysisof the numerical scheme is carried out meanwhile anumerical example with constant delay and time-varyingdelay is proposed to testify the effectiveness of the schemeThis algorithm can be used not only in the simulation ofdelayed fractional-order differential equations but also in thesimulation of delayed fractional-order control systems

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China under Grants 61004078 61273012 theFoundation of State Key Laboratory for Novel SoftwareTechnology of Nanjing University the China PostdoctoralScience Foundation Funded Project and the Special Fundsfor Postdoctoral Innovative Projects of Shandong Province

References

[1] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993

[2] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[3] P L Butzer and U Westphal ldquoAn introduction to fractionalcalculusrdquo in Applications of Fractional Calculus in Physics pp1ndash85 World Scientific Publishing River Edge NJ USA 2000

[4] Z Wang X Huang Y-X Li and X-N Song ldquoA new imageencryption algorithm based on fractional-order hyperchaoticLorenz systemrdquo Chinese Physics B vol 22 Article ID 0105042013

[5] K BOldhamand J SpanierTheFractional CalculusTheory andApplication of Differentiation and Integration to Arbitrary Ordervol 111 of Mathematics in Science and Engineering AcademicPress New York NY USA 1974

[6] R Hilfer Ed Applications of Fractional Calculus in PhysicsWorld Scientific Publishing River Edge NJ USA 2000

[7] P J Torvik andR L Bagley ldquoOn the appearance of the fractionalderivatives in the behaviour of realmaterialsrdquo Journal of AppliedMechanics vol 51 no 2 pp 294ndash298 1984

[8] F Mainardi ldquoFractional relaxation-oscillation and fractionaldiffusion-wave phenomenardquo Chaos Solitons amp Fractals vol 7no 9 pp 1461ndash1477 1996

[9] N J Ford and A C Simpson ldquoThe numerical solution of frac-tional differential equations speed versus accuracyrdquo NumericalAlgorithms vol 26 no 4 pp 333ndash346 2001

[10] V Daftardar-Gejji and A Babakhani ldquoAnalysis of a systemof fractional differential equationsrdquo Journal of MathematicalAnalysis and Applications vol 293 no 2 pp 511ndash522 2004

[11] K Diethelm and N J Ford ldquoAnalysis of fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 265 no 2 pp 229ndash248 2002

[12] 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

[13] KDiethelmN J Ford andAD Freed ldquoDetailed error analysisfor a fractional Adams methodrdquo Numerical Algorithms vol 36no 1 pp 31ndash52 2004

[14] K Diethelm N J Ford A D Freed and Yu Luchko ldquoAlgo-rithms for the fractional calculus a selection of numericalmethodsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 194 no 6-8 pp 743ndash773 2005

[15] K Diethelm and N J Ford ldquoMulti-order fractional differentialequations and their numerical solutionrdquo Applied Mathematicsand Computation vol 154 no 3 pp 621ndash640 2004

[16] Z Odibat ldquoApproximations of fractional integrals and Caputofractional derivativesrdquo Applied Mathematics and Computationvol 178 no 2 pp 527ndash533 2006

[17] J T Edwards N J Ford and A C Simpson ldquoThe numericalsolution of linear multi-term fractional differential equationssystems of equationsrdquo Journal of Computational and AppliedMathematics vol 148 no 2 pp 401ndash418 2002

[18] K Diethelm ldquoAn investigation of some nonclassical methodsfor the numerical approximation of Caputo-type fractionalderivativesrdquo Numerical Algorithms vol 47 no 4 pp 361ndash3902008

Journal of Applied Mathematics 7

[19] K Diethelm ldquoAn improvement of a nonclassical numericalmethod for the computation of fractional derivativesrdquo Journalof Vibration and Acoustics vol 131 no 1 Article ID 014502 4pages 2009

[20] C Li and C Tao ldquoOn the fractional AdamsmethodrdquoComputersamp Mathematics with Applications vol 58 no 8 pp 1573ndash15882009

[21] R Garrappa ldquoOn some explicit Adams multistep methods forfractional differential equationsrdquo Journal of Computational andApplied Mathematics vol 229 no 2 pp 392ndash399 2009

[22] L Galeone and R Garrappa ldquoFractional Adams-Moultonmethodsrdquo Mathematics and Computers in Simulation vol 79no 4 pp 1358ndash1367 2008

[23] L Galeone and R Garrappa ldquoExplicit methods for fractionaldifferential equations and their stability propertiesrdquo Journal ofComputational and Applied Mathematics vol 228 no 2 pp548ndash560 2009

[24] W Deng ldquoShort memory principle and a predictor-correctorapproach for fractional differential equationsrdquo Journal of Com-putational and AppliedMathematics vol 206 no 1 pp 174ndash1882007

[25] C Li and G Peng ldquoChaos in Chenrsquos system with a fractionalorderrdquo Chaos Solitons amp Fractals vol 22 no 2 pp 443ndash4502004

[26] XWu J Li and G Chen ldquoChaos in the fractional order unifiedsystem and its synchronizationrdquo Journal of the Franklin Institutevol 345 no 4 pp 392ndash401 2008

[27] M S Tavazoei andM Haeri ldquoChaotic attractors in incommen-surate fractional order systemsrdquo Physica D vol 237 no 20 pp2628ndash2637 2008

[28] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A vol 389 no12 pp 2434ndash2442 2010

[29] Y Chen and K L Moore ldquoAnalytical stability bound for aclass of delayed fractional-order dynamic systemsrdquo NonlinearDynamics vol 29 no 1ndash4 pp 191ndash200 2002

[30] P Lanusse H Benlaoukli D Nelson-Gruel and A OustaloupldquoFractional-order control and interval analysis of SISO systemswith time-delayed staterdquo IETControlTheoryampApplications vol2 no 1 pp 16ndash23 2008

[31] X Zhang ldquoSome results of linear fractional order time-delaysystemrdquo Applied Mathematics and Computation vol 197 no 1pp 407ndash411 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

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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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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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

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

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

Table 1 Error behavior at time 119905 = 119879 with analytical value with 120572 = 02 120591 = 01

119879 119864 ℎ = 110 ℎ = 120 ℎ = 140 ℎ = 180 ℎ = 1160 ℎ = 1320

5 119864119860

08037 02881 01074 00411 00160 00062119864119877

00402 00144 00054 00021 7980119890 minus 4 3090119890 minus 4

10 119864119860

1659 05964 02234 00861 00338 00134119864119877

00184 00066 00025 9562119890 minus 4 3757119890 minus 4 1494119890 minus 5

50 119864119860

8161 2953 1114 04322 01714 00690119864119877

00033 00012 4545119890 minus 4 17643119890 minus 4 69940119890 minus 5 28171119890 minus 5

of many researchers [29ndash31] To simulate these equations isan important technique in the research accordingly findingeffective numerical methods for the delayed FDEs is anecessary process However to the best of our knowledgethere are very few works devoted in the literature to suchwork so farThe aim of this paper is to generalize the Adams-Bashforth-Moulton algorithm introduced in [11ndash13] to thedelayed FDEs

The rest of this paper is organized as follows In Section 2the numerical scheme is devised based on the Adams-Bashforth-Moulton method and meanwhile the predictor-corrector scheme is developed in the case of constant andtime-varying delay respectively The detailed error analysisof the newly proposed algorithm is given in Section 3Numerical simulations are performed in Section 4 in orderto illustrate the effectiveness of the proposed scheme Finallysome concluding remarks are reported in Section 5

2 Formulation of the Numerical Method forDelayed FDEs

There are three commonly used definitions for fractionalderivatives [2] that is Riemann-Liouville Grunwald-Letnicov and Caputo definitions In pure mathematicsRiemann-Liouville derivative is more commonly used thanCaputo derivative However as we all know only the Caputodefinition has the same form as integer-order differentialequations in the initial conditions And therefore this paperis based on Caputo definition

21 Caputo Fractional Derivatives

Definition 1 (see [2]) The Caputo fractional derivative oforder 120572 of a continuous function 119891 119877

+rarr 119877 is defined

as follows

119863120572

119905119910 (119905) =

1

Γ (119898 minus 120572)int

119905

0

119891(119898)

(120591)

(119905 minus 120591)120572minus119898+1

119889120591 119898 minus 1 lt 120572 lt 119898

119889119898

119889119905119898119891 (119905) 120572 = 119898

(1)

where Γ is Γ-function and

Γ (119911) = int

infin

0

119890minus119905119905119911minus1

119889119905 Γ (119911 + 1) = 119911Γ (119911) (2)

22 The Procedure of Adams-Bashforth-Moulton AlgorithmThemain aim of this paper is to study a numerical scheme forthe approximate solution of delayed FDEs For this purposewe consider delayed FDEs described by

119863120572

119905119910 (119905) = 119891 (119905 119910 (119905) 119910 (119905 minus 120591)) 119905 ge 0 119898 minus 1 lt 120572 le 119898

(3)

119910 (119905) = 120593 (119905) 119905 le 0 (4)

where 120572 is the order of the differential equation 120593(119905) is theinitial value and 119898 is an integer

Before introducing the new algorithm for the delayedFDEs we first briefly recall the idea of the classical one-stepAdams-Bashforth-Moulton algorithm for the followingODE

119889119910 (119905)

119889119905= 119891 (119905 119910 (119905)) 119910 (0) = 119910

0 (5)

Let 119905 isin [0 119879] 119905119895

= 119895ℎ 119895 = 0 1 119873 and ℎ = 119879119873 bethe time step Assume that 119910

119895= 119910(119905119895) 119895 = 1 119899 then the

previous equation is equivalent to

119910 (119905119899+1

) = 119910 (119905119899) + int

119905119899+1

119905119899

119891 (119904 119910 (119904)) 119889119904 (6)

Since the integral in the right-hand side of (6) can beapproximated by trapezoidal quadrature formula

int

119905119899+1

119905119899

119891 (119904 119910 (119904)) 119889119904 asympℎ

2(119891 (119905119899 119910 (119905119899)) + 119891 (119905

119899+1 119910 (119905119899+1

)))

(7)

Thus we can get the approximation to 119910119899+1

as follows

119910119899+1

= 119910119899+

ℎ

2[119891 (119905119899 119910119899) + 119891 (119905

119899+1 119910119899+1

)] (8)

which is an implicit equation and cannot be solved for 119910119899+1

directly however one can adopt another numerical methodto approximate 119910

119899+1in the right-hand side preliminarily

The preliminary approximation 119910119901

119899+1is called predictor

and can be obtained by forward Euler method or any otherexplicit method here we take forward Euler method as anexample that is

119910119901

119899+1= 119910119899+ ℎ119891 (119905

119899 119910119899) (9)

Then the so-called one-step Adams-Bashforth-Moultonmethod is

119910119899+1

= 119910119899+

ℎ

2[119891 (119905119899 119910119899) + 119891 (119905

119899+1 119910119901

119899+1)] (10)

Journal of Applied Mathematics 3

Table 2 Error behavior at time 119905 = 119879 with analytical value with 120572 = 09 120591 = 01

119879 119864 ℎ = 110 ℎ = 120 ℎ = 140 ℎ = 180 ℎ = 1160 ℎ = 1320

5 119864119860

00534 00163 00038 minus75742119890 minus 4 minus00026 minus00033119864119877

00027 81293119890 minus 4 19072119890 minus 4 minus37871119890 minus 5 minus12853119890 minus 4 minus16663119890 minus 4

10 119864119860

00865 00247 00061 minus51223119890 minus 5 minus00022 minus00031119864119877

96095119890 minus 4 27404119890 minus 4 67297119890 minus 5 minus56910119890 minus 7 minus24824119890 minus 5 minus34191119890 minus 5

50 119864119860

03610 00955 00248 00051 minus53786119890 minus 4 minus00023119864119877

14736119890 minus 4 38962119890 minus 5 10103119890 minus 5 20997119890 minus 6 minus71954119890 minus 7 minus92911119890 minus 7

Table 3 Error behavior at time 119905 = 119879 with analytical value with 120572 = 02 120591 = 001 exp(minus119905)

119879 119864 ℎ = 110 ℎ = 120 ℎ = 140 ℎ = 180 ℎ = 1160 ℎ = 1320

5 119864119860

00425 00218 00133 00097 00081 00074119864119877

00021 00011 66680119890 minus 4 48592119890 minus 4 40585119890 minus 4 36960119890 minus 4

10 119864119860

00411 00217 00138 00104 00089 00082119864119877

45679119890 minus 4 24130119890 minus 4 15319119890 minus 4 11553119890 minus 4 98859119890 minus 5 91315119890 minus 5

