Research Article An Operational Matrix Technique for ...downloads.hindawi.com/journals/amp/2016/6345978.pdf · Research Article An Operational Matrix Technique for Solving Variable

Research ArticleAn Operational Matrix Technique for Solving VariableOrder Fractional Differential-Integral Equation Based onthe Second Kind of Chebyshev Polynomials

Jianping Liu Xia Li and Limeng Wu

Hebei Normal University of Science and Technology Qinhuangdao 066004 China

Correspondence should be addressed to Jianping Liu liujianping0408126com

Received 12 March 2016 Accepted 11 May 2016

Academic Editor Eugen Radu

Copyright copy 2016 Jianping Liu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

An operational matrix technique is proposed to solve variable order fractional differential-integral equation based on the secondkind of Chebyshev polynomials in this paperThe differential operational matrix and integral operational matrix are derived basedon the second kind of Chebyshev polynomials Using two types of operational matrixes the original equation is transformed intothe arithmetic product of several dependent matrixes which can be viewed as an algebraic system after adopting the collocationpoints Further numerical solution of original equation is obtained by solving the algebraic system Finally several examples showthat the numerical algorithm is computationally efficient

1 Introduction

Fractional differential equation (FDE) is an extension ofinteger-order model Compared with the classical integer-order differential equation FDE provides an excellent instru-ment for the description of memory and hereditary prop-erties of various materials and processes Therefore it maybe more accurate for modeling by FDE rather than integer-order differential equation Researchers have pointed outthat the fractional calculus plays an important role in mod-eling and many systems in interdisciplinary fields can beelegantly described with the help of fractional derivativesuch as dynamics of earthquakes [1] viscoelastic systems [2]biological systems [3 4] diffusion model [5] chaos [6] wavepropagation [7] and partial bed-load transport [8]

Many literatures study the numerical algorithms forsolving fractional calculus equations Ahmadian et al [9]proposed a computational method based on Jacobi polyno-mials for solving fuzzy linear FDE on interval [0 1] Fu etal [10] presented a Laplace transformed boundary particlemethod for numerical modeling of time fractional diffusionequations Pang et al [11] applied radial basis function mesh-less collocation method to the space-fractional advection-dispersion equations Bhrawy and Zaky [12] analyzed an

operational formulation based on Jacobi polynomials fortime-space FDEwith Dirichlet boundary conditions Bhrawyet al [13 14] introduced a fractional integral operator matrixbased on generalized Laguerre polynomials and shifted firstof Chebyshev polynomials for solving fractional integralequations

With further development of science research more andmore researchers have found that a variety of importantdynamical problems exhibit fractional order behavior thatmay vary with time or space This fact indicates that variableorder calculus is a natural candidate to provide an effectivemathematical framework for the description of complexdynamical problems The modeling and application of vari-able order FDE have been a front subject However since thekernel of the variable order operators has a variable exponentanalytical solution of variable order FDE is usually difficultto obtain The development of numerical algorithms to solvevariable order FDE is necessary

Only a few authors studied numericalmethods of variableorder FDE Shen et al [15] have given an approximatescheme for the variable order time fractional diffusion equa-tion Chen et al [16 17] paid their attention to Bernsteinpolynomials to solve variable order linear cable equationand variable order time fractional diffusion equation An

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2016 Article ID 6345978 9 pageshttpdxdoiorg10115520166345978

2 Advances in Mathematical Physics

alternating direct method for the two-dimensional variableorder fractional percolation equation was proposed in [18]Explicit and implicit Euler approximations for FDE wereintroduced in [19] A numerical method based on Legendrepolynomials was presented for a class of variable orderFDEs in [20] Chen et al [21] introduced the numericalsolution for a class of nonlinear variable order FDEs withLegendre wavelets In addition for most literatures theysolved variable order FDE which is defined on the interval[0 1] It is noteworthy that the Chebyshev polynomialsfamily have beneficial properties so that they are widelyused in approximation theory However the second kindof Chebyshev polynomials have been paid less attention forsolving variable order FDE Accordingly we will solve a kindof variable order fractional differential-integral equations(FDIEs) defined on the interval [0 119877] (119877 gt 0) based on thesecond kind of Chebyshev polynomials The FDIE is shownas follows

119863120572(119905)

119910 (119905) + 119886 (119905) int119905

0

119910 (119911) 119889119911 + 119887 (119905) 1199101015840

(119905) + 119888 (119905) 119910 (119905)

= 119892 (119905) 119910 (0) = 1199100 119905 isin [0 119877]

(1)

where 119863120572(119905)119910(119905) is fractional derivative of 119910(119905) in Caputorsquossense

The basic idea of this approach is that we derive thedifferential operational matrix and the integral operationalmatrix based on Chebyshev polynomials With the opera-tionalmatrixes (1) is transformed into the products of severaldependent matrixes which can be viewed as an algebraicsystem after taking the collocation points By solving thealgebraic system the numerical solution of (1) is acquiredSince the second kind of Chebyshev polynomials are orthog-onal to each other the operational matrixes based on Cheby-shev polynomials greatly reduce the size of computationalwork while accurately providing the series solution Fromsome numerical examples we can see that our results arein good agreement with exact solution Numerical resultdemonstrates the validity of this algorithm

The paper is organized as follows In Section 2 somenecessary preliminaries are introduced In Section 3 the basicdefinition and property of the second kind of Chebyshevpolynomials are given In Section 4 function approximationis given In Section 5 two kinds of operational matrixes arederived and we applied the operational matrixes to solve theequation as given at the beginning In Section 6 we presentsome numerical examples to demonstrate the efficiency of thealgorithm We end the paper with a few concluding remarksin Section 7

2 Preliminaries

There are several definitions for variable order fractionalderivatives such as the one in Riemann-Liouvillersquos sense andthe one in Caputorsquos sense [22] In this paper the definition inCaputorsquos sense is considered

Definition 1 Caputo fractional derivate with order 120572(119905) isdefined by

119863120572(119905)

119910 (119905) =1

Γ (1 minus 120572 (119905))int119905

0+

(119905 minus 120591)minus120572(119905)

1199101015840

(120591) 119889120591

+119910 (0+) minus 119910 (0minus)

Γ (1 minus 120572 (119905))119905minus120572(119905)

(2)

If we assume the starting time in a perfect situation we canget Definition 2 as follows

Definition 2 Consider the following

119863120572(119905)

119910 (119905) =1

Γ (1 minus 120572 (119905))int119905

0

(119905 minus 120591)minus120572(119905)

1199101015840

(120591) 119889120591 (3)

By Definition 2 we can get the following formula [16]

119863120572(119905)

119905119899

=

Γ (119899 + 1)

Γ (119899 + 1 minus 120572 (119905))119905119899minus120572(119905)

119899 = 1 2

0 119899 = 0

(4)

3 The Shifted Second Kind ofChebyshev Polynomials

Thesecondkind ofChebyshev polynomials are definedon theinterval 119868 = [minus1 1] They are orthogonal to each other withrespect to the weight function 120596(119909) = radic1 minus 1199092 They satisfythe following formulas

1198800(119909) = 1

1198801(119909) = 2119909

119880119899+1

(119909) = 2119909119880119899(119909) minus 119880

119899minus1(119909)

119899 = 1 2

int1

minus1

radic1 minus 1199092119880119899(119909)119880119898(119909) 119889119909 =

120587

2 119898 = 119899

0 119898 = 119899

(5)

For 119905 isin [0 119877] let 119909 = 2119905119877 minus 1 We can get shifted secondkind of Chebyshev polynomials

119899(119905) = 119880

119899(2119905119877 minus 1) which

are also orthogonal with respect to weight function 120596(119905) =

radic119905119877 minus 1199052 for 119905 isin [0 119877] They satisfy the following formulas

0(119905) = 1

1(119905) =

4119905

119877minus 2

119899+1

(119905) = 2 (2119905

119877minus 1)

119899(119905) minus

119899minus1(119905)

119899 = 1 2

int119877

0

radic119905119877 minus 1199052119899(119905) 119898(119905) 119889119905 =

120587

81198772

119898 = 119899

0 119898 = 119899

(6)

Advances in Mathematical Physics 3

The shifted second kind of Chebyshev polynomials canalso be expressed as

119899(119905)

=

1 119899 = 0

[1198992]

sum119896=0

(minus1)119896

(119899 minus 119896)

119896 (119899 minus 2119896)(4119905

119877minus 2)119899minus2119896

119899 ge 1

(7)

where [1198992] denotes the maximum integer which is no morethan 1198992

Let Ψ119899(119905) = [

0(119905) 1(119905)

119899(119905)]119879 and T

119899(119905) =

[1 119905 119905119899

]119879 Therefore

Ψ119899(119905) = AT

119899(119905) (8)

where

A = BC

C =

[[[[[[[[[[[[[[

[

1 0 0 sdot sdot sdot 0

1198620

1(minus

119877

2)1minus0

1198621

1(minus

119877

2)1minus1

0 sdot sdot sdot 0

1198620

2(minus

119877

2)2minus0

1198621

2(minus

119877

2)2minus1

1198622

2(minus

119877

2)2minus2

sdot sdot sdot 0

1198620

119899(minus

119877

2)119899minus0

1198621

119899(minus

119877

2)119899minus1

1198622

119899(minus

119877

2)119899minus2

sdot sdot sdot 119862119899

119899(minus

119877

2)119899minus119899

]]]]]]]]]]]]]]

]

(9)

If 119899 is an even number then

B

=

[[[[[[[[[[[[[[

[


0 (minus1)0

(1 minus 0)

0 (1 minus 0)(4

119877)1minus0

0 sdot sdot sdot 0

(minus1)1

(2 minus 1)

