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Journal of Applied Mathematics
Volume 2012, Article ID 542401, 19 pages
http://dx.doi.org/10.1155/2012/542401
Research Article

Wavelet Collocation Method for Solving Multiorder Fractional Differential Equations

Faculty of Mathematics, Yazd University, Yazd, Iran

Received 15 July 2011; Accepted 27 October 2011

Academic Editor: Md. Sazzad Chowdhury

Copyright © 2012 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.

Abstract

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 equations (FODEs), 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 processes [16]. Fractional differentiation and integration operators are also used for extensions of the diffusion and wave equations [7]. 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 (see [8] and references therein). Also there are different methods for solving MOFDEs as a special kind of FODEs [917]. However, few papers have reported applications of wavelets in solving fractional differential equations [8, 1821]. In view of successful application of wavelet operational matrices in numerical solution of integral and differential equations [2227], 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. In Section 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. In Section 5 the proposed method is applied to several numerical examples. Finally, a conclusion is given in Section 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 function 𝑓(𝑡),𝑡>0, is said to be in the space 𝐶𝜇, 𝜇 if there exists a real number 𝑝(>𝜇) such that 𝑓(𝑡)=𝑡𝑝𝑓1(𝑡), where 𝑓1(𝑡)𝐶[0,], and it is said to be in the space 𝐶𝑛𝜇 if 𝑓(𝑛)𝐶𝜇, 𝑛.

Definition 2.2. The Riemann-Liouville fractional integration operator of order 𝛼0, of a function 𝑓𝐶𝜇, 𝜇1, is defined as follows [4]: 𝐼𝛼1𝑓(𝑡)=Γ(𝛼)𝑡0(𝑡𝜏)𝛼1𝐼𝑓(𝜏)𝑑𝜏,0𝑓(𝑡)=𝑓(𝑡),(2.1) and according to [4], we have 𝐼𝛼𝐼𝛽𝑓(𝑡)=𝐼𝛼+𝛽𝐼𝑓(𝑡),𝛼𝐼𝛽𝑓(𝑡)=𝐼𝛽𝐼𝛼𝐼𝑓(𝑡),𝛼𝑡𝜗=Γ(𝜗+1)𝑡Γ(𝛼+𝜗+1)𝛼+𝜗,(2.2) where 𝛼,𝛽0, 𝑡>0 and 𝜗>1.

Definition 2.3. The fractional derivative of order 𝛼>0 in the Riemann-Liouville sense is defined as [4] 𝐷𝛼𝑑𝑓(𝑡)=𝑑𝑡𝑛𝐼𝑛𝛼𝑓(𝑡),(𝑛1<𝛼𝑛),(2.3) where 𝑛 is an integer and 𝑓𝐶𝑛1.
The Riemann-Liouville derivatives have certain disadvantages when trying to model real-world phenomena with fractional differential equations. Therefore, we will now introduce a modified fractional differential operator 𝐷𝛼 proposed by Caputo [5].

Definition 2.4. The fractional derivative of order 𝛼>0 in the Caputo sense is defined as [5] 𝐷𝛼1𝑓(𝑡)=Γ(𝑛𝛼)𝑡0(𝑡𝜏)𝑛𝛼1𝑓(𝑛)(𝜏)𝑑𝜏,(𝑛1<𝛼𝑛),(2.4) where 𝑛 is an integer 𝑡>0 and 𝑓𝐶𝑛1. Caputos integral operator has a useful property: 𝐼𝛼𝐷𝛼𝑓(𝑡)=𝑓(𝑡)𝑛1𝑘=0𝑓(𝑘)0+𝑡𝑘𝑘!,(𝑛1<𝛼𝑛),(2.5) where 𝑛 is an integer 𝑡>0 and 𝑓𝐶𝑛1.