50 119864119860

minus00351 00184 00116 00087 00074 00052119864119877

14317119890 minus 5 75036119890 minus 6 47294119890 minus 6 35467119890 minus 6 30242119890 minus 6 24322119890 minus 6

in which (9) is called the predictor and (10) is called thecorrector It is well known that thismethod is convergent thatis

max119895=01119873

10038161003816100381610038161003816119910 (119905119895) minus 119910119895

10038161003816100381610038161003816= 119874 (ℎ

2) (11)

23 The Representation of Algorithm for Delayed FDEs Hav-ing introduced the Adams-Bashforth-Moulton algorithmnow the key problem is to establish approximation to thedelayed term 119910(119905 minus 120591) which consists of the following twocases

Case I (when 120591 is constant) It can be seen that for anypositive constant 120591 119905

119895minus 120591 may not be a grid point 119905

119899for any

119899 Suppose that (119898 minus 120575)ℎ = 120591 and 0 le 120575 lt 1 When 120575 = 0119910(119905119899minus 120591) can be approximated by

119910 (119905119899minus 120591) asymp

119910119899minus119898

if 119899 gt 119898

120593119899 if 119899 le 119898

(12)

and when 0 lt 120575 lt 1 119910(119905119899minus 120591) cannot be calculated directly

For (119898 minus 1)ℎ lt 120591 lt 119898ℎ let V119899+1

be the approximation to119910(119905119899+1

minus120591) we interpolate it by the two nearest points that is

V119899+1

= 120575119910119899minus119898+2

+ (1 minus 120575) 119910119899minus119898+1 (13)

It can be seen from (13) that if119898 gt 1 the numerical equationis explicit and thus can be computed directly However when119898 = 1 and 120575 = 0 that is 120591 lt ℎ then the first term in the right-hand side of the previous equation is 120575119910

119899+1 It still needs to

predict in this case that is

V119899+1

= 120575119910119901

119899+1+ (1 minus 120575) 119910119899 (14)

Case II (when 120591 is time varying) If 120591 is time varying thatis 120591 = 120591(119905) then the approximation in this case is morecomplicated Let V

119899+1be the approximation to 119910(119905

119899+1minus 120591)

and the linear interpolation of 119910119895 at point 119905 = 119905

119899+1minus 120591(119905119899+1

)

is employed to approximate the delay term Let 120591(119905119899+1

) =

(119898119899+1

minus 120575119899+1

)ℎ where 119898119899+1

is a positive integer and 120575119899+1

isin

[0 1) then

V119899+1

= 120575119899+1

119910119899minus119898119899+1+2

+ (1 minus 120575119899+1

) 119910119899minus119898119899+1+1

(15)

It should be noted that when 120591 is constant for given ℎ and 120591we can judge whether 119898 = 1 or 119898 gt 1 holds at the beginningof the programme However when 120591 is time varying 119898 isalso time varying that is at one moment it is equal to 1 andat another moment it may greater than 1 If 119898

119899+1= 1 the

first term in the right-hand side of (15) still needs to predictotherwise when119898

119899+1gt 1 it needs not So in each step of the

computing it needs to test whether119898119899= 1 or not firstly then

computation of the first term in the right-hand side of (15)depends on whether it requires predicting or not in differentsteps

Now we begin the numerical algorithm for the delayedFDEs (3)-(4)

From the theory introduced in [10ndash14] we know that thedelayed initial value problem (3)-(4) is equivalent to Volterraintegral equation as follows

119910 (119905) =

119898minus1

sum

119896=0

120593 (119905)119905119896

119896+

1

Γ (120572)int

119905

0

119891 (119904 119910 (119904) 119910 (119904 minus 120591))

(119905 minus 119904)1minus120572

119889119904 (16)

Just as the method used in [13] the remaining problem isto compute the integral term in (16) Firstly the producttrapezoidal quadrature formula is applied to replace theintegrals term where nodes 119905

119895(119895 = 0 1 119899 + 1) are taken

with respect to the weight function (119905119899+1

minus sdot)120572minus1 That is we

employ the approximation

int

119905119899+1

0

(119905119896+1

minus 119911)120572minus1

119892 (119911) 119889119911 asymp int

119905119899+1

0

(119905119899+1

minus 119911)120572minus1

119892119899+1 (119911) 119889119911

(17)

4 Journal of Applied Mathematics

Table 4 Error behavior at time 119905 = 119879 with analytical value with 120572 = 09 120591 = 001 exp(minus119905)

119879 119864 ℎ = 110 ℎ = 120 ℎ = 140 ℎ = 180 ℎ = 1160 ℎ = 1320

5 119864119860

00031 00016 00012 00011 00011 00010119864119877

15375119890 minus 4 81279119890 minus 5 61399119890 minus 5 55480119890 minus 5 53282119890 minus 5 52136119890 minus 5

10 119864119860

00022 91466119890 minus 4 58474119890 minus 4 49919119890 minus 4 47680119890 minus 4 47115119890 minus 4

119864119877

24189119890 minus 5 10163119890 minus 5 64971119890 minus 6 55465119890 minus 6 52300119890 minus 6 52350119890 minus 6

50 119864119860

00016 49082119890 minus 4 19954119890 minus 4 12510119890 minus 4 10616119890 minus 4 70303119890 minus 4

119864119877

66311119890 minus 7 20033119890 minus 7 81447119890 minus 8 51062119890 minus 8 43330119890 minus 8 27081119890 minus 8

where 119892119899+1

(sdot) is the piecewise linear interpolation for 119892(sdot)

with nodes and knots chosen at 119905119895 119895 = 0 1 119899 + 1 Using

the standard technique of quadrature theory it can be foundthat we can write the integrals on the right-hand side of (17)as

int

119905119899+1

0

(119905119899+1

minus 119911)120572minus1

119892119899+1 (119911) 119889119911 =

119899+1

sum

119895=0

119886119895119899+1

119892 (119905119895) (18)

where119886119895119899+1

=ℎ120572

120572 (120572 + 1)

times

119899120572+1

minus(119899 minus 120572) (119899 + 1)120572 119895 = 0

(119899 minus 119895+2)120572+1

+(119899minus119895)120572+1

minus2(119899minus119895+1)120572+1

1 le 119895 le 119899

1 119895 = 119899 + 1

(19)

Consequently the numerical scheme for FDEs (3)-(4) can bedepicted as

119910119899+1

=

119898minus1

sum

119896=0

120593119896

119905119896

119899+1

119896+

ℎ120572

Γ (120572 + 2)119891 (119905119899+1

119910119901

119899+1 V119899+1

)

+1

Γ (120572)

119899

sum

119895=0

119886119895119899+1

119891 (119905119895 119910119895 V119895)

V119899+1

= 120575119910119899minus119898+2

+ (1 minus 120575) 119910119899minus119898+1 if 119898 gt 1

120575119910119901

119899+1+ (1 minus 120575) 119910119899 if 119898 = 1

119910119901

119899+1=

119898minus1

sum

119896=0

120593119896

119905119896

119899+1

119896+

1

Γ (120572)

119899

sum

119895=0

119887119895119899+1

119891 (119905119895 119910119895 V119895)

(20)

where

119887119895119899+1

=ℎ120572

120572((119899 minus 119895 + 1)

120572minus (119899 minus 119895)

120572) (21)

3 Error Analysis ofthe Newly-Proposed Algorithm

Before giving the main results we firstly introduce thefollowing two lemmas which will be used in the proof of thetheorem

Lemma 2 (see [13]) (i) Let 119911 isin 1198621[0 119879] then

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119911 (119905) 119889119905 minus

119899

sum

119895=0

119887119895119899+1

119911 (119905119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

120572

10038171003817100381710038171003817119911101584010038171003817100381710038171003817infin

119905120572

119899+1ℎ

(22)

(ii) Let 119911(119905) isin 1198622[0 119879] then

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119911 (119905) 119889119905 minus

119899+1

sum

119895=0

119886119895119899+1

119911 (119905119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862119879119903

120572

100381710038171003817100381710038171199111015840101584010038171003817100381710038171003817infin

119905120572

119899+1ℎ2

(23)

In this paper we further assume that 119891(sdot) in (3) satisfiesthe following Lipschitz conditionswith respect to its variablesas follows

1003816100381610038161003816119891 (119905 119910

1 119906) minus 119891 (119905 119910

2 119906)

1003816100381610038161003816le 1198711

10038161003816100381610038161199101minus 1199102

1003816100381610038161003816

1003816100381610038161003816119891 (119905 119910 119906

1) minus 119891 (119905 119910 119906

2)1003816100381610038161003816le 1198712

10038161003816100381610038161199061minus 1199062

1003816100381610038161003816

(24)

where 1198711 1198712are positive constants

Theorem 3 Suppose the solution 119910(119905) isin 1198622[0 119879] of the initial

value problem satisfies the following two conditions

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119863120572

119905119910 (119905) 119889119905 minus

119899

sum

119895=0

119887119895119896+1

119863120572

119905119910 (119905119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 1198621199051205741

119899+1ℎ1205751

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119863120572

119905119910 (119905) 119889119905 minus

119899

sum

119895=0

119886119895119896+1

119863120572

119905119910 (119905119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 1198621199051205742

119899+1ℎ1205752

(25)

with some 1205741 1205742

ge 0 and 1205751 1205752

gt 0 then for some suitable119879 gt 0 we have

max0le119869le119873

10038161003816100381610038161003816119910 (119905119895) minus 119910119895

10038161003816100381610038161003816le 119862ℎ119902 (26)

where 119902 = min1205751+ 120572 120575

2 119873 = [119879ℎ] and 119862 is a positive

constant

Journal of Applied Mathematics 5

Proof We will prove the result by virtue of mathematicalinduction Suppose that the conclusion is true for 119895 =

0 1 119899 thenwe have to prove that the inequality also holdsfor 119895 = 119899 + 1 To do this we first consider the error of thepredictor 119910

119901

119899+1

From the assumptions (24) we have

10038161003816100381610038161003816119891 (119905119895 119910 (119905119895) 119910 (119905

119895minus 120591)) minus 119891 (119905

119895 119910119895 V119895)10038161003816100381610038161003816

le10038161003816100381610038161003816119891 (119905119895 119910 (119905119895) 119910 (119905

119895minus 120591)) minus 119891 (119905

119895 119910119895 119910 (119905119895minus 120591))

10038161003816100381610038161003816

+10038161003816100381610038161003816119891 (119905119895 119910119895 119910 (119905119895minus 120591)) minus 119891 (119905

119895 119910119895 V119895)10038161003816100381610038161003816

le 1198711ℎ119902+ 1198712ℎ119902= (1198711+ 1198712) ℎ119902

(27)

So

10038161003816100381610038161003816119910 (119905119899+1

) minus 119910119901

119899+1

10038161003816100381610038161003816

=1

Γ (120572)

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119891 (119905 119910 (119905) 119910 (119905 minus 120591)) 119889119905

minus

119899

sum

119895=0

119887119895119896+1

119891 (119905119895 119910119895 V119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

Γ (120572)

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119863120572

119905119910 (119905) 119889119905 minus

119899

sum

119895=0

119887119895119899+1

119863120572

119905119910 (119905119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

+1

Γ (120572)

times

119899

sum

119895=0

119887119895119899+1

10038161003816100381610038161003816119891 (119905119895 119910 (119905119895) 119910 (119905

119895minus 120591))

minus119891 (119905119895 119910119895 119910 (119905119895minus 120591))

10038161003816100381610038161003816

+10038161003816100381610038161003816119891 (119905119895 119910119895 119910 (119905119895minus 120591)) minus 119891 (119905

119895 119910119895 V119895)10038161003816100381610038161003816

le1198621199051205741

119899+1

Γ (120572)ℎ1205751

+1

Γ (120572)

119899

sum

119895=0

119887119895119899+1

(1198711+ 1198712) 119862ℎ119902

(28)

and from

119899

sum

119895=0

119887119895119899+1

=

119899

sum

119895=0

int

(119895+1)ℎ

119895ℎ

(119905119899+1

minus 119905)120572minus1

119889119905 =1

120572119905120572

119899+1le

1

120572119879120572

(29)

we have

10038161003816100381610038161003816119910 (119905119899+1

) minus 119910119901

119899+1

10038161003816100381610038161003816le

1198621198791205741

Γ (120572)ℎ1205751

+119862119879120572

Γ (120572 + 1)ℎ119902 (30)

Since119899

sum

119895=0

119886119895119899+1

le

119899

sum

119895=0

ℎ120572

120572 (120572 + 1)

times [(119896 minus 119895 + 2)120572+1

minus (119896 minus 119895 + 1)120572+1

]

minus [(119896 minus 119895 + 1)120572+1

minus (119896 minus 119895)120572+1

]

= [int

119905119899+1

0

(119905119899+2

minus 119905)120572119889119905 minus int

119905119899+1

0

(119905119899+1

minus 119905)120572119889119905]

