M. H. Heydari, M. R. Hooshmandasl, F. M. Maalek Ghaini, and F. Mohammadi

(1)

Journal of Applied Mathematics Volume 2012, Article ID 542401,19pages doi:10.1155/2012/542401

Research Article

Wavelet Collocation Method for Solving Multiorder Fractional Differential Equations

M. H. Heydari, M. R. Hooshmandasl, F. M. Maalek Ghaini, and F. Mohammadi

Faculty of Mathematics, Yazd University, Yazd, Iran

Correspondence should be addressed to M. R. Hooshmandasl,hooshmandasl@yazduni.ac.ir Received 15 July 2011; Accepted 27 October 2011

Academic Editor: Md. Sazzad Chowdhury

Copyrightq2012 M. H. Heydari et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The operational matrices of fractional-order integration for the Legendre and Chebyshev wavelets are derived. Block pulse functions and collocation method are employed to derive a general procedure for forming these matrices for both the Legendre and the Chebyshev wavelets. Then numerical methods based on wavelet expansion and these operational matrices are proposed. In this proposed method, by a change of variables, the multiorder fractional diﬀerential equations MOFDEs with nonhomogeneous initial conditions are transformed to the MOFDEs with homogeneous initial conditions to obtain suitable numerical solution of these problems. Numerical examples are provided to demonstrate the applicability and simplicity of the numerical scheme based on the Legendre and Chebyshev wavelets.

1. Introduction

Fractional-order differential equationsFODEs, as generalizations of classical integer-order differential equations, are increasingly used to model some problems in fluid flow, mechanics, viscoelasticity, biology, physics, engineering, and other applications. Fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes1–6. Fractional differentiation and integration operators are also used for extensions of the diffusion and wave equations7. The solutions of FODEs are much involved, because in general, there exists no method that yields an exact solution for FODEs, and only approximate solutions can be derived using linearization or perturbation methods. Several methods have been suggested to solve fractional differential equations see8and references therein. Also there are different methods for solving MOFDEs as a special kind of FODEs9–17. However, few papers have reported applications of wavelets in solving fractional differential equations 8, 18–21. In view of successful application of

(2)

wavelet operational matrices in numerical solution of integral and diﬀerential equations22–

27, together with the characteristics of wavelet functions, we believe that they should be applicable in solving MOFDEs.

In this paper, the operational matrices of fractional-order integrations are derived and a general procedure based on collocation method and block pulse functions for forming these matrices is presented. Then, by application of these matrices, a numerical method for solving MOFDEs with nonhomogeneous conditions is presented. In the proposed method by a change of variables the MOFDEs with nonhomogeneous conditions transforms to the MOFDEs with homogeneous conditions. This way, we are able to obtain the exact solutions for such problems. In the proposed method, the Legendre and Chebyshev wavelet expansions along with operational matrices of fractional-order integrations are employed to reduce the MOFDE to systems of nonlinear algebraic equations. Illustrative examples of nonlinear types are given to demonstrate the eﬃciency and applicability of the proposed method. As numerical results show, the proposed method is eﬃcient and simple in implementation for both the Legendre and the Chebyshev wavelets. Moreover, for both these kinds of wavelets, numerical results have a good agreement with the exact solutions and the numerical results presented in other works.

The paper is organized as follows. InSection 2, we review some necessary definitions and mathematical preliminaries of fractional calculus and wavelets that are required for establishing our results. In Section 3 the Legendre and Chebyshev operational matrices of integration are derived. In Section 4 an application of the Legendre and Chebyshev operational matrices for solving the MOFDEs is presented. InSection 5the proposed method is applied to several numerical examples. Finally, a conclusion is given inSection 6.

2. Basic Definitions

2.1. Fractional Calculus

We give some basic definitions and properties of the fractional calculus theory which are used further in this paper.

Definition 2.1. A real functionft, t >0, is said to be in the spaceCμ,μ∈Rif there exists a real numberp> μsuch thatft t^pf1t, wheref1t∈C0,∞, and it is said to be in the spaceCⁿ_μiffⁿ∈Cμ,n∈N.

Definition 2.2. The Riemann-Liouville fractional integration operator of order α ≥ 0, of a functionf∈Cμ,μ≥ −1, is defined as follows4:

I^αft 1 Γα

_t

0

t−τ^α−1fτdτ, I⁰ft ft,

2.1

and according to4, we have

I^αI^βft I^αβft, I^αI^βft I^βI^αft, I^αt^ϑ Γϑ1

Γαϑ1t^αϑ,

2.2

whereα, β≥0,t >0 andϑ >−1.