1 (2 minus 2)(4

119877)2minus2

0 (minus1)0

(2 minus 0)

0 (2 minus 0)(4

119877)2minus0

sdot sdot sdot 0

(minus1)1198992

(119899 minus 1198992)

(1198992) (119899 minus 2 sdot (1198992))(4

119877)119899minus2sdot(1198992)

sdot sdot sdot (minus1)1198992minus1

(119899 minus 1198992 + 1)

(1198992 minus 1) [119899 minus 2 sdot (1198992 minus 1)](4

119877)119899minus2sdot(1198992minus1)

sdot sdot sdot (minus1)0

(119899 minus 0)

0 (119899 minus 0)(4

119877)119899minus0

]]]]]]]]]]]]]]

]

(10)

If 119899 is an odd number then

B

=

[[[[[[[[[[[[[[

[


0 (minus1)0

(1 minus 0)

0 (1 minus 0)(4

119877)1minus0

0 sdot sdot sdot 0

(minus1)1

(2 minus 1)

1 (2 minus 2)(4

119877)2minus2

0 (minus1)0

(2 minus 0)

0 (2 minus 0)(4

119877)2minus0

sdot sdot sdot 0

0 (minus1)(119899minus1)2

(119899 minus (119899 minus 1) 2)

((119899 minus 1) 2) (119899 minus 2 sdot ((119899 minus 1) 2))(4

119877)119899minus2sdot((119899minus1)2)

0 sdot sdot sdot (minus1)0

(119899 minus 0)

0 (119899 minus 0)(4

119877)119899minus0

]]]]]]]]]]]]]]

]

(11)

Therefore we can easily gain (12) as follows

T119899(119905) = Aminus1Ψ

119899(119905) (12)

4 Function Approximation

Theorem 3 Suppose that the function 119910(119905) is 119899 + 1 timescontinuously differentiable on the interval [0 119877] Let 119906

119899(119905) =


sum119899

119894=0120582119894119894(119905) = Λ

119879

Ψ119899(119905) be the best square approximation

function of 119910(119905) where Λ = [1205820 1205821 120582

119899]119879 Then

1003817100381710038171003817119910 (119905) minus 119906119899 (119905)10038171003817100381710038172 le

119872119878119899+1

119877

(119899 + 1)radic120587

8 (13)

where119872 = max119905isin[0119877]

119910119899+1

(119905) and 119878 = max119877 minus 1199050 1199050

Proof We consider the Taylor polynomials

119910 (119905) = 119910 (1199050) + 1199101015840

(1199050) (119905 minus 119905

0) + sdot sdot sdot

+ 119910(119899)

(1199050)(119905 minus 1199050)119899

119899+ 119910(119899+1)

(120578)(119905 minus 1199050)119899+1

(119899 + 1)

1199050isin [0 119877]

(14)

where 120578 is between 119905 and 1199050

Let

119901119899(119905) = 119910 (119905

0) + 1199101015840

(1199050) (119905 minus 119905

0) + sdot sdot sdot

+ 119910(119899)

(1199050)(119905 minus 1199050)119899

119899

(15)

Then

1003816100381610038161003816119910 (119905) minus 119901119899 (119905)1003816100381610038161003816 =

1003816100381610038161003816100381610038161003816100381610038161003816

119910(119899+1)

(120578)(119905 minus 1199050)119899+1

(119899 + 1)

1003816100381610038161003816100381610038161003816100381610038161003816

(16)

Since 119906119899(119905) = sum

119899

119894=0120582119894119894(119905) = Λ

119879

Ψ119899(119905) is the best square

approximation function of 119910(119905) we can gain

1003817100381710038171003817119910 (119905) minus 119906119899 (119905)10038171003817100381710038172

2le1003817100381710038171003817119910 (119905) minus 119901119899 (119905)

10038171003817100381710038172

2

= int119877

0

120596 (119905) [119910 (119905) minus 119901119899(119905)]2

119889119905

= int119877

0

120596 (119905) [119910(119899+1)

(120578)(119905 minus 1199050)119899+1

(119899 + 1)]

2

119889119905

le1198722

[(119899 + 1)]2int119877

0

120596 (119905) (119905 minus 1199050)2119899+2

119889119905

=1198722

[(119899 + 1)]2int119877

0

radic119905119877 minus 1199052 (119905 minus 1199050)2119899+2

119889119905

(17)

Let 119878 = max119877 minus 1199050 1199050 Therefore

1003817100381710038171003817119910 (119905) minus 119906119899 (119905)10038171003817100381710038172

2le

1198722

1198782119899+2

[(119899 + 1)]2int119877

0

radic119905119877 minus 1199052119889119905

=1198722

1198782119899+2

1198772

(119899 + 1) 2120587

8

(18)

Finally by taking the square roots Theorem 3 can beproved

5 Operational Matrix Techniquefor Solving Variable Order FractionalDifferential-Integral Equation

51 Differential Operational Matrix of 119863120572(119905)Ψ119899(119905) Consider-

ing (8) we obtain

119863120572(119905)

Ψ119899(119905) = 119863

120572(119905)

(AT119899(119905))

= A119863120572(119905) ([1 119905 119905119899]119879) (19)

Then according to (4) we can get

119863120572(119905)

([1 119905 119905119899

]119879

)

= [0Γ (2)

Γ (2 minus 120572 (119905))1199051minus120572(119905)

Γ (119899 + 1)

Γ (119899 + 1 minus 120572 (119905))119905119899minus120572(119905)

]

119879

=

[[[[[[[[[

[

0 0 sdot sdot sdot 0

0Γ (2)

Γ (2 minus 120572 (119905))119905minus120572(119905)

sdot sdot sdot 0

0 0 sdot sdot sdotΓ (119899 + 1)

Γ (119899 + 1 minus 120572 (119905))119905minus120572(119905)

]]]]]]]]]

]

[[[[[[

[

1

119905

119905119899

]]]]]]

]

= M120572(119905)

Aminus1Ψ119899(119905)

(20)

where

M120572(119905)

=

[[[[[[[[[

[


0Γ (2)

Γ (2 minus 120572 (119905))119905minus120572(119905)

sdot sdot sdot 0


Γ (119899 + 1 minus 120572 (119905))119905minus120572(119905)

]]]]]]]]]

]

(21)

Therefore

119863120572(119905)

Ψ119899(119905) = AM

120572(119905)Aminus1Ψ119899(119905) (22)

AM120572(119905)

Aminus1 is called the operational matrix of119863120572(119905)Ψ119899(119905)

In particular for 120572(119905) = 1 we can get

Ψ1015840

119899(119905) = AM

1Aminus1Ψ119899(119905) (23)

52 Integral Operational Matrix of int119905

0

Ψ119899(119911)119889119911 Let

int119905

0

Ψ119899(119911)119889119911 asymp PΨ

119899(119905) P is called the integral operational

matrixThe objective of this section is to generate this matrixP


According to (8) we have

int119905

0

Ψ119899(119911) 119889119911 = int

119905

0

AT119899(119911) 119889119911 = Aint

119905

0

T119899(119911) 119889119911

= Aint119905

0

[1 119911 119911119899

]119879

119889119911

= A

[[[[[[[[[

[

119905

1

21199052

1

119899 + 1119905119899+1

]]]]]]]]]

]

= A

[[[[[[[[[

[


01

2sdot sdot sdot 0

0 0 sdot sdot sdot1

119899 + 1

]]]]]]]]]

]

[[[[[[[

[

119905

1199052

119905119899+1

]]]]]]]

]

= A119901T119901

(24)

where

A119901= A

[[[[[[[[[

[


01

2sdot sdot sdot 0

0 0 sdot sdot sdot1

119899 + 1

]]]]]]]]]

]

T119901=

[[[[[[[

[

119905

1199052

119905119899+1

]]]]]]]

]

(25)

According to (8) we can get

119905119896

= 119860minus1

[119896+1]Ψ119899(119905) 119896 = 1 119899 (26)

where 119860minus1[119896]

is 119896 row of 119860minus1We approximate 119905119899+1 by the second kind of Chebyshev

polynomials as follows [22]

119905119899+1

asymp c119879119899+1

Ψ119899(119905) (27)

where

c119899+1

=8

1205871198772(119905119899+1

Ψ119899(119905))

=8

1205871198772int119877

0

radic119905119877 minus 1199052119905119899+1

Ψ119899(119905) 119889119905

(28)

Accordingly

P1=

[[[[[[[[[[

[

119860minus1

[2]

119860minus1

[3]

119860minus1

[119899+1]

c119879119899+1

]]]]]]]]]]

]

T119901asymp P1Ψ119899(119905)

(29)

Therefore

P = A119901P1 (30)

int119905

0

Ψ119899(119911) 119889119911 asymp A

119901P1Ψ119899(119905) (31)

53 SolveVariableOrder FractionalDifferential-Integral Equa-tion by Operational Matrix Technique Let 119910(119905) asymp 119906

119899(119905) =

sum119899

119894=0120582119894119894(119905) = Λ

119879

Ψ119899(119905) According to (22) (23) and (31)

original equation (1) is transformed into (32) as follows

Λ119879AM120572(119905)

Aminus1Ψ119899(119905) + 119886 (119905) Λ

119879PΨ119899(119905)

+ 119887 (119905) Λ119879AM1Aminus1Ψ119899(119905) + 119888 (119905) Λ

119879

Ψ119899(119905)

= 119892 (119905) 119905 isin [0 119877]

(32)

Taking the collocation points 119905119894= 119877((2119894+1)2(119899+1)) 119894 =