=1

120572int

119905119899+1

0

[(119905119899+1

minus 119905)120572]1015840

(119905) 119889119905 =1

120572119905120572

119899+1le 119879120572

(31)

we have1003816100381610038161003816119910 (119905119899+1

) minus 119910119899+1

1003816100381610038161003816

=1

Γ (120572)

10038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119891 (119905 119910 (119905)

119910 (119905 minus 120591)) 119889119905 minus

119899

sum

119895=0

119886119895119899+1

119891 (119905119895 119910119895 V119895)

minus 119886119899+1119899+1

119891 (119905119899+1

119910119901

119899+1 V119899+1

)

10038161003816100381610038161003816100381610038161003816

le1

Γ (120572)

10038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119891 (119905 119910 (119905) 119910 (119905 minus 120591))

minus

119899+1

sum

119895=0

119886119895119899+1

119891 (119905119895 119910 (119905119895) 119910 (119905

119895minus 120591))

10038161003816100381610038161003816100381610038161003816

+

119899

sum

119895=0

119886119895119899+1

10038161003816100381610038161003816119891 (119905119895 119910 (119905119895) 119910 (119905

119895minus120591)) minus 119891 (119905

119895 119910119895 V119895)10038161003816100381610038161003816

+ 119886119899+1119899+1

10038161003816100381610038161003816119891 (119905119899+1

119910 (119905119899+1

) 119910 (119905119899+1

minus 120591))

minus 119891 (119905119899+1

119910119901

119899+1 V119899+1

)10038161003816100381610038161003816

le1

Γ (120572)

[

[

1198621199051205742

119899+1ℎ1205752

+ 119862ℎ119902

119899

sum

119895=0

119886119895119899+1

+ 119886119899+1119899+1

119871(ℎ119902+

1198621198791205741

Γ (120572)ℎ1205751

+119862119879120572

Γ (120572 + 1)ℎ119902)]

]

le1

Γ (120572)[1198621199051205742

119899+1ℎ1205752

+119871ℎ119902+ℎ

120572 (120572 + 1)

+1198621198711198791205741

Γ (120572 + 2)+

119862119879120572ℎ120572+119902

120572Γ (120572 + 2)] = 119862ℎ

119902

(32)

which completes the proof

When 1205751

= 1 1205752

= 2 applying Theorem 3 it is notdifficult to derive the following result on the error bound

6 Journal of Applied Mathematics

Theorem 4 Suppose that 120572 gt 0 and 119891(119905 119910(119905) 119911(119905)) isin

1198621015840[0 119879] then

max0le119895le119873

10038161003816100381610038161003816119910 (119905119895) minus 119910119895

10038161003816100381610038161003816le 119862ℎ119902 (33)

where 119902 = min 1 + 120572 2 and 119862 is a positive constant

Remark 5 We can see from Cases I and II that if the solution119910(119905) is not smooth then the truncation error of V

119899+1 which

approximates to 119910(119905 minus 120591) will not be 119874(ℎ2) Therefore by the

error analysis elaborated in Section 3 we cannot yield thesame conclusion as in Theorem 4 We will consider such aproblem in our future work

4 A Numerical Example

In this section the following delayed FDE is considered

119863120572

119905119910 (119905) =

2

Γ (3 minus 120572)1199052minus120572

minus1

Γ (2 minus 120572)1199051minus120572

+ 2120591119905 minus 1205912

minus 120591 minus 119910 (119905) + 119910 (119905 minus 120591) 120572 isin (0 1)

119910 (119905) = 0 119905 le 0

(34)

Notice that the exact solution to this equation is

119910 (119905) = 1199052minus 119905 (35)

In accordance with delay 120591 being constant or time varyingthe numerical results are displayed in Tables 1 2 3 and4 respectively where 119864

119860denotes the absolute numerical

error and 119864119877denotes the relative numerical error From the

numerical results we can see that the computing errors are ingeneral acceptable for engineering

5 Conclusions

In this paper a numerical algorithm is formulated basedon Adams-Bashforth-Moulton method for fractional-orderdifferential equations with time delay The error analysisof the numerical scheme is carried out meanwhile anumerical example with constant delay and time-varyingdelay is proposed to testify the effectiveness of the schemeThis algorithm can be used not only in the simulation ofdelayed fractional-order differential equations but also in thesimulation of delayed fractional-order control systems

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China under Grants 61004078 61273012 theFoundation of State Key Laboratory for Novel SoftwareTechnology of Nanjing University the China PostdoctoralScience Foundation Funded Project and the Special Fundsfor Postdoctoral Innovative Projects of Shandong Province

References

[1] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993

[2] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[3] P L Butzer and U Westphal ldquoAn introduction to fractionalcalculusrdquo in Applications of Fractional Calculus in Physics pp1ndash85 World Scientific Publishing River Edge NJ USA 2000

[4] Z Wang X Huang Y-X Li and X-N Song ldquoA new imageencryption algorithm based on fractional-order hyperchaoticLorenz systemrdquo Chinese Physics B vol 22 Article ID 0105042013

[5] K BOldhamand J SpanierTheFractional CalculusTheory andApplication of Differentiation and Integration to Arbitrary Ordervol 111 of Mathematics in Science and Engineering AcademicPress New York NY USA 1974

[6] R Hilfer Ed Applications of Fractional Calculus in PhysicsWorld Scientific Publishing River Edge NJ USA 2000

[7] P J Torvik andR L Bagley ldquoOn the appearance of the fractionalderivatives in the behaviour of realmaterialsrdquo Journal of AppliedMechanics vol 51 no 2 pp 294ndash298 1984

[8] F Mainardi ldquoFractional relaxation-oscillation and fractionaldiffusion-wave phenomenardquo Chaos Solitons amp Fractals vol 7no 9 pp 1461ndash1477 1996

[9] N J Ford and A C Simpson ldquoThe numerical solution of frac-tional differential equations speed versus accuracyrdquo NumericalAlgorithms vol 26 no 4 pp 333ndash346 2001

[10] V Daftardar-Gejji and A Babakhani ldquoAnalysis of a systemof fractional differential equationsrdquo Journal of MathematicalAnalysis and Applications vol 293 no 2 pp 511ndash522 2004

[11] K Diethelm and N J Ford ldquoAnalysis of fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 265 no 2 pp 229ndash248 2002

[12] 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

[13] KDiethelmN J Ford andAD Freed ldquoDetailed error analysisfor a fractional Adams methodrdquo Numerical Algorithms vol 36no 1 pp 31ndash52 2004

[14] K Diethelm N J Ford A D Freed and Yu Luchko ldquoAlgo-rithms for the fractional calculus a selection of numericalmethodsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 194 no 6-8 pp 743ndash773 2005

[15] K Diethelm and N J Ford ldquoMulti-order fractional differentialequations and their numerical solutionrdquo Applied Mathematicsand Computation vol 154 no 3 pp 621ndash640 2004

[16] Z Odibat ldquoApproximations of fractional integrals and Caputofractional derivativesrdquo Applied Mathematics and Computationvol 178 no 2 pp 527ndash533 2006

[17] J T Edwards N J Ford and A C Simpson ldquoThe numericalsolution of linear multi-term fractional differential equationssystems of equationsrdquo Journal of Computational and AppliedMathematics vol 148 no 2 pp 401ndash418 2002

[18] K Diethelm ldquoAn investigation of some nonclassical methodsfor the numerical approximation of Caputo-type fractionalderivativesrdquo Numerical Algorithms vol 47 no 4 pp 361ndash3902008

Journal of Applied Mathematics 7

[19] K Diethelm ldquoAn improvement of a nonclassical numericalmethod for the computation of fractional derivativesrdquo Journalof Vibration and Acoustics vol 131 no 1 Article ID 014502 4pages 2009

[20] C Li and C Tao ldquoOn the fractional AdamsmethodrdquoComputersamp Mathematics with Applications vol 58 no 8 pp 1573ndash15882009

[21] R Garrappa ldquoOn some explicit Adams multistep methods forfractional differential equationsrdquo Journal of Computational andApplied Mathematics vol 229 no 2 pp 392ndash399 2009

[22] L Galeone and R Garrappa ldquoFractional Adams-Moultonmethodsrdquo Mathematics and Computers in Simulation vol 79no 4 pp 1358ndash1367 2008

[23] L Galeone and R Garrappa ldquoExplicit methods for fractionaldifferential equations and their stability propertiesrdquo Journal ofComputational and Applied Mathematics vol 228 no 2 pp548ndash560 2009

[24] W Deng ldquoShort memory principle and a predictor-correctorapproach for fractional differential equationsrdquo Journal of Com-putational and AppliedMathematics vol 206 no 1 pp 174ndash1882007

[25] C Li and G Peng ldquoChaos in Chenrsquos system with a fractionalorderrdquo Chaos Solitons amp Fractals vol 22 no 2 pp 443ndash4502004

[26] XWu J Li and G Chen ldquoChaos in the fractional order unifiedsystem and its synchronizationrdquo Journal of the Franklin Institutevol 345 no 4 pp 392ndash401 2008

[27] M S Tavazoei andM Haeri ldquoChaotic attractors in incommen-surate fractional order systemsrdquo Physica D vol 237 no 20 pp2628ndash2637 2008

[28] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A vol 389 no12 pp 2434ndash2442 2010

[29] Y Chen and K L Moore ldquoAnalytical stability bound for aclass of delayed fractional-order dynamic systemsrdquo NonlinearDynamics vol 29 no 1ndash4 pp 191ndash200 2002

[30] P Lanusse H Benlaoukli D Nelson-Gruel and A OustaloupldquoFractional-order control and interval analysis of SISO systemswith time-delayed staterdquo IETControlTheoryampApplications vol2 no 1 pp 16ndash23 2008

[31] X Zhang ldquoSome results of linear fractional order time-delaysystemrdquo Applied Mathematics and Computation vol 197 no 1pp 407ndash411 2008

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

Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

Table 2 Error behavior at time 119905 = 119879 with analytical value with 120572 = 09 120591 = 01

119879 119864 ℎ = 110 ℎ = 120 ℎ = 140 ℎ = 180 ℎ = 1160 ℎ = 1320

5 119864119860

00534 00163 00038 minus75742119890 minus 4 minus00026 minus00033119864119877

00027 81293119890 minus 4 19072119890 minus 4 minus37871119890 minus 5 minus12853119890 minus 4 minus16663119890 minus 4

10 119864119860

00865 00247 00061 minus51223119890 minus 5 minus00022 minus00031119864119877

96095119890 minus 4 27404119890 minus 4 67297119890 minus 5 minus56910119890 minus 7 minus24824119890 minus 5 minus34191119890 minus 5

50 119864119860

03610 00955 00248 00051 minus53786119890 minus 4 minus00023119864119877

14736119890 minus 4 38962119890 minus 5 10103119890 minus 5 20997119890 minus 6 minus71954119890 minus 7 minus92911119890 minus 7

Table 3 Error behavior at time 119905 = 119879 with analytical value with 120572 = 02 120591 = 001 exp(minus119905)

119879 119864 ℎ = 110 ℎ = 120 ℎ = 140 ℎ = 180 ℎ = 1160 ℎ = 1320

5 119864119860

00425 00218 00133 00097 00081 00074119864119877

00021 00011 66680119890 minus 4 48592119890 minus 4 40585119890 minus 4 36960119890 minus 4

10 119864119860

00411 00217 00138 00104 00089 00082119864119877

45679119890 minus 4 24130119890 minus 4 15319119890 minus 4 11553119890 minus 4 98859119890 minus 5 91315119890 minus 5

50 119864119860

minus00351 00184 00116 00087 00074 00052119864119877

14317119890 minus 5 75036119890 minus 6 47294119890 minus 6 35467119890 minus 6 30242119890 minus 6 24322119890 minus 6

in which (9) is called the predictor and (10) is called thecorrector It is well known that thismethod is convergent thatis

max119895=01119873

10038161003816100381610038161003816119910 (119905119895) minus 119910119895

10038161003816100381610038161003816= 119874 (ℎ

2) (11)

23 The Representation of Algorithm for Delayed FDEs Hav-ing introduced the Adams-Bashforth-Moulton algorithmnow the key problem is to establish approximation to thedelayed term 119910(119905 minus 120591) which consists of the following twocases

Case I (when 120591 is constant) It can be seen that for anypositive constant 120591 119905

119895minus 120591 may not be a grid point 119905

119899for any

119899 Suppose that (119898 minus 120575)ℎ = 120591 and 0 le 120575 lt 1 When 120575 = 0119910(119905119899minus 120591) can be approximated by