(3)

Definition 2.3. The fractional derivative of order α > 0 in the Riemann-Liouville sense is defined as4

D^αft d

dt n

I^n−αft, n−1< α≤n, 2.3

wherenis an integer andf∈Cⁿ₁.

The Riemann-Liouville derivatives have certain disadvantages when trying to model real-world phenomena with fractional diﬀerential equations. Therefore, we will now intro- duce a modified fractional diﬀerential operatorD^α_∗ proposed by Caputo5.

Definition 2.4. The fractional derivative of orderα >0 in the Caputo sense is defined as5

D_∗^αft 1 Γn−α

_t

0

t−τ^n−α−1fⁿτdτ, n−1< α≤n, 2.4

wherenis an integert >0 andf∈C₁ⁿ. Caputos integral operator has a useful property:

I^αD_∗^αft ft−ⁿ⁻¹

k0

f^k0t^k

k!, n−1< α≤n, 2.5

wherenis an integert >0 andf∈C₁ⁿ. 2.2. Wavelets

Wavelets constitute a family of functions constructed from dilations and translations of a single function called the mother wavelet ψt. When the dilation parameter a and the translation parameter b vary continuously, we have the following family of continuous wavelets24:

ψa,bt |a|^−1/2ψ t−b

a

, a, b∈R, a /0. 2.6

If we restrict the parametersaandbto discrete values asaa^−k₀ ,bnb0a^−k₀ ,a0 >1, b0>0, fornandkpositive integers, we have the following family of discrete wavelets:

ψk,nt |a0|^k/2ψ

a^k₀t−nb0

, 2.7

whereψk,ntforms a wavelet basis forL²R. In particular, whena0 2 andb0 1,ψk,nt forms an orthonormal basis. That isψk,nt, ψl,mt δklδnm.

(4)

2.2.1. The Legendre Wavelets

The Legendre waveletsψn,mt ψk,n, m, t have four arguments;k ∈N,n 1,2, . . . ,2^k−1, andn2n−1; moreover,mis the order of the Legendre polynomials andtis the normalized time, and they are defined on the interval0,1as

ψn,mt

⎧⎪

⎨

⎪⎩

m1 22^k/2pm

2^kt−n

, n−1

2^k ≤t < n 2^k,

0, otherwise,

2.8

where m 0,1, . . . , M−1 and M is a fixed positive integer. The coeﬃcient

m1/2 in 2.8is for orthonormality, the dilation parameter isa 2^−k, and the translation parameter isb n2 ^−k. Here,Pmtare the well-known Legendre polynomials of order m which are orthogonal with respect to the weight functionwt 1 on the interval−1,1and satisfy the following recursive formula:

p0t 1, p1t t, pm1t

2m1 m1

tpmt− m

m1

pm−1t, m1,2,3, . . . . 2.9

2.2.2. The Chebyshev Wavelets

The Chebyshev waveletsψn,mt ψk,n, m, t have four arguments;k∈N,n1,2, . . . ,2^k−1, andn2n−1; moreover,mis the order of the Chebyshev polynomials of the first kind andt is the normalized time, and they are defined on the interval0,1as

ψn,mt

⎧⎪

⎨

⎪⎩ 2^k/2Tm

2^kt−n

, n−1

2^k ≤t < n 2^k,

0, otherwise,

2.10

where

Tmt

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

√1

π, m0,

2

πTmt, m >0,

2.11

where m 0,1, . . . , M−1 and Mis a fixed positive integer. The coeﬃcients in2.11are used for orthonormality. Here,Tmtare the well-known Chebyshev polynomials of orderm which are orthogonal with respect to the weight functionwt 1/√

1−t² on the interval

−1,1and satisfy the following recursive formula:

T0t 1, T1t t, Tm1t 2tTmt−Tm−1, m1,2,3. . . . 2.12

(5)

We should note that in dealing with the Chebyshev polynomials the weight functionw w2t−1has to be dilated and translated as follows:

wnt w

2^kt−n

, 2.13

to get orthogonal wavelets.

2.2.3. Function Approximation

A functionftdefined over0,1may be expanded as follows:

ft ^∞

n1

∞

m0cn,mψn,mt, 2.14

by the Legendre or Chebyshev wavelets, wherecn,m ft, ψn,mtin which,denotes the inner product.