0 1 119899 to process (32) we can gain

Λ119879AM120572(119905119894)Aminus1Ψ119899(119905119894) + 119886 (119905

119894) Λ119879PΨ119899(119905119894)

+ 119887 (119905119894) Λ119879AM1Aminus1Ψ119899(119905119894) + 119888 (119905

119894) Λ119879

Ψ119899(119905119894)

= 119892 (119905119894)

(33)

Solving algebraic system (33) by Newton method [22] wecan gain the vector Λ = [120582

0 1205821 120582

119899]119879 Subsequently

numerical solution 119906119899(119905) = Λ

119879

Ψ119899(119905) is obtained

6 Numerical Examples and Result Analysis

In this section we verify the efficiency of operational matrixtechnique to support the above theoretical discussion Wecompare the numerical solution with the exact solution byusing our approach The results indicate that our algorithmis a powerful tool for solving variable order FDIE In thissection the notation

120576 = max119894=01119899

1003816100381610038161003816119910 (119905119894) minus 119906119899 (119905119894)1003816100381610038161003816 (34)

is used to show the precision of our proposed algorithmwhere 119905

119894= 119877((2119894 + 1)2(119899 + 1)) 119894 = 0 1 119899


Table 1 Computational results of Example 1 for 119877 = 1 and 119899 = 3

119905 Λ Exact solution Numerical solution Absolute error01250 90625 19531 19531 03952119890 minus 13

03750 50000 63281 63281 02220119890 minus 13

06250 03125 113281 113281 00711119890 minus 13

08750 00000 169531 169531 00355119890 minus 13



03000 50000 49500 49500 02336119890 minus 12

05000 03125 87500 87500 01350119890 minus 12

07000 00000 129500 129500 00480119890 minus 12

09000 minus00000 175500 175500 00249119890 minus 12

Table 3 Error 120576 of Example 1 for different 119877

119877 119899 = 3 119899 = 4 119899 = 5 119899 = 6

119877 = 2 17408119890 minus 13 52847119890 minus 13 69678119890 minus 13 52793119890 minus 12



Example 1 Consider the following nonlinear variable orderFDIE

119863120572(119905)

119910 (119905) + 6int119905

0

119910 (119911) 119889119911 + 21199051199101015840

(119905) + 119910 (119905) = 119892 (119905)

119910 (0) = 0 119905 isin [0 119877]

(35)

where

120572 (119905) =3 (sin 119905 + cos 119905)

5

119892 (119905) =10

Γ (3 minus 120572 (119905))1199052minus120572(119905)

+15

Γ (2 minus 120572 (119905))1199051minus120572(119905)

+ 101199053

+ 701199052

+ 45119905

(36)

The exact solution is 119910(119905) = 51199052

+ 15119905 We find thenumerical solution of Example 1 in MATLAB 2012 by ourtechnique

The computational results are shown in Tables 1ndash3 Asseen from Tables 1 and 2 the vector Λ = [120582

0 1205821 120582

119899]119879 is

mainly composed of three terms namely 1205820= 90625 120582

1=

50000 and 1205822= 03125 It is evident that the numerical solu-

tion converges to the exact solution Only a small number ofthe second Chebyshev polynomials are needed to reach highprecision which verifies the correction and high efficiencyof our algorithm In addition for 119877 = 1 with 119899 increasingabsolute errors of 119899 = 3 are a little bigger than those of119899 = 4 because of round-off error in MATLAB Therefore theapproximation effect of 119899 = 3 is better than that of 119899 = 4Furthermore our algorithm can obtain the solution not onlyfor 119877 = 1 but also for 119877 gt 1 We extend the interval from

[0 1] to [0 2119896] 119896 = 1 2 3 Similarly we also get the perfectresults shown in Table 3

In Table 3 we list 120576 for different 119877 and 119899 Also for thesame119877 with 119899 increasing 120576 is generally a little bigger becauseof round-off error in MATLAB On the other hand for thesame 119899 with 119877 increasing 120576 becomes slightly bigger whichis consistent withTheorem 3 However every error 120576 is smallenough tomeet the practical engineering application Figure 1also shows that the numerical solutions are very close toexact solutions It verifies the correction and efficiency of thealgorithm in this study

Example 2 Consider another nonlinear variable order FDIE

1198631199053

119910 (119905) + 119890119905

int119905

0

119910 (119911) 119889119911 + sin 1199051199101015840 (119905) + (119905 minus 1) 119910 (119905)

= 119892 (119905) 119910 (0) = 1 119905 isin [0 119877]

(37)

where

119892 (119905) =31199051minus1199053

(7119905 minus 6)

(18 minus 9119905 + 1199052) Γ (1 minus 1199053)+ 119890119905

(119905 minus1199052

2+1199053

3)

+ (2119905 minus 1) sin 119905 + (119905 minus 1) (1199052 minus 119905 + 1)

(38)


minus 119905 + 1 We stillobtain the solution as Example 1 by using our algorithmThecomputational results are shown in Tables 4ndash6 As seen fromTables 4 and 5 the vector Λ = [120582

0 1205821 120582

119899]119879 is mainly

composed of two terms namely 1205820= 08125 and 120582

2=

00625 It is evident that the numerical solution converges tothe exact solution In particular for 119877 = 1 and 119899 = 3 everyabsolute error at the collocation point is 0 because of round-off in MATLABTherefore the approximation effect of 119899 = 3

is better than that of 119899 = 4Finally we extend the interval from [0 1] to [0 119877] 119877 =

2 3 Similarly we also get the perfect results as shown inFigure 2 and Table 6 Figure 2 shows the exact solution andnumerical solution for different119877 at collocation points whichdemonstrates that the numerical solution is very close to



119905 Λ Exact solution Numerical solution Absolute error01250 08125 08906 08906 003750 00000 07656 07656 006250 00625 07656 07656 008750 00000 08906 08906 0



03000 minus00000 07900 07900 02220119890 minus 15

05000 00625 07500 07500 05551119890 minus 15

07000 00000 07900 07900 04441119890 minus 15

09000 00000 09100 09100 06661119890 minus 15

Exact solutionNumerical solution

R = 2 n = 3

0

10

20

30

40

50

y(t)

1 20 05 15

t

R = 8 n = 6

0

100

200

300

400

y(t)

2 4 6 80

t


Figure 1 Exact solution and numerical solution of Example 1

R = 2 n = 5 R = 3 n = 3


05

1

15

2

25

3

y(t)

05 1 15 20

t

0

1

2

3

4

5

6

y(t)

1 2 30

t





119877 119899 = 3 119899 = 4 119899 = 5 119899 = 6



the exact solution From Table 6 for the same 119877 with 119899

increasing 120576 is generally a little bigger because of round-offerror From another point of view for the same 119899 with 119877

increasing 120576 becomes slightly bigger which is consistent withTheorem 3 There is the same trend for the other 119877 Everyerror 120576 is small enough to meet the practical engineeringapplication

7 Conclusions

In this paper we present a numerical technique for solving thevariable order FDIE based on the second kind of Chebyshevpolynomials Taking advantage of the definition of the vari-able order fraction derivative and the simplicity of the secondkind of Chebyshev polynomials we transform the FDIE intoan algebraic system By solving the algebraic system thenumerical solution is acquired Numerical examples showthat the numerical solution is in very good accordance withthe exact solution The technique can be applied to solve theother variable order fractional differential problems

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is funded in part by the Natural Science Foun-dation of Hebei Province (no A2015407063) China andis funded in part by the Teaching Research Project ofHebei Normal University of Science and Technology (noJYZD201413) China The work is also funded by ScientificResearch Foundation of Hebei Normal University of Scienceand Technology

References

[1] A A Kilbas H M Srivastava and J J Trujillo Theoryand Application of Fractional Differential Equations vol 204Elsevier Science BV New York NY USA 2006

[2] Y A Rossikhin and M V Shitikova ldquoApplication of fractionalderivatives to the analysis of damped vibrations of viscoelasticsingle mass systemsrdquo Acta Mechanica vol 120 no 1 pp 109ndash125 1997

[3] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and its Applications vol 379 no 2 pp 607ndash614 2007

[4] N Zalp and E Demirci ldquoA fractional order SEIR model withvertical transmissionrdquo Mathematical and Computer Modellingvol 54 no 1-2 pp 1ndash6 2011

[5] H G Sun W Chen and Y Q Chen ldquoVariable-order fractionaldifferential operators in anomalous diffusion modelingrdquo Phys-ica A Statistical Mechanics and Its Applications vol 388 no 21pp 4586ndash4592 2009

[6] W Chen ldquoA speculative study of 23-order fractional Lapla-cian modeling of turbulence some thoughts and conjecturesrdquoChaos An Interdisciplinary Journal of Nonlinear Science vol 16no 2 Article ID 023126 2006

[7] W Chen H Sun X Zhang and D Korosak ldquoAnomalous diffu-sion modeling by fractal and fractional derivativesrdquo Computersamp Mathematics with Applications vol 59 no 5 pp 1754ndash17582010

[8] H G Sun D Chen Y Zhang and L Chen ldquoUnderstandingpartial bed-load transport experiments and stochastic modelanalysisrdquo Journal of Hydrology vol 521 pp 196ndash204 2015

[9] A Ahmadian M Suleiman S Salahshour and D Baleanu ldquoAJacobi operational matrix for solving a fuzzy linear fractionaldifferential equationrdquoAdvances inDifference Equations vol 104no 1 pp 1ndash29 2013

[10] Z J Fu W Chen and H T Yang ldquoBoundary particle methodfor Laplace transformed time fractional diffusion equationsrdquoJournal of Computational Physics vol 235 pp 52ndash66 2013