119910 (119905119899minus 120591) asymp

119910119899minus119898

if 119899 gt 119898

120593119899 if 119899 le 119898

(12)

and when 0 lt 120575 lt 1 119910(119905119899minus 120591) cannot be calculated directly

For (119898 minus 1)ℎ lt 120591 lt 119898ℎ let V119899+1

be the approximation to119910(119905119899+1

minus120591) we interpolate it by the two nearest points that is

V119899+1

= 120575119910119899minus119898+2

+ (1 minus 120575) 119910119899minus119898+1 (13)

It can be seen from (13) that if119898 gt 1 the numerical equationis explicit and thus can be computed directly However when119898 = 1 and 120575 = 0 that is 120591 lt ℎ then the first term in the right-hand side of the previous equation is 120575119910

119899+1 It still needs to

predict in this case that is

V119899+1

= 120575119910119901

119899+1+ (1 minus 120575) 119910119899 (14)

Case II (when 120591 is time varying) If 120591 is time varying thatis 120591 = 120591(119905) then the approximation in this case is morecomplicated Let V

119899+1be the approximation to 119910(119905

119899+1minus 120591)

and the linear interpolation of 119910119895 at point 119905 = 119905

119899+1minus 120591(119905119899+1

)

is employed to approximate the delay term Let 120591(119905119899+1

) =

(119898119899+1

minus 120575119899+1

)ℎ where 119898119899+1

is a positive integer and 120575119899+1

isin

[0 1) then

V119899+1

= 120575119899+1

119910119899minus119898119899+1+2

+ (1 minus 120575119899+1

) 119910119899minus119898119899+1+1

(15)

It should be noted that when 120591 is constant for given ℎ and 120591we can judge whether 119898 = 1 or 119898 gt 1 holds at the beginningof the programme However when 120591 is time varying 119898 isalso time varying that is at one moment it is equal to 1 andat another moment it may greater than 1 If 119898

119899+1= 1 the

first term in the right-hand side of (15) still needs to predictotherwise when119898

119899+1gt 1 it needs not So in each step of the

computing it needs to test whether119898119899= 1 or not firstly then

computation of the first term in the right-hand side of (15)depends on whether it requires predicting or not in differentsteps

Now we begin the numerical algorithm for the delayedFDEs (3)-(4)

From the theory introduced in [10ndash14] we know that thedelayed initial value problem (3)-(4) is equivalent to Volterraintegral equation as follows

119910 (119905) =

119898minus1

sum

119896=0

120593 (119905)119905119896

119896+

1

Γ (120572)int

119905

0

119891 (119904 119910 (119904) 119910 (119904 minus 120591))

(119905 minus 119904)1minus120572

119889119904 (16)

Just as the method used in [13] the remaining problem isto compute the integral term in (16) Firstly the producttrapezoidal quadrature formula is applied to replace theintegrals term where nodes 119905

119895(119895 = 0 1 119899 + 1) are taken

with respect to the weight function (119905119899+1

minus sdot)120572minus1 That is we

employ the approximation

int

119905119899+1

0

(119905119896+1

minus 119911)120572minus1

119892 (119911) 119889119911 asymp int

119905119899+1

0

(119905119899+1

minus 119911)120572minus1

119892119899+1 (119911) 119889119911

(17)

4 Journal of Applied Mathematics

Table 4 Error behavior at time 119905 = 119879 with analytical value with 120572 = 09 120591 = 001 exp(minus119905)

119879 119864 ℎ = 110 ℎ = 120 ℎ = 140 ℎ = 180 ℎ = 1160 ℎ = 1320

5 119864119860

00031 00016 00012 00011 00011 00010119864119877

15375119890 minus 4 81279119890 minus 5 61399119890 minus 5 55480119890 minus 5 53282119890 minus 5 52136119890 minus 5

10 119864119860

00022 91466119890 minus 4 58474119890 minus 4 49919119890 minus 4 47680119890 minus 4 47115119890 minus 4

119864119877

24189119890 minus 5 10163119890 minus 5 64971119890 minus 6 55465119890 minus 6 52300119890 minus 6 52350119890 minus 6

50 119864119860

00016 49082119890 minus 4 19954119890 minus 4 12510119890 minus 4 10616119890 minus 4 70303119890 minus 4

119864119877

66311119890 minus 7 20033119890 minus 7 81447119890 minus 8 51062119890 minus 8 43330119890 minus 8 27081119890 minus 8

where 119892119899+1

(sdot) is the piecewise linear interpolation for 119892(sdot)

with nodes and knots chosen at 119905119895 119895 = 0 1 119899 + 1 Using

the standard technique of quadrature theory it can be foundthat we can write the integrals on the right-hand side of (17)as

int

119905119899+1

0

(119905119899+1

minus 119911)120572minus1

119892119899+1 (119911) 119889119911 =

119899+1

sum

119895=0

119886119895119899+1

119892 (119905119895) (18)

where119886119895119899+1

=ℎ120572

120572 (120572 + 1)

times

119899120572+1

minus(119899 minus 120572) (119899 + 1)120572 119895 = 0

(119899 minus 119895+2)120572+1

+(119899minus119895)120572+1

minus2(119899minus119895+1)120572+1

1 le 119895 le 119899

1 119895 = 119899 + 1

(19)

Consequently the numerical scheme for FDEs (3)-(4) can bedepicted as

119910119899+1

=

119898minus1

sum

119896=0

120593119896

119905119896

119899+1

119896+

ℎ120572

Γ (120572 + 2)119891 (119905119899+1

119910119901

119899+1 V119899+1

)

+1

Γ (120572)

119899

sum

119895=0

119886119895119899+1

119891 (119905119895 119910119895 V119895)

V119899+1

= 120575119910119899minus119898+2

+ (1 minus 120575) 119910119899minus119898+1 if 119898 gt 1

120575119910119901

119899+1+ (1 minus 120575) 119910119899 if 119898 = 1

119910119901

119899+1=

119898minus1

sum

119896=0

120593119896

119905119896

119899+1

119896+

1

Γ (120572)

119899

sum

119895=0

119887119895119899+1

119891 (119905119895 119910119895 V119895)

(20)

where

119887119895119899+1

=ℎ120572

120572((119899 minus 119895 + 1)

120572minus (119899 minus 119895)

120572) (21)

3 Error Analysis ofthe Newly-Proposed Algorithm

Before giving the main results we firstly introduce thefollowing two lemmas which will be used in the proof of thetheorem

Lemma 2 (see [13]) (i) Let 119911 isin 1198621[0 119879] then

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119911 (119905) 119889119905 minus

119899

sum

119895=0

119887119895119899+1

119911 (119905119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

120572

10038171003817100381710038171003817119911101584010038171003817100381710038171003817infin

119905120572

119899+1ℎ

(22)

(ii) Let 119911(119905) isin 1198622[0 119879] then

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119911 (119905) 119889119905 minus

119899+1

sum

119895=0

119886119895119899+1

119911 (119905119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862119879119903

120572

100381710038171003817100381710038171199111015840101584010038171003817100381710038171003817infin

119905120572

119899+1ℎ2

(23)

In this paper we further assume that 119891(sdot) in (3) satisfiesthe following Lipschitz conditionswith respect to its variablesas follows

1003816100381610038161003816119891 (119905 119910

1 119906) minus 119891 (119905 119910

2 119906)

1003816100381610038161003816le 1198711

10038161003816100381610038161199101minus 1199102

1003816100381610038161003816

1003816100381610038161003816119891 (119905 119910 119906

1) minus 119891 (119905 119910 119906

2)1003816100381610038161003816le 1198712

10038161003816100381610038161199061minus 1199062

1003816100381610038161003816

(24)

where 1198711 1198712are positive constants

Theorem 3 Suppose the solution 119910(119905) isin 1198622[0 119879] of the initial

value problem satisfies the following two conditions

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119863120572

119905119910 (119905) 119889119905 minus

119899

sum

119895=0

119887119895119896+1

119863120572

119905119910 (119905119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 1198621199051205741

119899+1ℎ1205751

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119863120572

119905119910 (119905) 119889119905 minus

119899

sum

119895=0

119886119895119896+1

119863120572

119905119910 (119905119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 1198621199051205742

119899+1ℎ1205752

(25)

with some 1205741 1205742

ge 0 and 1205751 1205752

gt 0 then for some suitable119879 gt 0 we have

max0le119869le119873

10038161003816100381610038161003816119910 (119905119895) minus 119910119895

10038161003816100381610038161003816le 119862ℎ119902 (26)

where 119902 = min1205751+ 120572 120575

2 119873 = [119879ℎ] and 119862 is a positive

constant

Journal of Applied Mathematics 5

Proof We will prove the result by virtue of mathematicalinduction Suppose that the conclusion is true for 119895 =

0 1 119899 thenwe have to prove that the inequality also holdsfor 119895 = 119899 + 1 To do this we first consider the error of thepredictor 119910

119901

119899+1

From the assumptions (24) we have

10038161003816100381610038161003816119891 (119905119895 119910 (119905119895) 119910 (119905

119895minus 120591)) minus 119891 (119905

119895 119910119895 V119895)10038161003816100381610038161003816

le10038161003816100381610038161003816119891 (119905119895 119910 (119905119895) 119910 (119905

119895minus 120591)) minus 119891 (119905

119895 119910119895 119910 (119905119895minus 120591))

10038161003816100381610038161003816

+10038161003816100381610038161003816119891 (119905119895 119910119895 119910 (119905119895minus 120591)) minus 119891 (119905

119895 119910119895 V119895)10038161003816100381610038161003816

le 1198711ℎ119902+ 1198712ℎ119902= (1198711+ 1198712) ℎ119902

(27)

So

10038161003816100381610038161003816119910 (119905119899+1

) minus 119910119901

119899+1

10038161003816100381610038161003816

=1

Γ (120572)

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119891 (119905 119910 (119905) 119910 (119905 minus 120591)) 119889119905

minus

119899

sum

119895=0

119887119895119896+1

119891 (119905119895 119910119895 V119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

Γ (120572)

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119863120572

119905119910 (119905) 119889119905 minus

119899

sum

119895=0

119887119895119899+1

119863120572

119905119910 (119905119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

+1

Γ (120572)

times

119899

sum

119895=0

119887119895119899+1

10038161003816100381610038161003816119891 (119905119895 119910 (119905119895) 119910 (119905

119895minus 120591))

minus119891 (119905119895 119910119895 119910 (119905119895minus 120591))

10038161003816100381610038161003816

+10038161003816100381610038161003816119891 (119905119895 119910119895 119910 (119905119895minus 120591)) minus 119891 (119905

119895 119910119895 V119895)10038161003816100381610038161003816

le1198621199051205741

119899+1

Γ (120572)ℎ1205751

+1

Γ (120572)

119899

sum

119895=0

119887119895119899+1

(1198711+ 1198712) 119862ℎ119902

(28)

and from

119899

sum

119895=0

119887119895119899+1

=

119899

sum

119895=0

int

(119895+1)ℎ

119895ℎ

(119905119899+1

minus 119905)120572minus1

119889119905 =1

120572119905120572

119899+1le

1

120572119879120572

(29)

we have

10038161003816100381610038161003816119910 (119905119899+1

) minus 119910119901

119899+1

10038161003816100381610038161003816le

1198621198791205741

Γ (120572)ℎ1205751

+119862119879120572

Γ (120572 + 1)ℎ119902 (30)

Since119899

sum

119895=0

119886119895119899+1

le

119899

sum

119895=0

ℎ120572

120572 (120572 + 1)

times [(119896 minus 119895 + 2)120572+1

minus (119896 minus 119895 + 1)120572+1

]

minus [(119896 minus 119895 + 1)120572+1

minus (119896 minus 119895)120572+1

]

= [int

119905119899+1

0

(119905119899+2

minus 119905)120572119889119905 minus int

119905119899+1

0

(119905119899+1

minus 119905)120572119889119905]

=1

120572int

119905119899+1

0

[(119905119899+1

minus 119905)120572]1015840

(119905) 119889119905 =1

120572119905120572

119899+1le 119879120572

(31)

we have1003816100381610038161003816119910 (119905119899+1

) minus 119910119899+1

1003816100381610038161003816

=1

Γ (120572)

10038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119891 (119905 119910 (119905)

119910 (119905 minus 120591)) 119889119905 minus

119899

sum

119895=0

119886119895119899+1

119891 (119905119895 119910119895 V119895)

minus 119886119899+1119899+1

119891 (119905119899+1

119910119901

119899+1 V119899+1

)

10038161003816100381610038161003816100381610038161003816

le1

Γ (120572)

10038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119891 (119905 119910 (119905) 119910 (119905 minus 120591))

minus

119899+1

sum

119895=0

119886119895119899+1

119891 (119905119895 119910 (119905119895) 119910 (119905

119895minus 120591))