If the infinite series in2.14is truncated, then2.14can be written as

ft≈² k−1

n1 M−1

m0

cn,mψn,mt C^TΨt, 2.15

whereCandΨtare 2^k−1M×1 matrices given by

C c10, c11, . . . , c1M−1, c20, . . . , c2M−1, . . . , c2^k−10, . . . , c2^k−1M−1^T, Ψ

ψ10t, ψ11t, . . . , ψ1M−1t, ψ20t, . . . , ψ2M−1t, . . . , ψ2^k−10t, . . . , ψ2^k−1M−1t_T

. 2.16

Taking the collocation points

ti 2i−1

2^kM , i1,2, . . . , m, 2.17

wherem2^k−1M, we define the wavelet matrixΦ_m×mas Φ_m×m

Ψ

1 2m

,Ψ

3 2m

, . . . ,Ψ

2m−1 2m

. 2.18

IndeedΦm×mhas the following form:

Φm×m

⎡

⎢⎢

⎣

A 0 · · · 0 0 0 A 0 · · · 0 0 0 A · · · 0 ... ... . .. ... ...

0 0 · · · 0 A

⎤

⎥⎥

⎦

, 2.19

(6)

whereAis aM×Mmatrix given by

A

⎡

⎢⎢

⎢⎣ ψ10

1 2m

ψ10

3 2m

· · · ψ10

2M−1 2m

ψ11

1 2m

ψ11

3 2m

· · · ψ11

2M−1 2m

... ... ... ...

ψ1M−1

1 2m

ψ1M−1

3 2m

· · · ψ1M−1

2M−1 2m

⎤

⎥⎥

⎥⎦

. 2.20

For example, forM 4 and k 2, the Legendre and Chebyshev matrices can be expressed as

Φ8×8 A 0

0 A

, 2.21

where for the Legendre matrix we have

A

⎡

⎢⎢

⎣

1.41421 1.41421 1.41421 1.41421

−1.83712 −.612372 0.612372 1.83712 1.08703 −1.28468 −1.28468 1.08703 0.263085 1.25697 −1.25697 −.263085

⎤

⎥⎥

⎦, 2.22

and for the Chebyshev matrix we have

A

⎡

⎢⎢

⎣

1.12838 1.12838 1.12838 1.12838

−1.19683 −.398942 −.398942 1.19683 0.199471 −1.39630 −1.39630 0.199471 0.897621 1.09709 −1.09709 −.897621

⎤

⎥⎥

⎦. 2.23

3. Operational Matrix of Fractional Integration

The integration of the vectorΨtdefined in2.16can be obtained by _t

0Ψτdτ≈PΨt, 3.1

wherePis them×moperational matrix for integration24.

Now, we derive the wavelet operational matrix of fractional integration. For this purpose, we rewrite the Riemann-Liouville fractional integration as

I^αf

t 1 Γα

_t

0

t−τ^α−1fτdτ 1

Γαt^α−1∗ft, 3.2

(7)

whereα∈Ris the order of the integration andt^α−1∗ftdenotes the convolution product of t^α−1 andft. Now ifftis expanded by the Legendre and Chebyshev wavelets, as shown in2.15, the Riemann-Liouville fractional integration becomes

I^αf

t 1

Γαt^α−1∗ft≈C^T 1 Γα

t^α−1∗Ψt

. 3.3

So that, ift^α−1 ∗ftcan be integrated, then by expandingftin the Legendre and Chebyshev wavelets, the Riemann-Liouville fractional integration can be solved via the Legendre and Chebyshev wavelets.

Also, we define anm-set of block pulse functionsBPFsas

bit

⎧⎪

⎨

⎪⎩ 1, i

m ≤t < i1

m ,

0, otherwise,

3.4

wherei0,1,2, . . . ,m−1.

The functionsbitare disjoint and orthogonal, that is,

bitblt

0, i /l, bit, il, ₁

0

biτblτdτ

⎧⎪

⎨

⎪⎩

0, i /l, 1

m, il.

3.5

Similarly, the Legendre and Chebyshev wavelets may be expanded intom-term block pulse functionsBPFsas follows:

Ψmt Φm×mBmt, 3.6

whereBmt b0t, b1t, . . . , bit, . . . , b_m−1t^T.

In28, Kilicman and Al Zhour have given the block pulse operational matrix of the fractional integrationF^αas follows:

I^αBmt≈F^αBmt, 3.7

where

F^α 1 m^α

1 Γα2

⎡

⎢⎢

⎢⎣

1 ξ1 ξ2 · · · ξm−1

0 1 ξ1 · · · ξm−2

0 0 1 · · · ξm−3

0 0 0 . .. ...

0 0 0 0 1

⎤

⎥⎥

⎥⎦

, 3.8

(8)

andξi i1^α1−2i^α1 i−1^α1. Next, we derive the Legendre and Chebyshev wavelet operational matrices of the fractional integration. Let

I^αΨmt≈P_m×m^α Ψmt, 3.9

where the matrixP_m×m^α is called the wavelet operational matrix of the fractional integration.