[11] G F Pang W Chen and Z J Fu ldquoSpace-fractional advection-dispersion equations by the Kansa methodrdquo Journal of Compu-tational Physics vol 293 pp 280ndash296 2015

[12] A H Bhrawy and M A Zaky ldquoA method based on the Jacobitau approximation for solving multi-term time-space fractionalpartial differential equationsrdquo Journal of Computational Physicsvol 281 pp 876ndash895 2015

[13] A H Bhrawy M M Alghamdi and T M Taha ldquoA newmodified generalized Laguerre operational matrix of fractionalintegration for solving fractional differential equations on thehalf linerdquo Advances in Difference Equations vol 179 no 1 pp1ndash12 2012

[14] A H Bhrawy and A S Alofi ldquoThe operational matrix of frac-tional integration for shifted Chebyshev polynomialsrdquo AppliedMathematics Letters vol 26 no 1 pp 25ndash31 2013

[15] S Shen F Liu J Chen I Turner and V Anh ldquoNumericaltechniques for the variable order time fractional diffusionequationrdquo Applied Mathematics and Computation vol 218 no22 pp 10861ndash10870 2012

[16] Y M Chen L Q Liu B F Li and Y Sun ldquoNumerical solutionfor the variable order linear cable equation with Bernsteinpolynomialsrdquo Applied Mathematics and Computation vol 238pp 329ndash341 2014

[17] Y M Chen L Q Liu X Li and Y Sun ldquoNumerical solution forthe variable order time fractional diffusion equation with Bern-stein polynomialsrdquo CMES Computer Modeling in Engineeringamp Sciences vol 97 no 1 pp 81ndash100 2014

[18] S Chen F Liu and K Burrage ldquoNumerical simulation ofa new two-dimensional variable-order fractional percolationequation in non-homogeneous porous mediardquo Computers ampMathematics with Applications vol 68 no 12 pp 2133ndash21412014

[19] P Zhuang F Liu VAnh and I Turner ldquoNumericalmethods forthe variable-order fractional advection-diffusion equation witha nonlinear source termrdquo SIAM Journal on Numerical Analysisvol 47 no 3 pp 1760ndash1781 2009

[20] L F Wang Y P Ma and Y Q Yang ldquoLegendre polynomialsmethod for solving a class of variable order fractional differ-ential equationrdquo Computer Modeling in Engineering amp Sciencesvol 101 no 2 pp 97ndash111 2014


[21] Y-M Chen Y-Q Wei D-Y Liu and H Yu ldquoNumericalsolution for a class of nonlinear variable order fractional differ-ential equations with Legendre waveletsrdquo Applied MathematicsLetters vol 46 pp 83ndash88 2015

[22] Q Y Li N CWang and D Y YiNumerical Analysis TsinghuaUniversity Press Beijing China 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


alternating direct method for the two-dimensional variableorder fractional percolation equation was proposed in [18]Explicit and implicit Euler approximations for FDE wereintroduced in [19] A numerical method based on Legendrepolynomials was presented for a class of variable orderFDEs in [20] Chen et al [21] introduced the numericalsolution for a class of nonlinear variable order FDEs withLegendre wavelets In addition for most literatures theysolved variable order FDE which is defined on the interval[0 1] It is noteworthy that the Chebyshev polynomialsfamily have beneficial properties so that they are widelyused in approximation theory However the second kindof Chebyshev polynomials have been paid less attention forsolving variable order FDE Accordingly we will solve a kindof variable order fractional differential-integral equations(FDIEs) defined on the interval [0 119877] (119877 gt 0) based on thesecond kind of Chebyshev polynomials The FDIE is shownas follows

119863120572(119905)

119910 (119905) + 119886 (119905) int119905

0

119910 (119911) 119889119911 + 119887 (119905) 1199101015840

(119905) + 119888 (119905) 119910 (119905)

= 119892 (119905) 119910 (0) = 1199100 119905 isin [0 119877]

(1)

where 119863120572(119905)119910(119905) is fractional derivative of 119910(119905) in Caputorsquossense

The basic idea of this approach is that we derive thedifferential operational matrix and the integral operationalmatrix based on Chebyshev polynomials With the opera-tionalmatrixes (1) is transformed into the products of severaldependent matrixes which can be viewed as an algebraicsystem after taking the collocation points By solving thealgebraic system the numerical solution of (1) is acquiredSince the second kind of Chebyshev polynomials are orthog-onal to each other the operational matrixes based on Cheby-shev polynomials greatly reduce the size of computationalwork while accurately providing the series solution Fromsome numerical examples we can see that our results arein good agreement with exact solution Numerical resultdemonstrates the validity of this algorithm

The paper is organized as follows In Section 2 somenecessary preliminaries are introduced In Section 3 the basicdefinition and property of the second kind of Chebyshevpolynomials are given In Section 4 function approximationis given In Section 5 two kinds of operational matrixes arederived and we applied the operational matrixes to solve theequation as given at the beginning In Section 6 we presentsome numerical examples to demonstrate the efficiency of thealgorithm We end the paper with a few concluding remarksin Section 7

2 Preliminaries

There are several definitions for variable order fractionalderivatives such as the one in Riemann-Liouvillersquos sense andthe one in Caputorsquos sense [22] In this paper the definition inCaputorsquos sense is considered

Definition 1 Caputo fractional derivate with order 120572(119905) isdefined by

119863120572(119905)

119910 (119905) =1

Γ (1 minus 120572 (119905))int119905

0+

(119905 minus 120591)minus120572(119905)

1199101015840

(120591) 119889120591

+119910 (0+) minus 119910 (0minus)

Γ (1 minus 120572 (119905))119905minus120572(119905)

(2)

If we assume the starting time in a perfect situation we canget Definition 2 as follows

Definition 2 Consider the following

119863120572(119905)

119910 (119905) =1

Γ (1 minus 120572 (119905))int119905

0

(119905 minus 120591)minus120572(119905)

1199101015840

(120591) 119889120591 (3)

By Definition 2 we can get the following formula [16]

119863120572(119905)

119905119899

=

Γ (119899 + 1)

Γ (119899 + 1 minus 120572 (119905))119905119899minus120572(119905)

119899 = 1 2

0 119899 = 0

(4)

3 The Shifted Second Kind ofChebyshev Polynomials

Thesecondkind ofChebyshev polynomials are definedon theinterval 119868 = [minus1 1] They are orthogonal to each other withrespect to the weight function 120596(119909) = radic1 minus 1199092 They satisfythe following formulas

1198800(119909) = 1

1198801(119909) = 2119909

119880119899+1

(119909) = 2119909119880119899(119909) minus 119880

119899minus1(119909)

119899 = 1 2

int1

minus1

radic1 minus 1199092119880119899(119909)119880119898(119909) 119889119909 =

120587

2 119898 = 119899

0 119898 = 119899

(5)

For 119905 isin [0 119877] let 119909 = 2119905119877 minus 1 We can get shifted secondkind of Chebyshev polynomials

119899(119905) = 119880

119899(2119905119877 minus 1) which

are also orthogonal with respect to weight function 120596(119905) =

radic119905119877 minus 1199052 for 119905 isin [0 119877] They satisfy the following formulas

0(119905) = 1

1(119905) =

4119905

119877minus 2

119899+1

(119905) = 2 (2119905

119877minus 1)

119899(119905) minus

119899minus1(119905)

119899 = 1 2

int119877

0

radic119905119877 minus 1199052119899(119905) 119898(119905) 119889119905 =

120587

81198772

119898 = 119899

0 119898 = 119899

(6)



119899(119905)

=

1 119899 = 0

[1198992]

sum119896=0

(minus1)119896

(119899 minus 119896)

119896 (119899 minus 2119896)(4119905

119877minus 2)119899minus2119896

119899 ge 1

(7)


Let Ψ119899(119905) = [

0(119905) 1(119905)

119899(119905)]119879 and T

119899(119905) =

[1 119905 119905119899

]119879 Therefore

Ψ119899(119905) = AT

119899(119905) (8)

where

A = BC

C =

[[[[[[[[[[[[[[

[


1198620

1(minus

119877

2)1minus0

1198621

1(minus

119877

2)1minus1

0 sdot sdot sdot 0

1198620

2(minus

119877

2)2minus0

1198621

2(minus

119877

2)2minus1

1198622

2(minus

119877

2)2minus2

sdot sdot sdot 0

1198620

119899(minus

119877

2)119899minus0

1198621

119899(minus

119877

2)119899minus1

1198622

119899(minus

119877

2)119899minus2


119899(minus

119877

2)119899minus119899

]]]]]]]]]]]]]]

]

(9)


B

=

[[[[[[[[[[[[[[

[


0 (minus1)0

(1 minus 0)

0 (1 minus 0)(4

119877)1minus0

0 sdot sdot sdot 0

(minus1)1

(2 minus 1)

1 (2 minus 2)(4

119877)2minus2

0 (minus1)0

(2 minus 0)

0 (2 minus 0)(4

119877)2minus0

sdot sdot sdot 0

(minus1)1198992

(119899 minus 1198992)

(1198992) (119899 minus 2 sdot (1198992))(4

119877)119899minus2sdot(1198992)


(119899 minus 1198992 + 1)


119877)119899minus2sdot(1198992minus1)


(119899 minus 0)

0 (119899 minus 0)(4

119877)119899minus0

]]]]]]]]]]]]]]

]

(10)


B

=

[[[[[[[[[[[[[[

[


0 (minus1)0

(1 minus 0)

0 (1 minus 0)(4

119877)1minus0

0 sdot sdot sdot 0

(minus1)1

(2 minus 1)