10038161003816100381610038161003816100381610038161003816

+

119899

sum

119895=0

119886119895119899+1

10038161003816100381610038161003816119891 (119905119895 119910 (119905119895) 119910 (119905

119895minus120591)) minus 119891 (119905

119895 119910119895 V119895)10038161003816100381610038161003816

+ 119886119899+1119899+1

10038161003816100381610038161003816119891 (119905119899+1

119910 (119905119899+1

) 119910 (119905119899+1

minus 120591))

minus 119891 (119905119899+1

119910119901

119899+1 V119899+1

)10038161003816100381610038161003816

le1

Γ (120572)

[

[

1198621199051205742

119899+1ℎ1205752

+ 119862ℎ119902

119899

sum

119895=0

119886119895119899+1

+ 119886119899+1119899+1

119871(ℎ119902+

1198621198791205741

Γ (120572)ℎ1205751

+119862119879120572

Γ (120572 + 1)ℎ119902)]

]

le1

Γ (120572)[1198621199051205742

119899+1ℎ1205752

+119871ℎ119902+ℎ

120572 (120572 + 1)

+1198621198711198791205741

Γ (120572 + 2)+

119862119879120572ℎ120572+119902

120572Γ (120572 + 2)] = 119862ℎ

119902

(32)

which completes the proof

When 1205751

= 1 1205752

= 2 applying Theorem 3 it is notdifficult to derive the following result on the error bound

6 Journal of Applied Mathematics

Theorem 4 Suppose that 120572 gt 0 and 119891(119905 119910(119905) 119911(119905)) isin

1198621015840[0 119879] then

max0le119895le119873

10038161003816100381610038161003816119910 (119905119895) minus 119910119895

10038161003816100381610038161003816le 119862ℎ119902 (33)

where 119902 = min 1 + 120572 2 and 119862 is a positive constant

Remark 5 We can see from Cases I and II that if the solution119910(119905) is not smooth then the truncation error of V

119899+1 which

approximates to 119910(119905 minus 120591) will not be 119874(ℎ2) Therefore by the

error analysis elaborated in Section 3 we cannot yield thesame conclusion as in Theorem 4 We will consider such aproblem in our future work

4 A Numerical Example

In this section the following delayed FDE is considered

119863120572

119905119910 (119905) =

2

Γ (3 minus 120572)1199052minus120572

minus1

Γ (2 minus 120572)1199051minus120572

+ 2120591119905 minus 1205912

minus 120591 minus 119910 (119905) + 119910 (119905 minus 120591) 120572 isin (0 1)

119910 (119905) = 0 119905 le 0

(34)

Notice that the exact solution to this equation is

119910 (119905) = 1199052minus 119905 (35)

In accordance with delay 120591 being constant or time varyingthe numerical results are displayed in Tables 1 2 3 and4 respectively where 119864

119860denotes the absolute numerical

error and 119864119877denotes the relative numerical error From the

numerical results we can see that the computing errors are ingeneral acceptable for engineering

5 Conclusions

In this paper a numerical algorithm is formulated basedon Adams-Bashforth-Moulton method for fractional-orderdifferential equations with time delay The error analysisof the numerical scheme is carried out meanwhile anumerical example with constant delay and time-varyingdelay is proposed to testify the effectiveness of the schemeThis algorithm can be used not only in the simulation ofdelayed fractional-order differential equations but also in thesimulation of delayed fractional-order control systems

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China under Grants 61004078 61273012 theFoundation of State Key Laboratory for Novel SoftwareTechnology of Nanjing University the China PostdoctoralScience Foundation Funded Project and the Special Fundsfor Postdoctoral Innovative Projects of Shandong Province

References

[1] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993

[2] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[3] P L Butzer and U Westphal ldquoAn introduction to fractionalcalculusrdquo in Applications of Fractional Calculus in Physics pp1ndash85 World Scientific Publishing River Edge NJ USA 2000

[4] Z Wang X Huang Y-X Li and X-N Song ldquoA new imageencryption algorithm based on fractional-order hyperchaoticLorenz systemrdquo Chinese Physics B vol 22 Article ID 0105042013

[5] K BOldhamand J SpanierTheFractional CalculusTheory andApplication of Differentiation and Integration to Arbitrary Ordervol 111 of Mathematics in Science and Engineering AcademicPress New York NY USA 1974

[6] R Hilfer Ed Applications of Fractional Calculus in PhysicsWorld Scientific Publishing River Edge NJ USA 2000

[7] P J Torvik andR L Bagley ldquoOn the appearance of the fractionalderivatives in the behaviour of realmaterialsrdquo Journal of AppliedMechanics vol 51 no 2 pp 294ndash298 1984

[8] F Mainardi ldquoFractional relaxation-oscillation and fractionaldiffusion-wave phenomenardquo Chaos Solitons amp Fractals vol 7no 9 pp 1461ndash1477 1996

[9] N J Ford and A C Simpson ldquoThe numerical solution of frac-tional differential equations speed versus accuracyrdquo NumericalAlgorithms vol 26 no 4 pp 333ndash346 2001

[10] V Daftardar-Gejji and A Babakhani ldquoAnalysis of a systemof fractional differential equationsrdquo Journal of MathematicalAnalysis and Applications vol 293 no 2 pp 511ndash522 2004

[11] K Diethelm and N J Ford ldquoAnalysis of fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 265 no 2 pp 229ndash248 2002

[12] 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

[13] KDiethelmN J Ford andAD Freed ldquoDetailed error analysisfor a fractional Adams methodrdquo Numerical Algorithms vol 36no 1 pp 31ndash52 2004

[14] K Diethelm N J Ford A D Freed and Yu Luchko ldquoAlgo-rithms for the fractional calculus a selection of numericalmethodsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 194 no 6-8 pp 743ndash773 2005

[15] K Diethelm and N J Ford ldquoMulti-order fractional differentialequations and their numerical solutionrdquo Applied Mathematicsand Computation vol 154 no 3 pp 621ndash640 2004

[16] Z Odibat ldquoApproximations of fractional integrals and Caputofractional derivativesrdquo Applied Mathematics and Computationvol 178 no 2 pp 527ndash533 2006

[17] J T Edwards N J Ford and A C Simpson ldquoThe numericalsolution of linear multi-term fractional differential equationssystems of equationsrdquo Journal of Computational and AppliedMathematics vol 148 no 2 pp 401ndash418 2002

[18] K Diethelm ldquoAn investigation of some nonclassical methodsfor the numerical approximation of Caputo-type fractionalderivativesrdquo Numerical Algorithms vol 47 no 4 pp 361ndash3902008

Journal of Applied Mathematics 7

[19] K Diethelm ldquoAn improvement of a nonclassical numericalmethod for the computation of fractional derivativesrdquo Journalof Vibration and Acoustics vol 131 no 1 Article ID 014502 4pages 2009

[20] C Li and C Tao ldquoOn the fractional AdamsmethodrdquoComputersamp Mathematics with Applications vol 58 no 8 pp 1573ndash15882009

[21] R Garrappa ldquoOn some explicit Adams multistep methods forfractional differential equationsrdquo Journal of Computational andApplied Mathematics vol 229 no 2 pp 392ndash399 2009

[22] L Galeone and R Garrappa ldquoFractional Adams-Moultonmethodsrdquo Mathematics and Computers in Simulation vol 79no 4 pp 1358ndash1367 2008

[23] L Galeone and R Garrappa ldquoExplicit methods for fractionaldifferential equations and their stability propertiesrdquo Journal ofComputational and Applied Mathematics vol 228 no 2 pp548ndash560 2009

[24] W Deng ldquoShort memory principle and a predictor-correctorapproach for fractional differential equationsrdquo Journal of Com-putational and AppliedMathematics vol 206 no 1 pp 174ndash1882007

[25] C Li and G Peng ldquoChaos in Chenrsquos system with a fractionalorderrdquo Chaos Solitons amp Fractals vol 22 no 2 pp 443ndash4502004

[26] XWu J Li and G Chen ldquoChaos in the fractional order unifiedsystem and its synchronizationrdquo Journal of the Franklin Institutevol 345 no 4 pp 392ndash401 2008

[27] M S Tavazoei andM Haeri ldquoChaotic attractors in incommen-surate fractional order systemsrdquo Physica D vol 237 no 20 pp2628ndash2637 2008

[28] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A vol 389 no12 pp 2434ndash2442 2010

[29] Y Chen and K L Moore ldquoAnalytical stability bound for aclass of delayed fractional-order dynamic systemsrdquo NonlinearDynamics vol 29 no 1ndash4 pp 191ndash200 2002

[30] P Lanusse H Benlaoukli D Nelson-Gruel and A OustaloupldquoFractional-order control and interval analysis of SISO systemswith time-delayed staterdquo IETControlTheoryampApplications vol2 no 1 pp 16ndash23 2008

[31] X Zhang ldquoSome results of linear fractional order time-delaysystemrdquo Applied Mathematics and Computation vol 197 no 1pp 407ndash411 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

4 Journal of Applied Mathematics

Table 4 Error behavior at time 119905 = 119879 with analytical value with 120572 = 09 120591 = 001 exp(minus119905)

119879 119864 ℎ = 110 ℎ = 120 ℎ = 140 ℎ = 180 ℎ = 1160 ℎ = 1320

5 119864119860

00031 00016 00012 00011 00011 00010119864119877

15375119890 minus 4 81279119890 minus 5 61399119890 minus 5 55480119890 minus 5 53282119890 minus 5 52136119890 minus 5

10 119864119860

00022 91466119890 minus 4 58474119890 minus 4 49919119890 minus 4 47680119890 minus 4 47115119890 minus 4

119864119877

24189119890 minus 5 10163119890 minus 5 64971119890 minus 6 55465119890 minus 6 52300119890 minus 6 52350119890 minus 6

50 119864119860

00016 49082119890 minus 4 19954119890 minus 4 12510119890 minus 4 10616119890 minus 4 70303119890 minus 4

119864119877

66311119890 minus 7 20033119890 minus 7 81447119890 minus 8 51062119890 minus 8 43330119890 minus 8 27081119890 minus 8

where 119892119899+1

(sdot) is the piecewise linear interpolation for 119892(sdot)

with nodes and knots chosen at 119905119895 119895 = 0 1 119899 + 1 Using

the standard technique of quadrature theory it can be foundthat we can write the integrals on the right-hand side of (17)as

int

119905119899+1

0

(119905119899+1

minus 119911)120572minus1

119892119899+1 (119911) 119889119911 =

119899+1

sum

119895=0

119886119895119899+1

119892 (119905119895) (18)

where119886119895119899+1

=ℎ120572

120572 (120572 + 1)

times

119899120572+1

minus(119899 minus 120572) (119899 + 1)120572 119895 = 0

(119899 minus 119895+2)120572+1

+(119899minus119895)120572+1

minus2(119899minus119895+1)120572+1

1 le 119895 le 119899

1 119895 = 119899 + 1

(19)

Consequently the numerical scheme for FDEs (3)-(4) can bedepicted as

119910119899+1

=

119898minus1

sum

119896=0

120593119896

119905119896

119899+1

119896+

ℎ120572

Γ (120572 + 2)119891 (119905119899+1

119910119901

119899+1 V119899+1

)

+1

Γ (120572)

119899

sum

119895=0

119886119895119899+1

119891 (119905119895 119910119895 V119895)

V119899+1

= 120575119910119899minus119898+2

+ (1 minus 120575) 119910119899minus119898+1 if 119898 gt 1

120575119910119901

119899+1+ (1 minus 120575) 119910119899 if 119898 = 1

119910119901

119899+1=

119898minus1

sum

119896=0

120593119896

119905119896

119899+1

119896+

1

Γ (120572)

119899

sum

119895=0

119887119895119899+1

119891 (119905119895 119910119895 V119895)

(20)

where

119887119895119899+1

=ℎ120572

120572((119899 minus 119895 + 1)

120572minus (119899 minus 119895)

120572) (21)

3 Error Analysis ofthe Newly-Proposed Algorithm

Before giving the main results we firstly introduce thefollowing two lemmas which will be used in the proof of thetheorem

Lemma 2 (see [13]) (i) Let 119911 isin 1198621[0 119879] then

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119911 (119905) 119889119905 minus

119899

sum

119895=0

119887119895119899+1

119911 (119905119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

120572

10038171003817100381710038171003817119911101584010038171003817100381710038171003817infin

119905120572

119899+1ℎ

(22)

(ii) Let 119911(119905) isin 1198622[0 119879] then

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119911 (119905) 119889119905 minus

119899+1

sum

119895=0

119886119895119899+1

119911 (119905119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862119879119903