Using3.6and3.7, we have

I^αΨmt≈I^αΦ_m×mB_m×mt Φ_m×mI^αBmt≈Φ_m×mF^αBmt, 3.10

and from3.9and3.10, we get

P_m×m^α Ψmt≈P_m×m^α Φ_m×mBmt≈Φ_m×mF^αBmt. 3.11

Thus, the Legendre and Chebyshev wavelet operational matrices of the fractional inte- grationP_m×m^α can be approximately expressed by

P_m×m^α Φ_m×mF^αΦ⁻¹_m×m. 3.12

Also, from3.3and3.12, we obtain I^αf

t≈C^TΦ_m×mF^αBmt. 3.13

Here, for the Legendre and Chebyshev wavelets, we can obtainΦ_m×mF^αas follows:

Φ_m×mF^α

⎡

⎢⎢

⎣

B1 B2 B3 · · · Bk

0 B1 B2 · · · Bk−1

0 0 B1 · · · B_k−2 0 0 0 . .. ...

0 0 0 0 B1

⎤

⎥⎥

⎦

, 3.14

whereBr,r1,2, . . . ,kk2^k−1, areM×Mmatrices given by

Br b^r_ij

M×M, 3.15

such that forr1,

b¹_ij ^j

l1

ailξ_j−l, ξ₀1, 3.16

(9)

and forr2,3, . . . , M,

Br b^r_ij

M×M, l2,3, . . . ,k, b^r_ij ^M

l1

ailξ_r−1M−lj,

3.17

where

aij Ψ_1i−1 2j−1

2m

, 3.18

andξ_j 1/m^αΓα2ξj,j0,1, . . . m−1.

Now from3.12and3.14, we have

P_m×m^α

⎡

⎢⎢

⎢⎣

D1 D2 D3 · · · D_k 0 D1 D2 · · · Dk−1

0 0 D1 · · · Dk−2

0 0 0 . .. ...

0 0 0 0 D1

⎤

⎥⎥

⎥⎦

, 3.19

whereDiBiA⁻¹, i1,2, . . .k.

4. Applications and Results

In this section, the Legendre and Chebyshev wavelet expansions together with their operational matrices of fractional-order integration are used to obtain numerical solution of MOFDEs.

Consider the following nonlinear MOFDE:

β0tD^α_∗yt ^s¹

i1

βitD^α_∗ⁱyt ^s²

i1

γityⁱt ft, y^j0 cj, j 0,1, . . . n−1, 4.1

where 0 ≤ t < 1,n−1 < α≤ n, 0 < α1 < α2 < · · · < αs1 < α,n,s1ands2are fixed positive integers,D_∗^αdenotes the Caputo fractional derivative of orderα,fis a known function oft, cj,j0,1, . . . , n−1, are arbitrary constants, andyis an unknown function to be determined later. To solve this problem, we apply the following scheme.

Suppose

yt ⁿ⁻¹

j0

cjt^j

j! Yt. 4.2

(10)

Under this change of variable, we have

yⁱt

⎛

⎝ⁿ⁻¹

j0

cjt^j j!Yt

⎞

⎠

i

ⁱ

s0

i s

⎛⎝ⁿ⁻¹

j0

cjt^j j!

⎞

⎠

s

Yt^i−s,

D_∗^αyt D_∗^αYt,

D_∗^αⁱyt D^α_∗ⁱYt, n−1< αi< α, D^α_∗ⁱyt D^α_∗ⁱYt hit, αi≤n−1,

4.3

where

hit D_∗^αⁱ

⎛

⎝ⁿ⁻¹

j0

cjt^j j!

⎞

⎠. 4.4

Now, by substituting4.3into4.1, we transform the nonlinear MOFDE4.1with nonhomogeneous conditions to a nonlinear MOFDE with homogeneous conditions as follows:

β0tD_∗^αYt^s¹

i1

βitD^α_∗ⁱYt^s²

i1

i s0

γi,stY^i−st ft, Y^j0 0, j 0,1, . . . n−1, 4.5

where

β0t β0t,

βit βit, n−1< αi< α, βit βithit, αi≤n−1,

γi,st γit i

s

⎛⎝ⁿ⁻¹

j0

cjt^j j!

⎞

⎠

s

,

4.6

andft is a known function.