1 (2 minus 2)(4

119877)2minus2

0 (minus1)0

(2 minus 0)

0 (2 minus 0)(4

119877)2minus0

sdot sdot sdot 0


(119899 minus (119899 minus 1) 2)


119877)119899minus2sdot((119899minus1)2)


(119899 minus 0)

0 (119899 minus 0)(4

119877)119899minus0

]]]]]]]]]]]]]]

]

(11)


T119899(119905) = Aminus1Ψ

119899(119905) (12)



119899(119905) =


sum119899

119894=0120582119894119894(119905) = Λ

119879



119899]119879 Then

1003817100381710038171003817119910 (119905) minus 119906119899 (119905)10038171003817100381710038172 le

119872119878119899+1

119877

(119899 + 1)radic120587

8 (13)

where119872 = max119905isin[0119877]

119910119899+1

(119905) and 119878 = max119877 minus 1199050 1199050


119910 (119905) = 119910 (1199050) + 1199101015840

(1199050) (119905 minus 119905

0) + sdot sdot sdot

+ 119910(119899)

(1199050)(119905 minus 1199050)119899

119899+ 119910(119899+1)

(120578)(119905 minus 1199050)119899+1

(119899 + 1)

1199050isin [0 119877]

(14)


Let

119901119899(119905) = 119910 (119905

0) + 1199101015840

(1199050) (119905 minus 119905

0) + sdot sdot sdot

+ 119910(119899)

(1199050)(119905 minus 1199050)119899

119899

(15)

Then

1003816100381610038161003816119910 (119905) minus 119901119899 (119905)1003816100381610038161003816 =

1003816100381610038161003816100381610038161003816100381610038161003816

119910(119899+1)

(120578)(119905 minus 1199050)119899+1

(119899 + 1)

1003816100381610038161003816100381610038161003816100381610038161003816

(16)

Since 119906119899(119905) = sum

119899

119894=0120582119894119894(119905) = Λ

119879



1003817100381710038171003817119910 (119905) minus 119906119899 (119905)10038171003817100381710038172

2le1003817100381710038171003817119910 (119905) minus 119901119899 (119905)

10038171003817100381710038172

2

= int119877

0

120596 (119905) [119910 (119905) minus 119901119899(119905)]2

119889119905

= int119877

0

120596 (119905) [119910(119899+1)

(120578)(119905 minus 1199050)119899+1

(119899 + 1)]

2

119889119905

le1198722

[(119899 + 1)]2int119877

0

120596 (119905) (119905 minus 1199050)2119899+2

119889119905

=1198722

[(119899 + 1)]2int119877

0


119889119905

(17)


1003817100381710038171003817119910 (119905) minus 119906119899 (119905)10038171003817100381710038172

2le

1198722

1198782119899+2

[(119899 + 1)]2int119877

0

radic119905119877 minus 1199052119889119905

=1198722

1198782119899+2

1198772

(119899 + 1) 2120587

8

(18)




ing (8) we obtain

119863120572(119905)

Ψ119899(119905) = 119863

120572(119905)

(AT119899(119905))

= A119863120572(119905) ([1 119905 119905119899]119879) (19)


119863120572(119905)

([1 119905 119905119899

]119879

)

= [0Γ (2)

Γ (2 minus 120572 (119905))1199051minus120572(119905)

Γ (119899 + 1)

Γ (119899 + 1 minus 120572 (119905))119905119899minus120572(119905)

]

119879

=

[[[[[[[[[

[


0Γ (2)

Γ (2 minus 120572 (119905))119905minus120572(119905)

sdot sdot sdot 0


Γ (119899 + 1 minus 120572 (119905))119905minus120572(119905)

]]]]]]]]]

]

[[[[[[

[

1

119905

119905119899

]]]]]]

]

= M120572(119905)

Aminus1Ψ119899(119905)

(20)

where

M120572(119905)

=

[[[[[[[[[

[


0Γ (2)

Γ (2 minus 120572 (119905))119905minus120572(119905)

sdot sdot sdot 0


Γ (119899 + 1 minus 120572 (119905))119905minus120572(119905)

]]]]]]]]]

]

(21)

Therefore

119863120572(119905)

Ψ119899(119905) = AM

120572(119905)Aminus1Ψ119899(119905) (22)

AM120572(119905)



Ψ1015840

119899(119905) = AM

1Aminus1Ψ119899(119905) (23)


0

Ψ119899(119911)119889119911 Let

int119905

0

Ψ119899(119911)119889119911 asymp PΨ





int119905

0

Ψ119899(119911) 119889119911 = int

119905

0

AT119899(119911) 119889119911 = Aint

119905

0

T119899(119911) 119889119911

= Aint119905

0

[1 119911 119911119899

]119879

119889119911

= A

[[[[[[[[[

[

119905

1

21199052

1

119899 + 1119905119899+1

]]]]]]]]]

]

= A

[[[[[[[[[

[


01

2sdot sdot sdot 0

0 0 sdot sdot sdot1

119899 + 1

]]]]]]]]]

]

[[[[[[[

[

119905

1199052

119905119899+1

]]]]]]]

]

= A119901T119901

(24)

where

A119901= A

[[[[[[[[[

[


01

2sdot sdot sdot 0

0 0 sdot sdot sdot1

119899 + 1

]]]]]]]]]

]

T119901=

[[[[[[[

[

119905

1199052

119905119899+1

]]]]]]]

]

(25)


119905119896

= 119860minus1

[119896+1]Ψ119899(119905) 119896 = 1 119899 (26)

where 119860minus1[119896]



119905119899+1

asymp c119879119899+1

Ψ119899(119905) (27)

where

c119899+1

=8

1205871198772(119905119899+1

Ψ119899(119905))

=8

1205871198772int119877

0

radic119905119877 minus 1199052119905119899+1

Ψ119899(119905) 119889119905

(28)

Accordingly

P1=

[[[[[[[[[[

[

119860minus1

[2]

119860minus1

[3]

119860minus1

[119899+1]

c119879119899+1

]]]]]]]]]]

]

T119901asymp P1Ψ119899(119905)

(29)

Therefore

P = A119901P1 (30)

int119905

0

Ψ119899(119911) 119889119911 asymp A

119901P1Ψ119899(119905) (31)


119899(119905) =

sum119899

119894=0120582119894119894(119905) = Λ

119879



Λ119879AM120572(119905)

Aminus1Ψ119899(119905) + 119886 (119905) Λ

119879PΨ119899(119905)

+ 119887 (119905) Λ119879AM1Aminus1Ψ119899(119905) + 119888 (119905) Λ

119879

Ψ119899(119905)

= 119892 (119905) 119905 isin [0 119877]

(32)



Λ119879AM120572(119905119894)Aminus1Ψ119899(119905119894) + 119886 (119905

119894) Λ119879PΨ119899(119905119894)

+ 119887 (119905119894) Λ119879AM1Aminus1Ψ119899(119905119894) + 119888 (119905

119894) Λ119879

Ψ119899(119905119894)

= 119892 (119905119894)

(33)


0 1205821 120582



119879




120576 = max119894=01119899

1003816100381610038161003816119910 (119905119894) minus 119906119899 (119905119894)1003816100381610038161003816 (34)


119894= 119877((2119894 + 1)2(119899 + 1)) 119894 = 0 1 119899




03750 50000 63281 63281 02220119890 minus 13

06250 03125 113281 113281 00711119890 minus 13

08750 00000 169531 169531 00355119890 minus 13



03000 50000 49500 49500 02336119890 minus 12

05000 03125 87500 87500 01350119890 minus 12

07000 00000 129500 129500 00480119890 minus 12

09000 minus00000 175500 175500 00249119890 minus 12


119877 119899 = 3 119899 = 4 119899 = 5 119899 = 6





119863120572(119905)

119910 (119905) + 6int119905

0

119910 (119911) 119889119911 + 21199051199101015840

(119905) + 119910 (119905) = 119892 (119905)

119910 (0) = 0 119905 isin [0 119877]

(35)

where

120572 (119905) =3 (sin 119905 + cos 119905)

5

119892 (119905) =10

Γ (3 minus 120572 (119905))1199052minus120572(119905)

+15

Γ (2 minus 120572 (119905))1199051minus120572(119905)

+ 101199053

+ 701199052

+ 45119905

(36)




0 1205821 120582

119899]119879 is


1=






1198631199053

119910 (119905) + 119890119905

int119905

0

119910 (119911) 119889119911 + sin 1199051199101015840 (119905) + (119905 minus 1) 119910 (119905)

= 119892 (119905) 119910 (0) = 1 119905 isin [0 119877]

(37)

where

119892 (119905) =31199051minus1199053

(7119905 minus 6)

(18 minus 9119905 + 1199052) Γ (1 minus 1199053)+ 119890119905

(119905 minus1199052

2+1199053

3)


(38)



0 1205821 120582

119899]119879 is mainly


2=









03000 minus00000 07900 07900 02220119890 minus 15

05000 00625 07500 07500 05551119890 minus 15

07000 00000 07900 07900 04441119890 minus 15

09000 00000 09100 09100 06661119890 minus 15


R = 2 n = 3

0

10

20

30

40

50

y(t)

1 20 05 15

t

R = 8 n = 6

0

100

200

300

400

y(t)

2 4 6 80

t



R = 2 n = 5 R = 3 n = 3


05

1

15

2

25

3

y(t)

05 1 15 20

t

0

1

2

3

4

5

6

y(t)

1 2 30

t





119877 119899 = 3 119899 = 4 119899 = 5 119899 = 6






7 Conclusions


Competing Interests


Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119899(119905)

=

1 119899 = 0

[1198992]

sum119896=0

(minus1)119896

(119899 minus 119896)