2.2. Wavelets

Wavelets constitute a family of functions constructed from dilations and translations of a single function called the mother wavelet 𝜓(𝑡). When the dilation parameter 𝑎 and the translation parameter 𝑏 vary continuously, we have the following family of continuous wavelets [24]: 𝜓𝑎,𝑏(𝑡)=|𝑎|1/2𝜓𝑡𝑏𝑎,𝑎,𝑏,𝑎0.(2.6)

If we restrict the parameters 𝑎 and 𝑏 to discrete values as 𝑎=𝑎0𝑘, 𝑏=𝑛𝑏0𝑎0𝑘, 𝑎0>1, 𝑏0>0, for 𝑛 and 𝑘 positive integers, we have the following family of discrete wavelets: 𝜓𝑘,𝑛||𝑎(𝑡)=0||𝑘/2𝜓𝑎𝑘0𝑡𝑛𝑏0,(2.7) where 𝜓𝑘,𝑛(𝑡) forms a wavelet basis for 𝐿2(). In particular, when 𝑎0=2 and 𝑏0=1, 𝜓𝑘,𝑛(𝑡) forms an orthonormal basis. That is (𝜓𝑘,𝑛(𝑡),𝜓𝑙,𝑚(𝑡))=𝛿𝑘𝑙𝛿𝑛𝑚.

2.2.1. The Legendre Wavelets

The Legendre wavelets 𝜓𝑛,𝑚(𝑡)=𝜓(𝑘,̂𝑛,𝑚,𝑡) have four arguments; 𝑘, 𝑛=1,2,,2𝑘1, and ̂𝑛=2𝑛1; moreover, 𝑚 is the order of the Legendre polynomials and 𝑡 is the normalized time, and they are defined on the interval [0,1) as 𝜓𝑛,𝑚(𝑡)=1𝑚+22𝑘/2𝑝𝑚2𝑘,𝑡̂𝑛̂𝑛12𝑘𝑡<̂𝑛2𝑘,0,otherwise,(2.8) where 𝑚=0,1,,𝑀1 and 𝑀 is a fixed positive integer. The coefficient 𝑚+1/2 in (2.8) is for orthonormality, the dilation parameter is 𝑎=2𝑘, and the translation parameter is 𝑏=̂𝑛2𝑘. Here, 𝑃𝑚(𝑡) are the well-known Legendre polynomials of order 𝑚 which are orthogonal with respect to the weight function 𝑤(𝑡)=1 on the interval [1,1] and satisfy the following recursive formula: 𝑝0(𝑡)=1,𝑝1(𝑡)=𝑡,𝑝𝑚+1(𝑡)=2𝑚+1𝑚+1𝑡𝑝𝑚𝑚(𝑡)𝑝𝑚+1𝑚1(𝑡),𝑚=1,2,3,.(2.9)

2.2.2. The Chebyshev Wavelets

The Chebyshev wavelets 𝜓𝑛,𝑚(𝑡)=𝜓(𝑘,̂𝑛,𝑚,𝑡) have four arguments; 𝑘, 𝑛=1,2,,2𝑘1, and ̂𝑛=2𝑛1; moreover, 𝑚 is the order of the Chebyshev polynomials of the first kind and 𝑡 is the normalized time, and they are defined on the interval [0,1) as 𝜓𝑛,𝑚2(𝑡)=𝑘/2𝑇𝑚2𝑘,𝑡̂𝑛̂𝑛12𝑘𝑡<̂𝑛2𝑘,0,otherwise,(2.10) where 𝑇𝑚1(𝑡)=𝜋,𝑚=0,2𝜋𝑇𝑚(𝑡),𝑚>0,(2.11) where 𝑚=0,1,,𝑀1 and 𝑀 is a fixed positive integer. The coefficients in (2.11) are used for orthonormality. Here, 𝑇𝑚(𝑡) are the well-known Chebyshev polynomials of order 𝑚 which are orthogonal with respect to the weight function 𝑤(𝑡)=1/1𝑡2 on the interval [1,1] and satisfy the following recursive formula: 𝑇0(𝑡)=1,𝑇1(𝑡)=𝑡,𝑇𝑚+1(𝑡)=2𝑡𝑇𝑚(𝑡)𝑇𝑚1,𝑚=1,2,3.(2.12) We should note that in dealing with the Chebyshev polynomials the weight function 𝑤=𝑤(2𝑡1) has to be dilated and translated as follows: 𝑤𝑛2(𝑡)=𝑤𝑘𝑡̂𝑛,(2.13) to get orthogonal wavelets.

2.2.3. Function Approximation

A function 𝑓(𝑡) defined over [0,1) may be expanded as follows: 𝑓(𝑡)=𝑛=1𝑚=0𝑐𝑛,𝑚𝜓𝑛,𝑚(𝑡),(2.14) by the Legendre or Chebyshev wavelets, where 𝑐𝑛,𝑚=(𝑓(𝑡),𝜓𝑛,𝑚(𝑡)) in which (,) denotes the inner product.