We assume thatD_∗^αYtis given by

D^α_∗Yt K_m^TΨmt, 4.7

whereK^T_mis an unknown vector andΨmtis the vector which is defined in3.6. By using initial conditions and2.4, we have

D^α_∗ⁱYt K^T_mP_m×m^α−αⁱΨmt, i1,2, . . . , s1, 4.8

Yt K^T_mP_m×m^α Ψmt. 4.9

(11)

SinceΨm Φm×mBmt,4.9can be rewritten as follows:

Yt K_m^TP_m×m^α Φm×mBmt a1, a2, . . . , amBmt. 4.10 Now by using3.4, we obtain

Yt^i−s#

aî−s₁ , aî−s₂ , . . . , aî−s_m $

Bmt, i1,2, . . . , s2, s0,1, . . . , i. 4.11

Moreover, we expand functionsβit,γi,st, andft by wavelets as follows:

βit Ψmt^Tβi,m, i0,1, . . . , s1,

γi,st Ψmt^Tγi,s,m, i1,2, . . . , s2, s0,1, . . . , i, ft f_m^TΨmt,

4.12

whereβi,m,γi,m, andf_m^T are known vectors. By substituting4.7,4.8, and4.11-4.12into 4.5, we obtain

Ψmt^Tβ0,mK_m^TΨmt^s¹

i1

Ψmt^Tβi,mK^T_mP_m×m^α−αⁱΨmt^s²

i1

i s0

Ψmt^Tγi,s,m

#

aî−s₁ , aî−s₂ , . . . , aî−s_m $ Bmt Bmt^T1,1, . . . ,1^Tf_m^TΨmt.

4.13

Now, from3.12,4.10, and4.13, we have

Bmt^T

%

Φ^T_m×mβ0,ma1, a2, . . . , amF^α⁻¹^s¹

i1

Φ^T_m×mβi,ma1, a2, . . . , amF^α⁻¹F^α^α−^αi

^s²

i1

i s0

Φ^T_m×mγi,s,m

#aî−s₁ , aî−s₂ , . . . , aî−s_m $&

Bmt Bmt^T1,1, . . . ,1^Tf_m^TΦ_m×mBmt.

4.14

This is a nonlinear algebraic equation for unknown vectora1, a2, . . . , am. Here, by taking collocation points, expressed in2.17, we transform4.14into a nonlinear system of algebraic equations. This nonlinear system can be solved by the Newton iteration method for unknown vectora1, a2, . . . , am. Therefore,Ytas the solution of4.5is

Yt a1, a2, . . . , amBmt, 4.15

and finally,ytas the solution of4.1will be

yt ⁿ⁻¹

j0

cjt^j

j! a1, a2, . . . , amBmt. 4.16

(12)

In cases where all of coeﬃcientsβitandγi,st, are constants, that is,βit βiand

γi,st γi,swe can reduce4.13and4.14, respectively, as follows:

β0K^T_mΨmt ^s¹

i1

βiK^T_mP_m×m^α−αⁱΨmt ^s²

i1

i s0

γi,s

#

aî−s₁ , aî−s₂ , . . . , aî−s_m $

Bmt f_m^TΨmt,

%

a1, a2, . . . , am

%

β0F^α⁻¹F^α⁻¹^s¹

i1

βiF^α−αⁱ

&

^s²

i1

i s0

γi,s

#

aî−s₁ , aî−s₂ , . . . , aî−s_m $&

Bmt f_m^TΦm×mBmt.

4.17

This is a nonlinear system of algebraic equations for unknown vectora1, a2, . . . , am and can be solved directly without use of collocation points by the Newton iteration method.

Here, as before we obtain

Yt a1, a2, . . . , amBmt, 4.18

as the solution of4.5, and finally,ytas the solution of4.1will be

yt ⁿ⁻¹

j0

cjt^j

j! a1, a2, . . . , amBmt. 4.19

5. Numerical Examples

In this section we demonstrate the eﬃciency of the proposed wavelet collocation method for the numerical solution of MOFDEs. These examples are considered because either closed form solutions are available for them, or they have also been solved using other numerical schemes, by other authors.

Example 5.1. Consider the homogeneous Bagley-Torvik equation13,16:

D_∗²yt D^0.5_∗ yt yt 0, 5.1

subject to the following initial conditions:

y0 1, y0 0. 5.2

Here, we apply the method ofSection 4for solving this problem. By setting

yt 1Yt. 5.3

Equation5.1can be rewritten as

D_∗²Yt D_∗^0.5Yt Yt 10, 5.4

(13)

subject to the initial condition

Y0 Y0 0. 5.5

By using the proposed method, we get a system of algebraic equations for both the Legendre and the Chebyshev wavelets for5.4as follows:

a1, a2, . . . , am#

F²⁻¹F²⁻¹F^1.5$

a1, a2, . . . , am 1,1, . . . ,1 0. 5.6

Since the linear system of algebraic equations5.6is the same for both kinds of wavelets, we have the same numerical solution for the Legendre and Chebyshev wavelets. By solving this linear system, we get numerical solutions for5.4as follows:

Yt a1, a2, . . . , amBmt. 5.7

Finally the solution of the original problem5.1is

yt 1 a1, a2, . . . , amBmt. 5.8

Figure 1shows the behavior of the numerical solution form160M5, k6. AsFigure 1 shows, this method is very eﬃcient for numerical solution of this problem and the solution can be derived in a large interval0,40.