119896 (119899 minus 2119896)(4119905

119877minus 2)119899minus2119896

119899 ge 1

(7)


Let Ψ119899(119905) = [

0(119905) 1(119905)

119899(119905)]119879 and T

119899(119905) =

[1 119905 119905119899

]119879 Therefore

Ψ119899(119905) = AT

119899(119905) (8)

where

A = BC

C =

[[[[[[[[[[[[[[

[


1198620

1(minus

119877

2)1minus0

1198621

1(minus

119877

2)1minus1

0 sdot sdot sdot 0

1198620

2(minus

119877

2)2minus0

1198621

2(minus

119877

2)2minus1

1198622

2(minus

119877

2)2minus2

sdot sdot sdot 0

1198620

119899(minus

119877

2)119899minus0

1198621

119899(minus

119877

2)119899minus1

1198622

119899(minus

119877

2)119899minus2


119899(minus

119877

2)119899minus119899

]]]]]]]]]]]]]]

]

(9)


B

=

[[[[[[[[[[[[[[

[


0 (minus1)0

(1 minus 0)

0 (1 minus 0)(4

119877)1minus0

0 sdot sdot sdot 0

(minus1)1

(2 minus 1)

1 (2 minus 2)(4

119877)2minus2

0 (minus1)0

(2 minus 0)

0 (2 minus 0)(4

119877)2minus0

sdot sdot sdot 0

(minus1)1198992

(119899 minus 1198992)

(1198992) (119899 minus 2 sdot (1198992))(4

119877)119899minus2sdot(1198992)


(119899 minus 1198992 + 1)


119877)119899minus2sdot(1198992minus1)


(119899 minus 0)

0 (119899 minus 0)(4

119877)119899minus0

]]]]]]]]]]]]]]

]

(10)


B

=

[[[[[[[[[[[[[[

[


0 (minus1)0

(1 minus 0)

0 (1 minus 0)(4

119877)1minus0

0 sdot sdot sdot 0

(minus1)1

(2 minus 1)

1 (2 minus 2)(4

119877)2minus2

0 (minus1)0

(2 minus 0)

0 (2 minus 0)(4

119877)2minus0

sdot sdot sdot 0


(119899 minus (119899 minus 1) 2)


119877)119899minus2sdot((119899minus1)2)


(119899 minus 0)

0 (119899 minus 0)(4

119877)119899minus0

]]]]]]]]]]]]]]

]

(11)


T119899(119905) = Aminus1Ψ

119899(119905) (12)



119899(119905) =


sum119899

119894=0120582119894119894(119905) = Λ

119879



119899]119879 Then

1003817100381710038171003817119910 (119905) minus 119906119899 (119905)10038171003817100381710038172 le

119872119878119899+1

119877

(119899 + 1)radic120587

8 (13)

where119872 = max119905isin[0119877]

119910119899+1

(119905) and 119878 = max119877 minus 1199050 1199050


119910 (119905) = 119910 (1199050) + 1199101015840

(1199050) (119905 minus 119905

0) + sdot sdot sdot

+ 119910(119899)

(1199050)(119905 minus 1199050)119899

119899+ 119910(119899+1)

(120578)(119905 minus 1199050)119899+1

(119899 + 1)

1199050isin [0 119877]

(14)


Let

119901119899(119905) = 119910 (119905

0) + 1199101015840

(1199050) (119905 minus 119905

0) + sdot sdot sdot

+ 119910(119899)

(1199050)(119905 minus 1199050)119899

119899

(15)

Then

1003816100381610038161003816119910 (119905) minus 119901119899 (119905)1003816100381610038161003816 =

1003816100381610038161003816100381610038161003816100381610038161003816

119910(119899+1)

(120578)(119905 minus 1199050)119899+1

(119899 + 1)

1003816100381610038161003816100381610038161003816100381610038161003816

(16)

Since 119906119899(119905) = sum

119899

119894=0120582119894119894(119905) = Λ

119879



1003817100381710038171003817119910 (119905) minus 119906119899 (119905)10038171003817100381710038172

2le1003817100381710038171003817119910 (119905) minus 119901119899 (119905)

10038171003817100381710038172

2

= int119877

0

120596 (119905) [119910 (119905) minus 119901119899(119905)]2

119889119905

= int119877

0

120596 (119905) [119910(119899+1)

(120578)(119905 minus 1199050)119899+1

(119899 + 1)]

2

119889119905

le1198722

[(119899 + 1)]2int119877

0

120596 (119905) (119905 minus 1199050)2119899+2

119889119905

=1198722

[(119899 + 1)]2int119877

0


119889119905

(17)


1003817100381710038171003817119910 (119905) minus 119906119899 (119905)10038171003817100381710038172

2le

1198722

1198782119899+2

[(119899 + 1)]2int119877

0

radic119905119877 minus 1199052119889119905

=1198722

1198782119899+2

1198772

(119899 + 1) 2120587

8

(18)




ing (8) we obtain

119863120572(119905)

Ψ119899(119905) = 119863

120572(119905)

(AT119899(119905))

= A119863120572(119905) ([1 119905 119905119899]119879) (19)


119863120572(119905)

([1 119905 119905119899

]119879

)

= [0Γ (2)

Γ (2 minus 120572 (119905))1199051minus120572(119905)

Γ (119899 + 1)

Γ (119899 + 1 minus 120572 (119905))119905119899minus120572(119905)

]

119879

=

[[[[[[[[[

[


0Γ (2)

Γ (2 minus 120572 (119905))119905minus120572(119905)

sdot sdot sdot 0


Γ (119899 + 1 minus 120572 (119905))119905minus120572(119905)

]]]]]]]]]

]

[[[[[[

[

1

119905

119905119899

]]]]]]

]

= M120572(119905)

Aminus1Ψ119899(119905)

(20)

where

M120572(119905)

=

[[[[[[[[[

[


0Γ (2)

Γ (2 minus 120572 (119905))119905minus120572(119905)

sdot sdot sdot 0


Γ (119899 + 1 minus 120572 (119905))119905minus120572(119905)

]]]]]]]]]

]

(21)

Therefore

119863120572(119905)

Ψ119899(119905) = AM

120572(119905)Aminus1Ψ119899(119905) (22)

AM120572(119905)



Ψ1015840

119899(119905) = AM

1Aminus1Ψ119899(119905) (23)


0

Ψ119899(119911)119889119911 Let

int119905

0

Ψ119899(119911)119889119911 asymp PΨ





int119905

0

Ψ119899(119911) 119889119911 = int

119905

0

AT119899(119911) 119889119911 = Aint

119905

0

T119899(119911) 119889119911

= Aint119905

0

[1 119911 119911119899

]119879

119889119911

= A

[[[[[[[[[

[

119905

1

21199052

1

119899 + 1119905119899+1

]]]]]]]]]

]

= A

[[[[[[[[[

[


01

2sdot sdot sdot 0

0 0 sdot sdot sdot1

119899 + 1

]]]]]]]]]

]

[[[[[[[

[

119905

1199052

119905119899+1

]]]]]]]

]

= A119901T119901

(24)

where

A119901= A

[[[[[[[[[

[


01

2sdot sdot sdot 0

0 0 sdot sdot sdot1

119899 + 1

]]]]]]]]]

]

T119901=

[[[[[[[

[

119905

1199052

119905119899+1

]]]]]]]

]

(25)


119905119896

= 119860minus1

[119896+1]Ψ119899(119905) 119896 = 1 119899 (26)

where 119860minus1[119896]



119905119899+1

asymp c119879119899+1

Ψ119899(119905) (27)

where

c119899+1

=8

1205871198772(119905119899+1

Ψ119899(119905))

=8

1205871198772int119877

0

radic119905119877 minus 1199052119905119899+1

Ψ119899(119905) 119889119905

(28)

Accordingly

P1=

[[[[[[[[[[

[

119860minus1

[2]

119860minus1

[3]

119860minus1

[119899+1]

c119879119899+1

]]]]]]]]]]

]

T119901asymp P1Ψ119899(119905)

(29)

Therefore

P = A119901P1 (30)

int119905

0

Ψ119899(119911) 119889119911 asymp A

119901P1Ψ119899(119905) (31)


119899(119905) =

sum119899

119894=0120582119894119894(119905) = Λ

119879



Λ119879AM120572(119905)

Aminus1Ψ119899(119905) + 119886 (119905) Λ

119879PΨ119899(119905)

+ 119887 (119905) Λ119879AM1Aminus1Ψ119899(119905) + 119888 (119905) Λ

119879

Ψ119899(119905)

= 119892 (119905) 119905 isin [0 119877]

(32)



Λ119879AM120572(119905119894)Aminus1Ψ119899(119905119894) + 119886 (119905

119894) Λ119879PΨ119899(119905119894)

+ 119887 (119905119894) Λ119879AM1Aminus1Ψ119899(119905119894) + 119888 (119905

119894) Λ119879

Ψ119899(119905119894)

= 119892 (119905119894)

(33)


0 1205821 120582



119879




120576 = max119894=01119899

1003816100381610038161003816119910 (119905119894) minus 119906119899 (119905119894)1003816100381610038161003816 (34)


119894= 119877((2119894 + 1)2(119899 + 1)) 119894 = 0 1 119899




03750 50000 63281 63281 02220119890 minus 13

06250 03125 113281 113281 00711119890 minus 13

08750 00000 169531 169531 00355119890 minus 13



03000 50000 49500 49500 02336119890 minus 12

05000 03125 87500 87500 01350119890 minus 12

07000 00000 129500 129500 00480119890 minus 12

09000 minus00000 175500 175500 00249119890 minus 12


119877 119899 = 3 119899 = 4 119899 = 5 119899 = 6





119863120572(119905)