If the infinite series in (2.14) is truncated, then (2.14) can be written as 𝑓(𝑡)2𝑘1𝑛=1𝑀1𝑚=0𝑐𝑛,𝑚𝜓𝑛,𝑚(𝑡)=𝐶𝑇Ψ(𝑡),(2.15) where 𝐶 and Ψ(𝑡) are 2𝑘1𝑀×1 matrices given by 𝑐𝐶=10,𝑐11,,𝑐1𝑀1,𝑐20,,𝑐2𝑀1,,𝑐2𝑘10,,𝑐2𝑘1𝑀1𝑇,𝜓Ψ=10(𝑡),𝜓11(𝑡),,𝜓1𝑀1(𝑡),𝜓20(𝑡),,𝜓2𝑀1(𝑡),,𝜓2𝑘10(𝑡),,𝜓2𝑘1𝑀1(𝑡)𝑇.(2.16)

Taking the collocation points 𝑡𝑖=(2𝑖1)2𝑘𝑀,𝑖=1,2,,𝑚,(2.17) where 𝑚=2𝑘1𝑀, we define the wavelet matrix Φ𝑚×𝑚 as Φ𝑚×𝑚=Ψ132𝑚,Ψ2𝑚,,Ψ2𝑚12𝑚.(2.18)

Indeed Φ𝑚×𝑚 has the following form: Φ𝑚×𝑚=𝐴0000𝐴0000𝐴0000𝐴,(2.19) where 𝐴 is a 𝑀×𝑀 matrix given by 𝜓𝐴=101𝜓2𝑚1032𝑚𝜓102𝑀1𝜓2𝑚111𝜓2𝑚1132𝑚𝜓112𝑀1𝜓2𝑚1𝑀11𝜓2𝑚1𝑀132𝑚𝜓1𝑀12𝑀12𝑚.(2.20)

For example, for 𝑀=4 and 𝑘=2, the Legendre and Chebyshev matrices can be expressed as Φ8×8=𝐴00𝐴,(2.21) where for the Legendre matrix we have 𝐴=1.414211.414211.414211.414211.83712.6123720.6123721.837121.087031.284681.284681.087030.2630851.256971.25697.263085,(2.22) and for the Chebyshev matrix we have 𝐴=1.128381.128381.128381.128381.19683.398942.3989421.196830.1994711.396301.396300.1994710.8976211.097091.09709.897621.(2.23)

3. Operational Matrix of Fractional Integration

The integration of the vector Ψ(𝑡) defined in (2.16) can be obtained by 𝑡0Ψ(𝜏)𝑑𝜏𝑃Ψ(𝑡),(3.1) where 𝑃 is the 𝑚×𝑚 operational matrix for integration [24].

Now, we derive the wavelet operational matrix of fractional integration. For this purpose, we rewrite the Riemann-Liouville fractional integration as (𝐼𝛼1𝑓)(𝑡)=Γ(𝛼)𝑡0(𝑡𝜏)𝛼11𝑓(𝜏)𝑑𝜏=𝑡Γ(𝛼)𝛼1𝑓(𝑡),(3.2) where 𝛼 is the order of the integration and 𝑡𝛼1𝑓(𝑡) denotes the convolution product of 𝑡𝛼1 and 𝑓(𝑡). Now if 𝑓(𝑡) is expanded by the Legendre and Chebyshev wavelets, as shown in (2.15), the Riemann-Liouville fractional integration becomes (𝐼𝛼1𝑓)(𝑡)=Γ𝑡(𝛼)𝛼1𝑓(𝑡)𝐶𝑇1Γ𝑡(𝛼)𝛼1Ψ(𝑡).(3.3)

So that, if 𝑡𝛼1𝑓(𝑡) can be integrated, then by expanding 𝑓(𝑡) in the Legendre and Chebyshev wavelets, the Riemann-Liouville fractional integration can be solved via the Legendre and Chebyshev wavelets.

Also, we define an 𝑚-set of block pulse functions (BPFs) as 𝑏𝑖𝑖(𝑡)=1,𝑚𝑡<(𝑖+1)𝑚,0,otherwise,(3.4) where 𝑖=0,1,2,,(𝑚1).