120572

100381710038171003817100381710038171199111015840101584010038171003817100381710038171003817infin

119905120572

119899+1ℎ2

(23)

In this paper we further assume that 119891(sdot) in (3) satisfiesthe following Lipschitz conditionswith respect to its variablesas follows

1003816100381610038161003816119891 (119905 119910

1 119906) minus 119891 (119905 119910

2 119906)

1003816100381610038161003816le 1198711

10038161003816100381610038161199101minus 1199102

1003816100381610038161003816

1003816100381610038161003816119891 (119905 119910 119906

1) minus 119891 (119905 119910 119906

2)1003816100381610038161003816le 1198712

10038161003816100381610038161199061minus 1199062

1003816100381610038161003816

(24)

where 1198711 1198712are positive constants

Theorem 3 Suppose the solution 119910(119905) isin 1198622[0 119879] of the initial

value problem satisfies the following two conditions

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119863120572

119905119910 (119905) 119889119905 minus

119899

sum

119895=0

119887119895119896+1

119863120572

119905119910 (119905119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 1198621199051205741

119899+1ℎ1205751

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119863120572

119905119910 (119905) 119889119905 minus

119899

sum

119895=0

119886119895119896+1

119863120572

119905119910 (119905119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 1198621199051205742

119899+1ℎ1205752

(25)

with some 1205741 1205742

ge 0 and 1205751 1205752

gt 0 then for some suitable119879 gt 0 we have

max0le119869le119873

10038161003816100381610038161003816119910 (119905119895) minus 119910119895

10038161003816100381610038161003816le 119862ℎ119902 (26)

where 119902 = min1205751+ 120572 120575

2 119873 = [119879ℎ] and 119862 is a positive

constant

Journal of Applied Mathematics 5

Proof We will prove the result by virtue of mathematicalinduction Suppose that the conclusion is true for 119895 =

0 1 119899 thenwe have to prove that the inequality also holdsfor 119895 = 119899 + 1 To do this we first consider the error of thepredictor 119910

119901

119899+1

From the assumptions (24) we have

10038161003816100381610038161003816119891 (119905119895 119910 (119905119895) 119910 (119905

119895minus 120591)) minus 119891 (119905

119895 119910119895 V119895)10038161003816100381610038161003816

le10038161003816100381610038161003816119891 (119905119895 119910 (119905119895) 119910 (119905

119895minus 120591)) minus 119891 (119905

119895 119910119895 119910 (119905119895minus 120591))

10038161003816100381610038161003816

+10038161003816100381610038161003816119891 (119905119895 119910119895 119910 (119905119895minus 120591)) minus 119891 (119905

119895 119910119895 V119895)10038161003816100381610038161003816

le 1198711ℎ119902+ 1198712ℎ119902= (1198711+ 1198712) ℎ119902

(27)

So

10038161003816100381610038161003816119910 (119905119899+1

) minus 119910119901

119899+1

10038161003816100381610038161003816

=1

Γ (120572)

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119891 (119905 119910 (119905) 119910 (119905 minus 120591)) 119889119905

minus

119899

sum

119895=0

119887119895119896+1

119891 (119905119895 119910119895 V119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

Γ (120572)

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119863120572

119905119910 (119905) 119889119905 minus

119899

sum

119895=0

119887119895119899+1

119863120572

119905119910 (119905119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

+1

Γ (120572)

times

119899

sum

119895=0

119887119895119899+1

10038161003816100381610038161003816119891 (119905119895 119910 (119905119895) 119910 (119905

119895minus 120591))

minus119891 (119905119895 119910119895 119910 (119905119895minus 120591))

10038161003816100381610038161003816

+10038161003816100381610038161003816119891 (119905119895 119910119895 119910 (119905119895minus 120591)) minus 119891 (119905

119895 119910119895 V119895)10038161003816100381610038161003816

le1198621199051205741

119899+1

Γ (120572)ℎ1205751

+1

Γ (120572)

119899

sum

119895=0

119887119895119899+1

(1198711+ 1198712) 119862ℎ119902

(28)

and from

119899

sum

119895=0

119887119895119899+1

=

119899

sum

119895=0

int

(119895+1)ℎ

119895ℎ

(119905119899+1

minus 119905)120572minus1

119889119905 =1

120572119905120572

119899+1le

1

120572119879120572

(29)

we have

10038161003816100381610038161003816119910 (119905119899+1

) minus 119910119901

119899+1

10038161003816100381610038161003816le

1198621198791205741

Γ (120572)ℎ1205751

+119862119879120572

Γ (120572 + 1)ℎ119902 (30)

Since119899

sum

119895=0

119886119895119899+1

le

119899

sum

119895=0

ℎ120572

120572 (120572 + 1)

times [(119896 minus 119895 + 2)120572+1

minus (119896 minus 119895 + 1)120572+1

]

minus [(119896 minus 119895 + 1)120572+1

minus (119896 minus 119895)120572+1

]

= [int

119905119899+1

0

(119905119899+2

minus 119905)120572119889119905 minus int

119905119899+1

0

(119905119899+1

minus 119905)120572119889119905]

=1

120572int

119905119899+1

0

[(119905119899+1

minus 119905)120572]1015840

(119905) 119889119905 =1

120572119905120572

119899+1le 119879120572

(31)

we have1003816100381610038161003816119910 (119905119899+1

) minus 119910119899+1

1003816100381610038161003816

=1

Γ (120572)

10038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119891 (119905 119910 (119905)

119910 (119905 minus 120591)) 119889119905 minus

119899

sum

119895=0

119886119895119899+1

119891 (119905119895 119910119895 V119895)

minus 119886119899+1119899+1

119891 (119905119899+1

119910119901

119899+1 V119899+1

)

10038161003816100381610038161003816100381610038161003816

le1

Γ (120572)

10038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119891 (119905 119910 (119905) 119910 (119905 minus 120591))

minus

119899+1

sum

119895=0

119886119895119899+1

119891 (119905119895 119910 (119905119895) 119910 (119905

119895minus 120591))

10038161003816100381610038161003816100381610038161003816

+

119899

sum

119895=0

119886119895119899+1

10038161003816100381610038161003816119891 (119905119895 119910 (119905119895) 119910 (119905

119895minus120591)) minus 119891 (119905

119895 119910119895 V119895)10038161003816100381610038161003816

+ 119886119899+1119899+1

10038161003816100381610038161003816119891 (119905119899+1

119910 (119905119899+1

) 119910 (119905119899+1

minus 120591))

minus 119891 (119905119899+1

119910119901

119899+1 V119899+1

)10038161003816100381610038161003816

le1

Γ (120572)

[

[

1198621199051205742

119899+1ℎ1205752

+ 119862ℎ119902

119899

sum

119895=0

119886119895119899+1

+ 119886119899+1119899+1

119871(ℎ119902+

1198621198791205741

Γ (120572)ℎ1205751

+119862119879120572

Γ (120572 + 1)ℎ119902)]

]

le1

Γ (120572)[1198621199051205742

119899+1ℎ1205752

+119871ℎ119902+ℎ

120572 (120572 + 1)

+1198621198711198791205741

Γ (120572 + 2)+

119862119879120572ℎ120572+119902

120572Γ (120572 + 2)] = 119862ℎ

119902

(32)

which completes the proof

When 1205751

= 1 1205752

= 2 applying Theorem 3 it is notdifficult to derive the following result on the error bound

6 Journal of Applied Mathematics

Theorem 4 Suppose that 120572 gt 0 and 119891(119905 119910(119905) 119911(119905)) isin

1198621015840[0 119879] then

max0le119895le119873

10038161003816100381610038161003816119910 (119905119895) minus 119910119895

10038161003816100381610038161003816le 119862ℎ119902 (33)

where 119902 = min 1 + 120572 2 and 119862 is a positive constant

Remark 5 We can see from Cases I and II that if the solution119910(119905) is not smooth then the truncation error of V

119899+1 which

approximates to 119910(119905 minus 120591) will not be 119874(ℎ2) Therefore by the

error analysis elaborated in Section 3 we cannot yield thesame conclusion as in Theorem 4 We will consider such aproblem in our future work

4 A Numerical Example

In this section the following delayed FDE is considered

119863120572

119905119910 (119905) =

2

Γ (3 minus 120572)1199052minus120572

minus1

Γ (2 minus 120572)1199051minus120572

+ 2120591119905 minus 1205912

minus 120591 minus 119910 (119905) + 119910 (119905 minus 120591) 120572 isin (0 1)

119910 (119905) = 0 119905 le 0

(34)

Notice that the exact solution to this equation is

119910 (119905) = 1199052minus 119905 (35)

In accordance with delay 120591 being constant or time varyingthe numerical results are displayed in Tables 1 2 3 and4 respectively where 119864

119860denotes the absolute numerical

error and 119864119877denotes the relative numerical error From the

numerical results we can see that the computing errors are ingeneral acceptable for engineering

5 Conclusions

In this paper a numerical algorithm is formulated basedon Adams-Bashforth-Moulton method for fractional-orderdifferential equations with time delay The error analysisof the numerical scheme is carried out meanwhile anumerical example with constant delay and time-varyingdelay is proposed to testify the effectiveness of the schemeThis algorithm can be used not only in the simulation ofdelayed fractional-order differential equations but also in thesimulation of delayed fractional-order control systems

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China under Grants 61004078 61273012 theFoundation of State Key Laboratory for Novel SoftwareTechnology of Nanjing University the China PostdoctoralScience Foundation Funded Project and the Special Fundsfor Postdoctoral Innovative Projects of Shandong Province

References

[1] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993

[2] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[3] P L Butzer and U Westphal ldquoAn introduction to fractionalcalculusrdquo in Applications of Fractional Calculus in Physics pp1ndash85 World Scientific Publishing River Edge NJ USA 2000

[4] Z Wang X Huang Y-X Li and X-N Song ldquoA new imageencryption algorithm based on fractional-order hyperchaoticLorenz systemrdquo Chinese Physics B vol 22 Article ID 0105042013

[5] K BOldhamand J SpanierTheFractional CalculusTheory andApplication of Differentiation and Integration to Arbitrary Ordervol 111 of Mathematics in Science and Engineering AcademicPress New York NY USA 1974

[6] R Hilfer Ed Applications of Fractional Calculus in PhysicsWorld Scientific Publishing River Edge NJ USA 2000

[7] P J Torvik andR L Bagley ldquoOn the appearance of the fractionalderivatives in the behaviour of realmaterialsrdquo Journal of AppliedMechanics vol 51 no 2 pp 294ndash298 1984

[8] F Mainardi ldquoFractional relaxation-oscillation and fractionaldiffusion-wave phenomenardquo Chaos Solitons amp Fractals vol 7no 9 pp 1461ndash1477 1996

[9] N J Ford and A C Simpson ldquoThe numerical solution of frac-tional differential equations speed versus accuracyrdquo NumericalAlgorithms vol 26 no 4 pp 333ndash346 2001

[10] V Daftardar-Gejji and A Babakhani ldquoAnalysis of a systemof fractional differential equationsrdquo Journal of MathematicalAnalysis and Applications vol 293 no 2 pp 511ndash522 2004

[11] K Diethelm and N J Ford ldquoAnalysis of fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 265 no 2 pp 229ndash248 2002

[12] 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

[13] KDiethelmN J Ford andAD Freed ldquoDetailed error analysisfor a fractional Adams methodrdquo Numerical Algorithms vol 36no 1 pp 31ndash52 2004

[14] K Diethelm N J Ford A D Freed and Yu Luchko ldquoAlgo-rithms for the fractional calculus a selection of numericalmethodsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 194 no 6-8 pp 743ndash773 2005

[15] K Diethelm and N J Ford ldquoMulti-order fractional differentialequations and their numerical solutionrdquo Applied Mathematicsand Computation vol 154 no 3 pp 621ndash640 2004

[16] Z Odibat ldquoApproximations of fractional integrals and Caputofractional derivativesrdquo Applied Mathematics and Computationvol 178 no 2 pp 527ndash533 2006

[17] J T Edwards N J Ford and A C Simpson ldquoThe numericalsolution of linear multi-term fractional differential equationssystems of equationsrdquo Journal of Computational and AppliedMathematics vol 148 no 2 pp 401ndash418 2002

[18] K Diethelm ldquoAn investigation of some nonclassical methodsfor the numerical approximation of Caputo-type fractionalderivativesrdquo Numerical Algorithms vol 47 no 4 pp 361ndash3902008

Journal of Applied Mathematics 7

[19] K Diethelm ldquoAn improvement of a nonclassical numericalmethod for the computation of fractional derivativesrdquo Journalof Vibration and Acoustics vol 131 no 1 Article ID 014502 4pages 2009