119910 (119905) + 6int119905

0

119910 (119911) 119889119911 + 21199051199101015840

(119905) + 119910 (119905) = 119892 (119905)

119910 (0) = 0 119905 isin [0 119877]

(35)

where

120572 (119905) =3 (sin 119905 + cos 119905)

5

119892 (119905) =10

Γ (3 minus 120572 (119905))1199052minus120572(119905)

+15

Γ (2 minus 120572 (119905))1199051minus120572(119905)

+ 101199053

+ 701199052

+ 45119905

(36)




0 1205821 120582

119899]119879 is


1=






1198631199053

119910 (119905) + 119890119905

int119905

0

119910 (119911) 119889119911 + sin 1199051199101015840 (119905) + (119905 minus 1) 119910 (119905)

= 119892 (119905) 119910 (0) = 1 119905 isin [0 119877]

(37)

where

119892 (119905) =31199051minus1199053

(7119905 minus 6)

(18 minus 9119905 + 1199052) Γ (1 minus 1199053)+ 119890119905

(119905 minus1199052

2+1199053

3)


(38)



0 1205821 120582

119899]119879 is mainly


2=









03000 minus00000 07900 07900 02220119890 minus 15

05000 00625 07500 07500 05551119890 minus 15

07000 00000 07900 07900 04441119890 minus 15

09000 00000 09100 09100 06661119890 minus 15


R = 2 n = 3

0

10

20

30

40

50

y(t)

1 20 05 15

t

R = 8 n = 6

0

100

200

300

400

y(t)

2 4 6 80

t



R = 2 n = 5 R = 3 n = 3


05

1

15

2

25

3

y(t)

05 1 15 20

t

0

1

2

3

4

5

6

y(t)

1 2 30

t





119877 119899 = 3 119899 = 4 119899 = 5 119899 = 6






7 Conclusions


Competing Interests


Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










sum119899

119894=0120582119894119894(119905) = Λ

119879



119899]119879 Then

1003817100381710038171003817119910 (119905) minus 119906119899 (119905)10038171003817100381710038172 le

119872119878119899+1

119877

(119899 + 1)radic120587

8 (13)

where119872 = max119905isin[0119877]

119910119899+1

(119905) and 119878 = max119877 minus 1199050 1199050


119910 (119905) = 119910 (1199050) + 1199101015840

(1199050) (119905 minus 119905

0) + sdot sdot sdot

+ 119910(119899)

(1199050)(119905 minus 1199050)119899

119899+ 119910(119899+1)

(120578)(119905 minus 1199050)119899+1

(119899 + 1)

1199050isin [0 119877]

(14)


Let

119901119899(119905) = 119910 (119905

0) + 1199101015840

(1199050) (119905 minus 119905

0) + sdot sdot sdot

+ 119910(119899)

(1199050)(119905 minus 1199050)119899

119899

(15)

Then

1003816100381610038161003816119910 (119905) minus 119901119899 (119905)1003816100381610038161003816 =

1003816100381610038161003816100381610038161003816100381610038161003816

119910(119899+1)

(120578)(119905 minus 1199050)119899+1

(119899 + 1)

1003816100381610038161003816100381610038161003816100381610038161003816

(16)

Since 119906119899(119905) = sum

119899

119894=0120582119894119894(119905) = Λ

119879



1003817100381710038171003817119910 (119905) minus 119906119899 (119905)10038171003817100381710038172

2le1003817100381710038171003817119910 (119905) minus 119901119899 (119905)

10038171003817100381710038172

2

= int119877

0

120596 (119905) [119910 (119905) minus 119901119899(119905)]2

119889119905

= int119877

0

120596 (119905) [119910(119899+1)

(120578)(119905 minus 1199050)119899+1

(119899 + 1)]

2

119889119905

le1198722

[(119899 + 1)]2int119877

0

120596 (119905) (119905 minus 1199050)2119899+2

119889119905

=1198722

[(119899 + 1)]2int119877

0


119889119905

(17)


1003817100381710038171003817119910 (119905) minus 119906119899 (119905)10038171003817100381710038172

2le

1198722

1198782119899+2

[(119899 + 1)]2int119877

0

radic119905119877 minus 1199052119889119905

=1198722

1198782119899+2

1198772

(119899 + 1) 2120587

8

(18)




ing (8) we obtain

119863120572(119905)

Ψ119899(119905) = 119863

120572(119905)

(AT119899(119905))

= A119863120572(119905) ([1 119905 119905119899]119879) (19)


119863120572(119905)

([1 119905 119905119899

]119879

)

= [0Γ (2)

Γ (2 minus 120572 (119905))1199051minus120572(119905)

Γ (119899 + 1)

Γ (119899 + 1 minus 120572 (119905))119905119899minus120572(119905)

]

119879

=

[[[[[[[[[

[


0Γ (2)

Γ (2 minus 120572 (119905))119905minus120572(119905)

sdot sdot sdot 0


Γ (119899 + 1 minus 120572 (119905))119905minus120572(119905)

]]]]]]]]]

]

[[[[[[

[

1

119905

119905119899

]]]]]]

]

= M120572(119905)

Aminus1Ψ119899(119905)

(20)

where

M120572(119905)

=

[[[[[[[[[

[


0Γ (2)

Γ (2 minus 120572 (119905))119905minus120572(119905)

sdot sdot sdot 0


Γ (119899 + 1 minus 120572 (119905))119905minus120572(119905)

]]]]]]]]]

]

(21)

Therefore

119863120572(119905)

Ψ119899(119905) = AM

120572(119905)Aminus1Ψ119899(119905) (22)

AM120572(119905)



Ψ1015840

119899(119905) = AM

1Aminus1Ψ119899(119905) (23)


0

Ψ119899(119911)119889119911 Let

int119905

0

Ψ119899(119911)119889119911 asymp PΨ





int119905

0

Ψ119899(119911) 119889119911 = int

119905

0

AT119899(119911) 119889119911 = Aint

119905

0

T119899(119911) 119889119911

= Aint119905

0

[1 119911 119911119899

]119879

119889119911

= A

[[[[[[[[[

[

119905

1

21199052

1

119899 + 1119905119899+1

]]]]]]]]]

]

= A

[[[[[[[[[

[


01

2sdot sdot sdot 0

0 0 sdot sdot sdot1

119899 + 1

]]]]]]]]]

]

[[[[[[[

[

119905

1199052

119905119899+1

]]]]]]]

]

= A119901T119901

(24)

where

A119901= A

[[[[[[[[[

[


01

2sdot sdot sdot 0

0 0 sdot sdot sdot1

119899 + 1

]]]]]]]]]

]

T119901=

[[[[[[[

[

119905

1199052

119905119899+1

]]]]]]]

]

(25)


119905119896

= 119860minus1

[119896+1]Ψ119899(119905) 119896 = 1 119899 (26)

where 119860minus1[119896]



119905119899+1

asymp c119879119899+1

Ψ119899(119905) (27)

where

c119899+1

=8

1205871198772(119905119899+1

Ψ119899(119905))

=8

1205871198772int119877

0

radic119905119877 minus 1199052119905119899+1

Ψ119899(119905) 119889119905

(28)

Accordingly

P1=

[[[[[[[[[[

[

119860minus1

[2]

119860minus1

[3]

119860minus1

[119899+1]

c119879119899+1

]]]]]]]]]]

]

T119901asymp P1Ψ119899(119905)

(29)

Therefore

P = A119901P1 (30)

int119905

0

Ψ119899(119911) 119889119911 asymp A

119901P1Ψ119899(119905) (31)


119899(119905) =

sum119899

119894=0120582119894119894(119905) = Λ

119879



Λ119879AM120572(119905)

Aminus1Ψ119899(119905) + 119886 (119905) Λ

119879PΨ119899(119905)

+ 119887 (119905) Λ119879AM1Aminus1Ψ119899(119905) + 119888 (119905) Λ

119879

Ψ119899(119905)

= 119892 (119905) 119905 isin [0 119877]

(32)



Λ119879AM120572(119905119894)Aminus1Ψ119899(119905119894) + 119886 (119905

119894) Λ119879PΨ119899(119905119894)

+ 119887 (119905119894) Λ119879AM1Aminus1Ψ119899(119905119894) + 119888 (119905

119894) Λ119879

Ψ119899(119905119894)

= 119892 (119905119894)

(33)


0 1205821 120582



119879




120576 = max119894=01119899

1003816100381610038161003816119910 (119905119894) minus 119906119899 (119905119894)1003816100381610038161003816 (34)


119894= 119877((2119894 + 1)2(119899 + 1)) 119894 = 0 1 119899




03750 50000 63281 63281 02220119890 minus 13

06250 03125 113281 113281 00711119890 minus 13

08750 00000 169531 169531 00355119890 minus 13



03000 50000 49500 49500 02336119890 minus 12

05000 03125 87500 87500 01350119890 minus 12

07000 00000 129500 129500 00480119890 minus 12

09000 minus00000 175500 175500 00249119890 minus 12


119877 119899 = 3 119899 = 4 119899 = 5 119899 = 6





119863120572(119905)

119910 (119905) + 6int119905

0

119910 (119911) 119889119911 + 21199051199101015840

(119905) + 119910 (119905) = 119892 (119905)

119910 (0) = 0 119905 isin [0 119877]

(35)

where

120572 (119905) =3 (sin 119905 + cos 119905)

5

119892 (119905) =10

Γ (3 minus 120572 (119905))1199052minus120572(119905)

+15

Γ (2 minus 120572 (119905))1199051minus120572(119905)