The functions 𝑏𝑖(𝑡) are disjoint and orthogonal, that is, 𝑏𝑖(𝑡)𝑏𝑙𝑏(𝑡)=0,𝑖𝑙,𝑖(𝑡),𝑖=𝑙,10𝑏𝑖(𝜏)𝑏𝑙1(𝜏)𝑑𝜏=0,𝑖𝑙,𝑚,𝑖=𝑙.(3.5)

Similarly, the Legendre and Chebyshev wavelets may be expanded into 𝑚-term block pulse functions (BPFs) as follows: Ψ𝑚(𝑡)=Φ𝑚×𝑚𝐵𝑚(𝑡),(3.6) where 𝐵𝑚(𝑡)=[𝑏0(𝑡),𝑏1(𝑡),,𝑏𝑖(𝑡),,𝑏𝑚1(𝑡)]𝑇.

In [28], Kilicman and Al Zhour have given the block pulse operational matrix of the fractional integration 𝐹𝛼 as follows: 𝐼𝛼𝐵𝑚(𝑡)𝐹𝛼𝐵𝑚(𝑡),(3.7) where 𝐹𝛼=1𝑚𝛼1Γ(𝛼+2)1𝜉1𝜉2𝜉𝑚101𝜉1𝜉𝑚2001𝜉𝑚300000001,(3.8) and 𝜉𝑖=(𝑖+1)𝛼+12𝑖𝛼+1+(𝑖1)𝛼+1. Next, we derive the Legendre and Chebyshev wavelet operational matrices of the fractional integration. Let 𝐼𝛼Ψ𝑚(𝑡)𝑃𝛼𝑚×𝑚Ψ𝑚(𝑡),(3.9) where the matrix 𝑃𝛼𝑚×𝑚 is called the wavelet operational matrix of the fractional integration. Using (3.6) and (3.7), we have 𝐼𝛼Ψ𝑚𝐼(𝑡)𝛼Φ𝑚×𝑚𝐵𝑚×𝑚(𝑡)=Φ𝑚×𝑚𝐼𝛼𝐵𝑚(𝑡)Φ𝑚×𝑚𝐹𝛼𝐵𝑚(𝑡),(3.10) and from (3.9) and (3.10), we get 𝑃𝛼𝑚×𝑚Ψ𝑚(𝑡)𝑃𝛼𝑚×𝑚Φ𝑚×𝑚𝐵𝑚(𝑡)Φ𝑚×𝑚𝐹𝛼𝐵𝑚(𝑡).(3.11)

Thus, the Legendre and Chebyshev wavelet operational matrices of the fractional integration 𝑃𝛼𝑚×𝑚 can be approximately expressed by 𝑃𝛼𝑚×𝑚=Φ𝑚×𝑚𝐹𝛼Φ1𝑚×𝑚.(3.12) Also, from (3.3) and (3.12), we obtain (𝐼𝛼𝑓)(𝑡)𝐶𝑇Φ𝑚×𝑚𝐹𝛼𝐵𝑚(𝑡).(3.13) Here, for the Legendre and Chebyshev wavelets, we can obtain Φ𝑚×𝑚𝐹𝛼 as follows: Φ𝑚×𝑚𝐹𝛼=𝐵1𝐵2𝐵3̂𝑘𝐵0𝐵1𝐵2̂𝑘𝐵100𝐵1̂𝑘𝐵20000000𝐵1,(3.14) where 𝐵𝑟, ̂̂𝑟=1,2,,𝑘(𝑘=2𝑘1), are 𝑀×𝑀 matrices given by 𝐵𝑟=𝑏(𝑟)𝑖𝑗𝑀×𝑀,(3.15) such that for 𝑟=1, 𝑏(1)𝑖𝑗=𝑗𝑙=1𝑎𝑖𝑙𝜉𝑗𝑙,𝜉0=1,(3.16) and for 𝑟=2,3,,𝑀,𝐵𝑟=𝑏(𝑟)𝑖𝑗𝑀×𝑀̂𝑏,𝑙=2,3,,𝑘,(𝑟)𝑖𝑗=𝑀𝑙=1𝑎𝑖𝑙𝜉(𝑟1)𝑀𝑙+𝑗,(3.17) where 𝑎𝑖𝑗=Ψ1(𝑖1)2𝑗12𝑚,(3.18) and 𝜉𝑗=(1/𝑚𝛼Γ(𝛼+2))𝜉𝑗, 𝑗=0,1,𝑚1.