Example 5.2. Consider the nonhomogeneous Bagley-Torvik equation10,13–15,17,20:

aD²_∗yt bD^1.5_∗ yt cyt ft, 5.9

where

ft c1t, 5.10

subject to the initial condition

y0 y0 1, 5.11

which has the exact solution

yt 1t. 5.12

Here, we solve this problem by applying the method ofSection 4. By setting

yt 1tYt. 5.13

(14)

0 10 20 30 40 t

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

y(t)

Figure 1: Numerical solution by the Legendre and Chebyshev wavelets.

Equation5.9can be rewritten as

aD²_∗Yt bD^1.5_∗ Yt cYt 0, 5.14

Y0 Y0 0. 5.15

The system of algebraic equations for both the Legendre and the Chebyshev wavelets for 5.14has the following form:

a1, a2, . . . , am#

aF²⁻¹bF²⁻¹F^0.5cI_m×m$

0. 5.16

Here, the linear system of algebraic equations5.16is nonsingular and so has only the trivial solution, that is,Yt 0. Then the solution of the original problem5.9is

yt 1t, 5.17

which is the exact solution. It is mentionable that this equation has been solved by Diethelm and Luchko10 forabc1with error 1.60E−4, Diethelm and Ford17 fora1, bc0.5with error 5.62E−3, while in20Li and Zhao obtained approximation solution by the Haar wavelet.

Example 5.3. Consider the following nonhomogenous MOFDE13:

aD^α_∗yt bD_∗^α²yt cD_∗^α¹yt eyt ft, 0< α1≤1, 1< α2≤2, 3< α <4, 5.18

(15)

where

ft 2b

Γ3−α2t^2−α² 2c

Γ3−α1t^2−α¹e t²−t

, 5.19

y0 0, y0 −1, y0 2, y0 0, 5.20

yt t²−t. 5.21

yt −tt²Yt. 5.22

Equation5.18can be rewritten as follows:

aD_∗^αYt bD^α_∗²Yt cD^α_∗¹Yt eYt 0, 0< α1≤1, 1< α2≤2, 3< α <4, 5.23

subject to the initial conditions

Y0 0, Y 0 0, Y0 0, Y0 0. 5.24

The system of algebraic equations corresponding to the Legendre and Chebyshev wavelets for5.23has the following form:

a1, a2, . . . , am#

aF^α⁻¹bF^α⁻¹F^α−α²cF^α⁻¹F^α−α¹eIm×m

$0. 5.25

Here, the linear system of algebraic5.25is nonsingular and so has only the trivial solution, that is,Yt 0. Then the solution of the original problem5.18is

yt t²−t, 5.26

which is the exact solution. This equation has been solved by El-Sayed et al.13 forab ce1,α10.77,α21.44, andα0.91with error 9.53E−4.

Example 5.4. Consider the following nonlinear MOFDE15:

D³_∗yt D^2.5_∗ yt yt₂

t⁴, 5.27

(16)

y0 y0 0, y0 2, 5.28

yt t². 5.29

yt t²Yt. 5.30

D³_∗Yt D^2.5_∗ Yt Yt²2t²Yt 0, 5.31

Y0 0, Y0 0, Y0 0. 5.32

The nonlinear algebraic equation corresponding to the Legendre and Chebyshev wavelets for 5.31has the following form:

Bmt^T#

1,1, . . . ,1^Ta1, a2, . . . , am#

F³⁻¹F³⁻¹F^0.5$

1,1, . . . ,1^T#

a²₁, a²₂, . . . , a²_m$ Φ^T_m×mγma1, a2, . . . , am$

Bmt 0, 5.33

whereγm is a known constant vector corresponding to the kind of wavelet expansions and Φ_m×m is the wavelet matrix. By taking collocation points expressed in2.17, we transform 5.33 into a nonlinear system of algebraic equations. Then applying Newton iteration method for solving this nonlinear system, we obtain only trivial solution, that is,Yt 0.

Then the solution of the original problem5.27is

yt t², 5.34

which is the exact solution.