[20] C Li and C Tao ldquoOn the fractional AdamsmethodrdquoComputersamp Mathematics with Applications vol 58 no 8 pp 1573ndash15882009

[21] R Garrappa ldquoOn some explicit Adams multistep methods forfractional differential equationsrdquo Journal of Computational andApplied Mathematics vol 229 no 2 pp 392ndash399 2009

[22] L Galeone and R Garrappa ldquoFractional Adams-Moultonmethodsrdquo Mathematics and Computers in Simulation vol 79no 4 pp 1358ndash1367 2008

[23] L Galeone and R Garrappa ldquoExplicit methods for fractionaldifferential equations and their stability propertiesrdquo Journal ofComputational and Applied Mathematics vol 228 no 2 pp548ndash560 2009

[24] W Deng ldquoShort memory principle and a predictor-correctorapproach for fractional differential equationsrdquo Journal of Com-putational and AppliedMathematics vol 206 no 1 pp 174ndash1882007

[25] C Li and G Peng ldquoChaos in Chenrsquos system with a fractionalorderrdquo Chaos Solitons amp Fractals vol 22 no 2 pp 443ndash4502004

[26] XWu J Li and G Chen ldquoChaos in the fractional order unifiedsystem and its synchronizationrdquo Journal of the Franklin Institutevol 345 no 4 pp 392ndash401 2008

[27] M S Tavazoei andM Haeri ldquoChaotic attractors in incommen-surate fractional order systemsrdquo Physica D vol 237 no 20 pp2628ndash2637 2008

[28] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A vol 389 no12 pp 2434ndash2442 2010

[29] Y Chen and K L Moore ldquoAnalytical stability bound for aclass of delayed fractional-order dynamic systemsrdquo NonlinearDynamics vol 29 no 1ndash4 pp 191ndash200 2002

[30] P Lanusse H Benlaoukli D Nelson-Gruel and A OustaloupldquoFractional-order control and interval analysis of SISO systemswith time-delayed staterdquo IETControlTheoryampApplications vol2 no 1 pp 16ndash23 2008

[31] X Zhang ldquoSome results of linear fractional order time-delaysystemrdquo Applied Mathematics and Computation vol 197 no 1pp 407ndash411 2008

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

Journal of Applied Mathematics 5

Proof We will prove the result by virtue of mathematicalinduction Suppose that the conclusion is true for 119895 =

0 1 119899 thenwe have to prove that the inequality also holdsfor 119895 = 119899 + 1 To do this we first consider the error of thepredictor 119910

119901

119899+1

From the assumptions (24) we have

10038161003816100381610038161003816119891 (119905119895 119910 (119905119895) 119910 (119905

119895minus 120591)) minus 119891 (119905

119895 119910119895 V119895)10038161003816100381610038161003816

le10038161003816100381610038161003816119891 (119905119895 119910 (119905119895) 119910 (119905

119895minus 120591)) minus 119891 (119905

119895 119910119895 119910 (119905119895minus 120591))

10038161003816100381610038161003816

+10038161003816100381610038161003816119891 (119905119895 119910119895 119910 (119905119895minus 120591)) minus 119891 (119905

119895 119910119895 V119895)10038161003816100381610038161003816

le 1198711ℎ119902+ 1198712ℎ119902= (1198711+ 1198712) ℎ119902

(27)

So

10038161003816100381610038161003816119910 (119905119899+1

) minus 119910119901

119899+1

10038161003816100381610038161003816

=1

Γ (120572)

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119891 (119905 119910 (119905) 119910 (119905 minus 120591)) 119889119905

minus

119899

sum

119895=0

119887119895119896+1

119891 (119905119895 119910119895 V119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

Γ (120572)

10038161003816100381610038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119863120572

119905119910 (119905) 119889119905 minus

119899

sum

119895=0

119887119895119899+1

119863120572

119905119910 (119905119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

+1

Γ (120572)

times

119899

sum

119895=0

119887119895119899+1

10038161003816100381610038161003816119891 (119905119895 119910 (119905119895) 119910 (119905

119895minus 120591))

minus119891 (119905119895 119910119895 119910 (119905119895minus 120591))

10038161003816100381610038161003816

+10038161003816100381610038161003816119891 (119905119895 119910119895 119910 (119905119895minus 120591)) minus 119891 (119905

119895 119910119895 V119895)10038161003816100381610038161003816

le1198621199051205741

119899+1

Γ (120572)ℎ1205751

+1

Γ (120572)

119899

sum

119895=0

119887119895119899+1

(1198711+ 1198712) 119862ℎ119902

(28)

and from

119899

sum

119895=0

119887119895119899+1

=

119899

sum

119895=0

int

(119895+1)ℎ

119895ℎ

(119905119899+1

minus 119905)120572minus1

119889119905 =1

120572119905120572

119899+1le

1

120572119879120572

(29)

we have

10038161003816100381610038161003816119910 (119905119899+1

) minus 119910119901

119899+1

10038161003816100381610038161003816le

1198621198791205741

Γ (120572)ℎ1205751

+119862119879120572

Γ (120572 + 1)ℎ119902 (30)

Since119899

sum

119895=0

119886119895119899+1

le

119899

sum

119895=0

ℎ120572

120572 (120572 + 1)

times [(119896 minus 119895 + 2)120572+1

minus (119896 minus 119895 + 1)120572+1

]

minus [(119896 minus 119895 + 1)120572+1

minus (119896 minus 119895)120572+1

]

= [int

119905119899+1

0

(119905119899+2

minus 119905)120572119889119905 minus int

119905119899+1

0

(119905119899+1

minus 119905)120572119889119905]

=1

120572int

119905119899+1

0

[(119905119899+1

minus 119905)120572]1015840

(119905) 119889119905 =1

120572119905120572

119899+1le 119879120572

(31)

we have1003816100381610038161003816119910 (119905119899+1

) minus 119910119899+1

1003816100381610038161003816

=1

Γ (120572)

10038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119891 (119905 119910 (119905)

119910 (119905 minus 120591)) 119889119905 minus

119899

sum

119895=0

119886119895119899+1

119891 (119905119895 119910119895 V119895)

minus 119886119899+1119899+1

119891 (119905119899+1

119910119901

119899+1 V119899+1

)

10038161003816100381610038161003816100381610038161003816

le1

Γ (120572)

10038161003816100381610038161003816100381610038161003816

int

119905119899+1

0

(119905119899+1

minus 119905)120572minus1

119891 (119905 119910 (119905) 119910 (119905 minus 120591))

minus

119899+1

sum

119895=0

119886119895119899+1

119891 (119905119895 119910 (119905119895) 119910 (119905

119895minus 120591))

10038161003816100381610038161003816100381610038161003816

+

119899

sum

119895=0

119886119895119899+1

10038161003816100381610038161003816119891 (119905119895 119910 (119905119895) 119910 (119905

119895minus120591)) minus 119891 (119905

119895 119910119895 V119895)10038161003816100381610038161003816

+ 119886119899+1119899+1

10038161003816100381610038161003816119891 (119905119899+1

119910 (119905119899+1

) 119910 (119905119899+1

minus 120591))

minus 119891 (119905119899+1

119910119901

119899+1 V119899+1

)10038161003816100381610038161003816

le1

Γ (120572)

[

[

1198621199051205742

119899+1ℎ1205752

+ 119862ℎ119902

119899

sum

119895=0

119886119895119899+1

+ 119886119899+1119899+1

119871(ℎ119902+

1198621198791205741

Γ (120572)ℎ1205751

+119862119879120572

Γ (120572 + 1)ℎ119902)]

]

le1

Γ (120572)[1198621199051205742

119899+1ℎ1205752

+119871ℎ119902+ℎ

120572 (120572 + 1)

+1198621198711198791205741

Γ (120572 + 2)+

119862119879120572ℎ120572+119902

120572Γ (120572 + 2)] = 119862ℎ

119902

(32)

which completes the proof

When 1205751

= 1 1205752

= 2 applying Theorem 3 it is notdifficult to derive the following result on the error bound

6 Journal of Applied Mathematics

Theorem 4 Suppose that 120572 gt 0 and 119891(119905 119910(119905) 119911(119905)) isin

1198621015840[0 119879] then

max0le119895le119873

10038161003816100381610038161003816119910 (119905119895) minus 119910119895

10038161003816100381610038161003816le 119862ℎ119902 (33)

where 119902 = min 1 + 120572 2 and 119862 is a positive constant

Remark 5 We can see from Cases I and II that if the solution119910(119905) is not smooth then the truncation error of V

119899+1 which

approximates to 119910(119905 minus 120591) will not be 119874(ℎ2) Therefore by the

error analysis elaborated in Section 3 we cannot yield thesame conclusion as in Theorem 4 We will consider such aproblem in our future work

4 A Numerical Example

In this section the following delayed FDE is considered

119863120572

119905119910 (119905) =

2

Γ (3 minus 120572)1199052minus120572

minus1

Γ (2 minus 120572)1199051minus120572

+ 2120591119905 minus 1205912

minus 120591 minus 119910 (119905) + 119910 (119905 minus 120591) 120572 isin (0 1)

119910 (119905) = 0 119905 le 0

(34)

Notice that the exact solution to this equation is

119910 (119905) = 1199052minus 119905 (35)

In accordance with delay 120591 being constant or time varyingthe numerical results are displayed in Tables 1 2 3 and4 respectively where 119864

119860denotes the absolute numerical

error and 119864119877denotes the relative numerical error From the

numerical results we can see that the computing errors are ingeneral acceptable for engineering

5 Conclusions

In this paper a numerical algorithm is formulated basedon Adams-Bashforth-Moulton method for fractional-orderdifferential equations with time delay The error analysisof the numerical scheme is carried out meanwhile anumerical example with constant delay and time-varyingdelay is proposed to testify the effectiveness of the schemeThis algorithm can be used not only in the simulation ofdelayed fractional-order differential equations but also in thesimulation of delayed fractional-order control systems

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China under Grants 61004078 61273012 theFoundation of State Key Laboratory for Novel SoftwareTechnology of Nanjing University the China PostdoctoralScience Foundation Funded Project and the Special Fundsfor Postdoctoral Innovative Projects of Shandong Province

References

[1] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993

[2] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[3] P L Butzer and U Westphal ldquoAn introduction to fractionalcalculusrdquo in Applications of Fractional Calculus in Physics pp1ndash85 World Scientific Publishing River Edge NJ USA 2000

[4] Z Wang X Huang Y-X Li and X-N Song ldquoA new imageencryption algorithm based on fractional-order hyperchaoticLorenz systemrdquo Chinese Physics B vol 22 Article ID 0105042013

[5] K BOldhamand J SpanierTheFractional CalculusTheory andApplication of Differentiation and Integration to Arbitrary Ordervol 111 of Mathematics in Science and Engineering AcademicPress New York NY USA 1974

[6] R Hilfer Ed Applications of Fractional Calculus in PhysicsWorld Scientific Publishing River Edge NJ USA 2000

[7] P J Torvik andR L Bagley ldquoOn the appearance of the fractionalderivatives in the behaviour of realmaterialsrdquo Journal of AppliedMechanics vol 51 no 2 pp 294ndash298 1984

[8] F Mainardi ldquoFractional relaxation-oscillation and fractionaldiffusion-wave phenomenardquo Chaos Solitons amp Fractals vol 7no 9 pp 1461ndash1477 1996

[9] N J Ford and A C Simpson ldquoThe numerical solution of frac-tional differential equations speed versus accuracyrdquo NumericalAlgorithms vol 26 no 4 pp 333ndash346 2001

[10] V Daftardar-Gejji and A Babakhani ldquoAnalysis of a systemof fractional differential equationsrdquo Journal of MathematicalAnalysis and Applications vol 293 no 2 pp 511ndash522 2004

[11] K Diethelm and N J Ford ldquoAnalysis of fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 265 no 2 pp 229ndash248 2002

[12] 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

[13] KDiethelmN J Ford andAD Freed ldquoDetailed error analysisfor a fractional Adams methodrdquo Numerical Algorithms vol 36no 1 pp 31ndash52 2004

[14] K Diethelm N J Ford A D Freed and Yu Luchko ldquoAlgo-rithms for the fractional calculus a selection of numericalmethodsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 194 no 6-8 pp 743ndash773 2005

[15] K Diethelm and N J Ford ldquoMulti-order fractional differentialequations and their numerical solutionrdquo Applied Mathematicsand Computation vol 154 no 3 pp 621ndash640 2004

[16] Z Odibat ldquoApproximations of fractional integrals and Caputofractional derivativesrdquo Applied Mathematics and Computationvol 178 no 2 pp 527ndash533 2006

[17] J T Edwards N J Ford and A C Simpson ldquoThe numericalsolution of linear multi-term fractional differential equationssystems of equationsrdquo Journal of Computational and AppliedMathematics vol 148 no 2 pp 401ndash418 2002