+ 101199053

+ 701199052

+ 45119905

(36)




0 1205821 120582

119899]119879 is


1=






1198631199053

119910 (119905) + 119890119905

int119905

0

119910 (119911) 119889119911 + sin 1199051199101015840 (119905) + (119905 minus 1) 119910 (119905)

= 119892 (119905) 119910 (0) = 1 119905 isin [0 119877]

(37)

where

119892 (119905) =31199051minus1199053

(7119905 minus 6)

(18 minus 9119905 + 1199052) Γ (1 minus 1199053)+ 119890119905

(119905 minus1199052

2+1199053

3)


(38)



0 1205821 120582

119899]119879 is mainly


2=









03000 minus00000 07900 07900 02220119890 minus 15

05000 00625 07500 07500 05551119890 minus 15

07000 00000 07900 07900 04441119890 minus 15

09000 00000 09100 09100 06661119890 minus 15


R = 2 n = 3

0

10

20

30

40

50

y(t)

1 20 05 15

t

R = 8 n = 6

0

100

200

300

400

y(t)

2 4 6 80

t



R = 2 n = 5 R = 3 n = 3


05

1

15

2

25

3

y(t)

05 1 15 20

t

0

1

2

3

4

5

6

y(t)

1 2 30

t





119877 119899 = 3 119899 = 4 119899 = 5 119899 = 6






7 Conclusions


Competing Interests


Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











int119905

0

Ψ119899(119911) 119889119911 = int

119905

0

AT119899(119911) 119889119911 = Aint

119905

0

T119899(119911) 119889119911

= Aint119905

0

[1 119911 119911119899

]119879

119889119911

= A

[[[[[[[[[

[

119905

1

21199052

1

119899 + 1119905119899+1

]]]]]]]]]

]

= A

[[[[[[[[[

[


01

2sdot sdot sdot 0

0 0 sdot sdot sdot1

119899 + 1

]]]]]]]]]

]

[[[[[[[

[

119905

1199052

119905119899+1

]]]]]]]

]

= A119901T119901

(24)

where

A119901= A

[[[[[[[[[

[


01

2sdot sdot sdot 0

0 0 sdot sdot sdot1

119899 + 1

]]]]]]]]]

]

T119901=

[[[[[[[

[

119905

1199052

119905119899+1

]]]]]]]

]

(25)


119905119896

= 119860minus1

[119896+1]Ψ119899(119905) 119896 = 1 119899 (26)

where 119860minus1[119896]



119905119899+1

asymp c119879119899+1

Ψ119899(119905) (27)

where

c119899+1

=8

1205871198772(119905119899+1

Ψ119899(119905))

=8

1205871198772int119877

0

radic119905119877 minus 1199052119905119899+1

Ψ119899(119905) 119889119905

(28)

Accordingly

P1=

[[[[[[[[[[

[

119860minus1

[2]

119860minus1

[3]

119860minus1

[119899+1]

c119879119899+1

]]]]]]]]]]

]

T119901asymp P1Ψ119899(119905)

(29)

Therefore

P = A119901P1 (30)

int119905

0

Ψ119899(119911) 119889119911 asymp A

119901P1Ψ119899(119905) (31)


119899(119905) =

sum119899

119894=0120582119894119894(119905) = Λ

119879



Λ119879AM120572(119905)

Aminus1Ψ119899(119905) + 119886 (119905) Λ

119879PΨ119899(119905)

+ 119887 (119905) Λ119879AM1Aminus1Ψ119899(119905) + 119888 (119905) Λ

119879

Ψ119899(119905)

= 119892 (119905) 119905 isin [0 119877]

(32)



Λ119879AM120572(119905119894)Aminus1Ψ119899(119905119894) + 119886 (119905

119894) Λ119879PΨ119899(119905119894)

+ 119887 (119905119894) Λ119879AM1Aminus1Ψ119899(119905119894) + 119888 (119905

119894) Λ119879

Ψ119899(119905119894)

= 119892 (119905119894)

(33)


0 1205821 120582



119879




120576 = max119894=01119899

1003816100381610038161003816119910 (119905119894) minus 119906119899 (119905119894)1003816100381610038161003816 (34)


119894= 119877((2119894 + 1)2(119899 + 1)) 119894 = 0 1 119899




03750 50000 63281 63281 02220119890 minus 13

06250 03125 113281 113281 00711119890 minus 13

08750 00000 169531 169531 00355119890 minus 13



03000 50000 49500 49500 02336119890 minus 12

05000 03125 87500 87500 01350119890 minus 12

07000 00000 129500 129500 00480119890 minus 12

09000 minus00000 175500 175500 00249119890 minus 12


119877 119899 = 3 119899 = 4 119899 = 5 119899 = 6





119863120572(119905)

119910 (119905) + 6int119905

0

119910 (119911) 119889119911 + 21199051199101015840

(119905) + 119910 (119905) = 119892 (119905)

119910 (0) = 0 119905 isin [0 119877]

(35)

where

120572 (119905) =3 (sin 119905 + cos 119905)

5

119892 (119905) =10

Γ (3 minus 120572 (119905))1199052minus120572(119905)

+15

Γ (2 minus 120572 (119905))1199051minus120572(119905)

+ 101199053

+ 701199052

+ 45119905

(36)




0 1205821 120582

119899]119879 is


1=






1198631199053

119910 (119905) + 119890119905

int119905

0

119910 (119911) 119889119911 + sin 1199051199101015840 (119905) + (119905 minus 1) 119910 (119905)

= 119892 (119905) 119910 (0) = 1 119905 isin [0 119877]

(37)

where

119892 (119905) =31199051minus1199053

(7119905 minus 6)

(18 minus 9119905 + 1199052) Γ (1 minus 1199053)+ 119890119905

(119905 minus1199052

2+1199053

3)


(38)



0 1205821 120582

119899]119879 is mainly


2=









03000 minus00000 07900 07900 02220119890 minus 15

05000 00625 07500 07500 05551119890 minus 15

07000 00000 07900 07900 04441119890 minus 15

09000 00000 09100 09100 06661119890 minus 15


R = 2 n = 3

0

10

20

30

40

50

y(t)

1 20 05 15

t

R = 8 n = 6

0

100

200

300

400

y(t)

2 4 6 80

t



R = 2 n = 5 R = 3 n = 3


05

1

15

2

25

3

y(t)

05 1 15 20

t

0

1

2

3

4

5

6

y(t)

1 2 30

t





119877 119899 = 3 119899 = 4 119899 = 5 119899 = 6






7 Conclusions


Competing Interests


Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












03750 50000 63281 63281 02220119890 minus 13

06250 03125 113281 113281 00711119890 minus 13

08750 00000 169531 169531 00355119890 minus 13



03000 50000 49500 49500 02336119890 minus 12

05000 03125 87500 87500 01350119890 minus 12

07000 00000 129500 129500 00480119890 minus 12

09000 minus00000 175500 175500 00249119890 minus 12


119877 119899 = 3 119899 = 4 119899 = 5 119899 = 6





119863120572(119905)

119910 (119905) + 6int119905

0

119910 (119911) 119889119911 + 21199051199101015840

(119905) + 119910 (119905) = 119892 (119905)

119910 (0) = 0 119905 isin [0 119877]

(35)

where

120572 (119905) =3 (sin 119905 + cos 119905)

5

119892 (119905) =10

Γ (3 minus 120572 (119905))1199052minus120572(119905)

+15

Γ (2 minus 120572 (119905))1199051minus120572(119905)

+ 101199053

+ 701199052

+ 45119905

(36)




0 1205821 120582

119899]119879 is


1=






1198631199053

119910 (119905) + 119890119905

int119905

0

119910 (119911) 119889119911 + sin 1199051199101015840 (119905) + (119905 minus 1) 119910 (119905)

= 119892 (119905) 119910 (0) = 1 119905 isin [0 119877]

(37)

where

119892 (119905) =31199051minus1199053

(7119905 minus 6)

(18 minus 9119905 + 1199052) Γ (1 minus 1199053)+ 119890119905

(119905 minus1199052

2+1199053

3)


(38)



0 1205821 120582

119899]119879 is mainly


2=









03000 minus00000 07900 07900 02220119890 minus 15

05000 00625 07500 07500 05551119890 minus 15

07000 00000 07900 07900 04441119890 minus 15

09000 00000 09100 09100 06661119890 minus 15


R = 2 n = 3

0

10

20

30

40

50

y(t)

1 20 05 15

t

R = 8 n = 6

0

100

200

300

400

y(t)

2 4 6 80

t



R = 2 n = 5 R = 3 n = 3


05

1

15

2

25

3

y(t)

05 1 15 20

t

0

1

2

3

4

5

6

y(t)

1 2 30

t





119877 119899 = 3 119899 = 4 119899 = 5 119899 = 6






7 Conclusions


Competing Interests


Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














03000 minus00000 07900 07900 02220119890 minus 15

05000 00625 07500 07500 05551119890 minus 15

07000 00000 07900 07900 04441119890 minus 15

09000 00000 09100 09100 06661119890 minus 15


R = 2 n = 3

0

10

20

30

40

50

y(t)

1 20 05 15

t

R = 8 n = 6

0

100

200

300

400

y(t)

2 4 6 80

t



R = 2 n = 5 R = 3 n = 3


05

1

15

2

25

3

y(t)

05 1 15 20

t

0

1

2

3

4

5

6

y(t)

1 2 30

t





119877 119899 = 3 119899 = 4 119899 = 5 119899 = 6






7 Conclusions


Competing Interests


Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119877 119899 = 3 119899 = 4 119899 = 5 119899 = 6






7 Conclusions


Competing Interests


Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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