Example 5.5. Consider the following nonlinear MOFDE14:

aD³_∗yt bD_∗^α²yt cD_∗^α¹yt e yt₂

ft, 0< α1≤1, 1< α2<2, 5.35

where

ft ct^1−α¹

Γ2−α1et², 5.36

(17)

y0 0, y0 1, y0 0, 5.37

yt t. 5.38

yt tYt. 5.39

aD_∗³Yt bD_∗^α²Yt cD_∗^α¹Yt eYt²2etYt 0, 5.40

Y0 0, Y0 0, Y0 0. 5.41

The nonlinear algebraic equation corresponding to the Legendre and Chebyshev wavelets for 5.40has the following form:

Bmt^T#

1,1, . . . ,1^Ta1, a2, . . . , am#

aF³⁻¹bF³⁻¹F^3−α²cF³⁻¹F^3−α¹$ 1,1, . . . ,1^T#

a²₁, a²₂, . . . , a²_m$

Φ^T_m×mγma1, a2, . . . , am$

Bmt 0,

5.42

whereγm is a known constant vector corresponding to the kind of wavelet expansion and Φm×m is the wavelet matrix. By taking collocation points expressed in2.17, we transform 5.42 into a nonlinear system of algebraic equations. By applying the Newton iteration method for solving this nonlinear system, we obtain only trivial solution, that is,Yt 0.

Then the solution of the original problem5.35is

yt t, 5.43

which is the exact solution.

6. Conclusion

In this paper a general formulation for the Legendre and Chebyshev wavelet operational matrices of fractional-order integration has been derived. Then a numerical method based on Legendre and Chebyshev wavelets expansions together with these matrices are proposed to obtain the numerical solutions of MOFDEs. In this proposed method, by a change of variables, the MOFDEs with nonhomogeneous conditions are transformed to the MOFDEs

(18)

with homogeneous conditions. The exact solutions for some MOFDEs are obtained by our method. The proposed method is very simple in implementation for both the Legendre and the Chebyshev wavelets. As the numerical results show, the method is very eﬃcient for the numerical solution of MOFDEs and only a few number of wavelet expansion terms are needed to obtain a good approximate solution for these problems.

References

1 A. Carpinteri and F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, Springer, New York, NY, USA, 1997.

2 K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Diﬀerential Equations, John Wiley & Sons, New York, NY, USA, 1993.

3 K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974.

4 I. Podlubny, Fractional Diﬀerential Equations, vol. 198, Academic Press, San Diego, Calif, USA, 1999.

5 I. Podlubny, “Fractional-order systems and fractional-order controllers,” Report UEF-03-94, Slovak Academy of Sciences, Institute of Experimental Physics, Kosice, Slovakia, 1994.

6 R. Gorenflo and F. Mainardi, “Fractional calculus: integral and diﬀerential equations of fractional order,” in Fractals and Fractional Calculus in Continuum Mechanics, vol. 378, pp. 223–276, Springer, Vienna, Austria, 1997.

7 W. R. Schneider and W. Wyss, “Fractional diﬀusion and wave equations,” Journal of Mathematical Physics, vol. 30, no. 1, pp. 134–144, 1989.

8 Y. Li, “Solving a nonlinear fractional diﬀerential equation using Chebyshev wavelets,” Communica- tions in Nonlinear Science and Numerical Simulation, vol. 15, no. 9, pp. 2284–2292, 2010.

9 K. Diethelm and N. J. Ford, “Multi-order fractional diﬀerential equations and their numerical solution,” Applied Mathematics and Computation, vol. 154, no. 3, pp. 621–640, 2004.

10 K. Diethelm and Y. Luchko, “Numerical solution of linear multi-term initial value problems of fractional order,” Journal of Computational Analysis and Applications, vol. 6, no. 3, pp. 243–263, 2004.

11 K. Diethelm, “Multi-term fractional differential equations multi-order fractional differential systems and their numerical solution,” Journal Européen des Systèmes Automatisés, vol. 42, pp. 665–676, 2008.

12 A. M. A. El-Sayed, A. E. M. El-Mesiry, and H. A. A. El-Saka, “Numerical solution for multi-term fractionalarbitraryorders diﬀerential equations,” Computational & Applied Mathematics, vol. 23, no.

1, pp. 33–54, 2004.

13 A. M. A. El-Sayed, A. E. M. El-Mesiry, and H. A. A. El-Saka, “Numerical methods for multi-term fractionalarbitraryorders diﬀerential equations,” Applied Mathematics and Computation, vol. 160, no. 3, pp. 683–699, 2005.