[18] K Diethelm ldquoAn investigation of some nonclassical methodsfor the numerical approximation of Caputo-type fractionalderivativesrdquo Numerical Algorithms vol 47 no 4 pp 361ndash3902008

Journal of Applied Mathematics 7

[19] K Diethelm ldquoAn improvement of a nonclassical numericalmethod for the computation of fractional derivativesrdquo Journalof Vibration and Acoustics vol 131 no 1 Article ID 014502 4pages 2009

[20] C Li and C Tao ldquoOn the fractional AdamsmethodrdquoComputersamp Mathematics with Applications vol 58 no 8 pp 1573ndash15882009

[21] R Garrappa ldquoOn some explicit Adams multistep methods forfractional differential equationsrdquo Journal of Computational andApplied Mathematics vol 229 no 2 pp 392ndash399 2009

[22] L Galeone and R Garrappa ldquoFractional Adams-Moultonmethodsrdquo Mathematics and Computers in Simulation vol 79no 4 pp 1358ndash1367 2008

[23] L Galeone and R Garrappa ldquoExplicit methods for fractionaldifferential equations and their stability propertiesrdquo Journal ofComputational and Applied Mathematics vol 228 no 2 pp548ndash560 2009

[24] W Deng ldquoShort memory principle and a predictor-correctorapproach for fractional differential equationsrdquo Journal of Com-putational and AppliedMathematics vol 206 no 1 pp 174ndash1882007

[25] C Li and G Peng ldquoChaos in Chenrsquos system with a fractionalorderrdquo Chaos Solitons amp Fractals vol 22 no 2 pp 443ndash4502004

[26] XWu J Li and G Chen ldquoChaos in the fractional order unifiedsystem and its synchronizationrdquo Journal of the Franklin Institutevol 345 no 4 pp 392ndash401 2008

[27] M S Tavazoei andM Haeri ldquoChaotic attractors in incommen-surate fractional order systemsrdquo Physica D vol 237 no 20 pp2628ndash2637 2008

[28] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A vol 389 no12 pp 2434ndash2442 2010

[29] Y Chen and K L Moore ldquoAnalytical stability bound for aclass of delayed fractional-order dynamic systemsrdquo NonlinearDynamics vol 29 no 1ndash4 pp 191ndash200 2002

[30] P Lanusse H Benlaoukli D Nelson-Gruel and A OustaloupldquoFractional-order control and interval analysis of SISO systemswith time-delayed staterdquo IETControlTheoryampApplications vol2 no 1 pp 16ndash23 2008

[31] X Zhang ldquoSome results of linear fractional order time-delaysystemrdquo Applied Mathematics and Computation vol 197 no 1pp 407ndash411 2008

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 Journal of Applied Mathematics

Theorem 4 Suppose that 120572 gt 0 and 119891(119905 119910(119905) 119911(119905)) isin

1198621015840[0 119879] then

max0le119895le119873

10038161003816100381610038161003816119910 (119905119895) minus 119910119895

10038161003816100381610038161003816le 119862ℎ119902 (33)

where 119902 = min 1 + 120572 2 and 119862 is a positive constant

Remark 5 We can see from Cases I and II that if the solution119910(119905) is not smooth then the truncation error of V

119899+1 which

approximates to 119910(119905 minus 120591) will not be 119874(ℎ2) Therefore by the

error analysis elaborated in Section 3 we cannot yield thesame conclusion as in Theorem 4 We will consider such aproblem in our future work

4 A Numerical Example

In this section the following delayed FDE is considered

119863120572

119905119910 (119905) =

2

Γ (3 minus 120572)1199052minus120572

minus1

Γ (2 minus 120572)1199051minus120572

+ 2120591119905 minus 1205912

minus 120591 minus 119910 (119905) + 119910 (119905 minus 120591) 120572 isin (0 1)

119910 (119905) = 0 119905 le 0

(34)

Notice that the exact solution to this equation is

119910 (119905) = 1199052minus 119905 (35)

In accordance with delay 120591 being constant or time varyingthe numerical results are displayed in Tables 1 2 3 and4 respectively where 119864

119860denotes the absolute numerical

error and 119864119877denotes the relative numerical error From the

numerical results we can see that the computing errors are ingeneral acceptable for engineering

5 Conclusions

In this paper a numerical algorithm is formulated basedon Adams-Bashforth-Moulton method for fractional-orderdifferential equations with time delay The error analysisof the numerical scheme is carried out meanwhile anumerical example with constant delay and time-varyingdelay is proposed to testify the effectiveness of the schemeThis algorithm can be used not only in the simulation ofdelayed fractional-order differential equations but also in thesimulation of delayed fractional-order control systems

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China under Grants 61004078 61273012 theFoundation of State Key Laboratory for Novel SoftwareTechnology of Nanjing University the China PostdoctoralScience Foundation Funded Project and the Special Fundsfor Postdoctoral Innovative Projects of Shandong Province

References

[1] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993

[2] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[3] P L Butzer and U Westphal ldquoAn introduction to fractionalcalculusrdquo in Applications of Fractional Calculus in Physics pp1ndash85 World Scientific Publishing River Edge NJ USA 2000

[4] Z Wang X Huang Y-X Li and X-N Song ldquoA new imageencryption algorithm based on fractional-order hyperchaoticLorenz systemrdquo Chinese Physics B vol 22 Article ID 0105042013

[5] K BOldhamand J SpanierTheFractional CalculusTheory andApplication of Differentiation and Integration to Arbitrary Ordervol 111 of Mathematics in Science and Engineering AcademicPress New York NY USA 1974

[6] R Hilfer Ed Applications of Fractional Calculus in PhysicsWorld Scientific Publishing River Edge NJ USA 2000

[7] P J Torvik andR L Bagley ldquoOn the appearance of the fractionalderivatives in the behaviour of realmaterialsrdquo Journal of AppliedMechanics vol 51 no 2 pp 294ndash298 1984

[8] F Mainardi ldquoFractional relaxation-oscillation and fractionaldiffusion-wave phenomenardquo Chaos Solitons amp Fractals vol 7no 9 pp 1461ndash1477 1996

[9] N J Ford and A C Simpson ldquoThe numerical solution of frac-tional differential equations speed versus accuracyrdquo NumericalAlgorithms vol 26 no 4 pp 333ndash346 2001

[10] V Daftardar-Gejji and A Babakhani ldquoAnalysis of a systemof fractional differential equationsrdquo Journal of MathematicalAnalysis and Applications vol 293 no 2 pp 511ndash522 2004

[11] K Diethelm and N J Ford ldquoAnalysis of fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 265 no 2 pp 229ndash248 2002

[12] 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

[13] KDiethelmN J Ford andAD Freed ldquoDetailed error analysisfor a fractional Adams methodrdquo Numerical Algorithms vol 36no 1 pp 31ndash52 2004

[14] K Diethelm N J Ford A D Freed and Yu Luchko ldquoAlgo-rithms for the fractional calculus a selection of numericalmethodsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 194 no 6-8 pp 743ndash773 2005

[15] K Diethelm and N J Ford ldquoMulti-order fractional differentialequations and their numerical solutionrdquo Applied Mathematicsand Computation vol 154 no 3 pp 621ndash640 2004

[16] Z Odibat ldquoApproximations of fractional integrals and Caputofractional derivativesrdquo Applied Mathematics and Computationvol 178 no 2 pp 527ndash533 2006

[17] J T Edwards N J Ford and A C Simpson ldquoThe numericalsolution of linear multi-term fractional differential equationssystems of equationsrdquo Journal of Computational and AppliedMathematics vol 148 no 2 pp 401ndash418 2002

[18] K Diethelm ldquoAn investigation of some nonclassical methodsfor the numerical approximation of Caputo-type fractionalderivativesrdquo Numerical Algorithms vol 47 no 4 pp 361ndash3902008

Journal of Applied Mathematics 7

[19] K Diethelm ldquoAn improvement of a nonclassical numericalmethod for the computation of fractional derivativesrdquo Journalof Vibration and Acoustics vol 131 no 1 Article ID 014502 4pages 2009

[20] C Li and C Tao ldquoOn the fractional AdamsmethodrdquoComputersamp Mathematics with Applications vol 58 no 8 pp 1573ndash15882009

[21] R Garrappa ldquoOn some explicit Adams multistep methods forfractional differential equationsrdquo Journal of Computational andApplied Mathematics vol 229 no 2 pp 392ndash399 2009

[22] L Galeone and R Garrappa ldquoFractional Adams-Moultonmethodsrdquo Mathematics and Computers in Simulation vol 79no 4 pp 1358ndash1367 2008

[23] L Galeone and R Garrappa ldquoExplicit methods for fractionaldifferential equations and their stability propertiesrdquo Journal ofComputational and Applied Mathematics vol 228 no 2 pp548ndash560 2009

[24] W Deng ldquoShort memory principle and a predictor-correctorapproach for fractional differential equationsrdquo Journal of Com-putational and AppliedMathematics vol 206 no 1 pp 174ndash1882007

[25] C Li and G Peng ldquoChaos in Chenrsquos system with a fractionalorderrdquo Chaos Solitons amp Fractals vol 22 no 2 pp 443ndash4502004

[26] XWu J Li and G Chen ldquoChaos in the fractional order unifiedsystem and its synchronizationrdquo Journal of the Franklin Institutevol 345 no 4 pp 392ndash401 2008

[27] M S Tavazoei andM Haeri ldquoChaotic attractors in incommen-surate fractional order systemsrdquo Physica D vol 237 no 20 pp2628ndash2637 2008

[28] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A vol 389 no12 pp 2434ndash2442 2010

[29] Y Chen and K L Moore ldquoAnalytical stability bound for aclass of delayed fractional-order dynamic systemsrdquo NonlinearDynamics vol 29 no 1ndash4 pp 191ndash200 2002

[30] P Lanusse H Benlaoukli D Nelson-Gruel and A OustaloupldquoFractional-order control and interval analysis of SISO systemswith time-delayed staterdquo IETControlTheoryampApplications vol2 no 1 pp 16ndash23 2008

[31] X Zhang ldquoSome results of linear fractional order time-delaysystemrdquo Applied Mathematics and Computation vol 197 no 1pp 407ndash411 2008

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

Journal of Applied Mathematics 7

[19] K Diethelm ldquoAn improvement of a nonclassical numericalmethod for the computation of fractional derivativesrdquo Journalof Vibration and Acoustics vol 131 no 1 Article ID 014502 4pages 2009

[20] C Li and C Tao ldquoOn the fractional AdamsmethodrdquoComputersamp Mathematics with Applications vol 58 no 8 pp 1573ndash15882009

[21] R Garrappa ldquoOn some explicit Adams multistep methods forfractional differential equationsrdquo Journal of Computational andApplied Mathematics vol 229 no 2 pp 392ndash399 2009

[22] L Galeone and R Garrappa ldquoFractional Adams-Moultonmethodsrdquo Mathematics and Computers in Simulation vol 79no 4 pp 1358ndash1367 2008

[23] L Galeone and R Garrappa ldquoExplicit methods for fractionaldifferential equations and their stability propertiesrdquo Journal ofComputational and Applied Mathematics vol 228 no 2 pp548ndash560 2009

[24] W Deng ldquoShort memory principle and a predictor-correctorapproach for fractional differential equationsrdquo Journal of Com-putational and AppliedMathematics vol 206 no 1 pp 174ndash1882007

[25] C Li and G Peng ldquoChaos in Chenrsquos system with a fractionalorderrdquo Chaos Solitons amp Fractals vol 22 no 2 pp 443ndash4502004

[26] XWu J Li and G Chen ldquoChaos in the fractional order unifiedsystem and its synchronizationrdquo Journal of the Franklin Institutevol 345 no 4 pp 392ndash401 2008

[27] M S Tavazoei andM Haeri ldquoChaotic attractors in incommen-surate fractional order systemsrdquo Physica D vol 237 no 20 pp2628ndash2637 2008

[28] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A vol 389 no12 pp 2434ndash2442 2010

[29] Y Chen and K L Moore ldquoAnalytical stability bound for aclass of delayed fractional-order dynamic systemsrdquo NonlinearDynamics vol 29 no 1ndash4 pp 191ndash200 2002

[30] P Lanusse H Benlaoukli D Nelson-Gruel and A OustaloupldquoFractional-order control and interval analysis of SISO systemswith time-delayed staterdquo IETControlTheoryampApplications vol2 no 1 pp 16ndash23 2008

[31] X Zhang ldquoSome results of linear fractional order time-delaysystemrdquo Applied Mathematics and Computation vol 197 no 1pp 407ndash411 2008

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