14 A. M. A. El-Sayed, I. L. El-Kalla, and E. A. A. Ziada, “Analytical and numerical solutions of multi- term nonlinear fractional orders diﬀerential equations,” Applied Numerical Mathematics, vol. 60, no. 8, pp. 788–797, 2010.

15 V. Daftardar-Gejji and H. Jafari, “Solving a multi-order fractional diﬀerential equation using Adomian decomposition,” Applied Mathematics and Computation, vol. 189, no. 1, pp. 541–548, 2007.

16 J. T. Edwards, N. J. Ford, and A. C. Simpson, “The numerical solution of linear multiterm fractional diﬀerential equations: systems of equations,” Manchester Center for Numerical Computational Mathematics, 2002.

17 K. Diethelm and N. J. Ford, “Numerical solution of the Bagley-Torvik equation,” BIT. Numerical Mathematics, vol. 42, no. 3, pp. 490–507, 2002.

18 J. L. Wu, “A wavelet operational method for solving fractional partial diﬀerential equations numeri- cally,” Applied Mathematics and Computation, vol. 214, no. 1, pp. 31–40, 2009.

19 U. Lepik, “Solving fractional integral equations by the Haar wavelet method,” Applied Mathematics¨ and Computation, vol. 214, no. 2, pp. 468–478, 2009.

20 Y. Li and W. Zhao, “Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order diﬀerential equations,” Applied Mathematics and Computation, vol. 216, no. 8, pp. 2276–2285, 2010.

21 M. U. Rehman and R. A. Khan, “The legendre wavelet method for solving fractional diﬀerential equations,” Communication in Nonlinear Science and Numerical Simulation, vol. 227, no. 2, pp. 234–244, 2009.

(19)

22 N. M. Bujurke, S. C. Shiralashetti, and C. S. Salimath, “An application of single-term Haar wavelet series in the solution of nonlinear oscillator equations,” Journal of Computational and Applied Mathematics, vol. 227, no. 2, pp. 234–244, 2009.

23 E. Babolian, Z. Masouri, and S. Hatamzadeh-Varmazyar, “Numerical solution of nonlinear Volterra- Fredholm integro-diﬀerential equations via direct method using triangular functions,” Computers &

Mathematics with Applications, vol. 58, no. 2, pp. 239–247, 2009.

24 M. Tavassoli Kajani, A. Hadi Vencheh, and M. Ghasemi, “The Chebyshev wavelets operational matrix of integration and product operation matrix,” International Journal of Computer Mathematics, vol. 86, no.

7, pp. 1118–1125, 2009.

25 M. H. Reihani and Z. Abadi, “Rationalized Haar functions method for solving Fredholm and Volterra integral equations,” Journal of Computational and Applied Mathematics, vol. 200, no. 1, pp. 12–20, 2007.

26 F. Khellat and S. A. Yousefi, “The linear Legendre mother wavelets operational matrix of integration and its application,” Journal of the Franklin Institute, vol. 343, no. 2, pp. 181–190, 2006.

27 M. Razzaghi and S. Yousefi, “The Legendre wavelets operational matrix of integration,” International Journal of Systems Science, vol. 32, no. 4, pp. 495–502, 2001.

28 A. Kilicman and Z. A. Al Zhour, “Kronecker operational matrices for fractional calculus and some applications,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 250–265, 2007.

(20)

Submit your manuscripts at http://www.hindawi.com

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematics

^{Journal of}

in Engineering

Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations

International Journal of

Volume 2014

Applied Mathematics^{Journal of}

Probability and Statistics

Journal of

Mathematical PhysicsAdvances in

Complex Analysis

^{Journal of}

Optimization

^{Journal of}

Combinatorics

Operations Research

Journal of

Function Spaces

Abstract and Applied Analysis

International Journal of Mathematics and Mathematical Sciences

The Scientific World Journal

Algebra

Discrete Dynamics in Nature and Society

Decision Sciences

Discrete Mathematics

^{Journal of}

M. H. Heydari, M. R. Hooshmandasl, F. M. Maalek Ghaini, and F. Mohammadi

Research Article

Wavelet Collocation Method for Solving Multiorder Fractional Differential Equations

M. H. Heydari, M. R. Hooshmandasl, F. M. Maalek Ghaini, and F. Mohammadi

1. Introduction

2. Basic Definitions

3. Operational Matrix of Fractional Integration

4. Applications and Results

5. Numerical Examples

6. Conclusion

References

Submit your manuscripts at http://www.hindawi.com

Mathematics

Complex Analysis

Optimization

Combinatorics

Function Spaces

The Scientific World Journal

Algebra

Decision Sciences

Discrete Mathematics

Stochastic Analysis