Now from (3.12) and (3.14), we have 𝑃𝛼𝑚×𝑚=𝐷1𝐷2𝐷3̂𝑘𝐷0𝐷1𝐷2̂𝑘𝐷100𝐷1̂𝑘𝐷20000000𝐷1,(3.19) where 𝐷𝑖=𝐵𝑖𝐴1̂𝑘,𝑖=1,2,.

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: 𝛽0(𝑡)𝐷𝛼𝑦(𝑡)+𝑠1𝑖=1𝛽𝑖(𝑡)𝐷𝛼𝑖𝑦(𝑡)+𝑠2𝑖=1𝛾𝑖(𝑡)𝑦𝑖(𝑡)=𝑓(𝑡),𝑦(𝑗)(0)=𝑐𝑗,𝑗=0,1,𝑛1,(4.1) where 0𝑡<1, 𝑛1<𝛼𝑛, 0<𝛼1<𝛼2<<𝛼𝑠1<𝛼, 𝑛, 𝑠1 and 𝑠2 are fixed positive integers, 𝐷𝛼 denotes the Caputo fractional derivative of order 𝛼, 𝑓 is a known function of 𝑡, 𝑐𝑗, 𝑗=0,1,,𝑛1, are arbitrary constants, and 𝑦 is an unknown function to be determined later. To solve this problem, we apply the following scheme.

Suppose 𝑦(𝑡)=𝑛1𝑗=0𝑐𝑗𝑡𝑗𝑗!+𝑌(𝑡).(4.2)

Under this change of variable, we have 𝑦𝑖(𝑡)=𝑛1𝑗=0𝑐𝑗𝑡𝑗𝑗!+𝑌(𝑡)𝑖=𝑖𝑠=0𝑖𝑠𝑛1𝑗=0𝑐𝑗𝑡𝑗𝑗!𝑠𝑌(𝑡)𝑖𝑠,𝐷𝛼𝑦(𝑡)=𝐷𝛼𝐷𝑌(𝑡),𝛼𝑖𝑦(𝑡)=𝐷𝛼𝑖𝑌(𝑡),𝑛1<𝛼𝑖𝐷<𝛼,𝛼𝑖𝑦(𝑡)=𝐷𝛼𝑖𝑌(𝑡)+𝑖(𝑡),𝛼𝑖𝑛1,(4.3) where 𝑖(𝑡)=𝐷𝛼𝑖𝑛1𝑗=0𝑐𝑗𝑡𝑗𝑗!.(4.4)

Now, by substituting (4.3) into (4.1), we transform the nonlinear MOFDE (4.1) with nonhomogeneous conditions to a nonlinear MOFDE with homogeneous conditions as follows: ̂𝛽0(𝑡)𝐷𝛼𝑌(𝑡)+𝑠1𝑖=1̂𝛽𝑖(𝑡)𝐷𝛼𝑖𝑌(𝑡)+𝑠2𝑖𝑖=1𝑠=0̂𝛾𝑖,𝑠(𝑡)𝑌𝑖𝑠(𝑡)=𝑓(𝑡),𝑌(𝑗)(0)=0,𝑗=0,1,𝑛1,(4.5) where ̂𝛽0(𝑡)=𝛽0̂𝛽(𝑡),𝑖(𝑡)=𝛽𝑖(𝑡),𝑛1<𝛼𝑖̂𝛽<𝛼,𝑖(𝑡)=𝛽𝑖(𝑡)𝑖(𝑡),𝛼𝑖𝑛1,̂𝛾𝑖,𝑠(𝑡)=𝛾𝑖𝑖𝑠(𝑡)𝑛1𝑗=0𝑐𝑗𝑡𝑗𝑗!𝑠,(4.6) and 𝑓(𝑡) is a known function.

We assume that 𝐷𝛼𝑌(𝑡) is given by 𝐷𝛼𝑌(𝑡)=𝐾𝑇𝑚Ψ𝑚(𝑡),(4.7) where 𝐾𝑇𝑚 is an unknown vector and Ψ𝑚(𝑡) is the vector which is defined in (3.6). By using initial conditions and (2.4), we have 𝐷𝛼𝑖𝑌(𝑡)=𝐾𝑇𝑚𝑃𝛼𝛼𝑖𝑚×𝑚Ψ𝑚(𝑡),𝑖=1,2,,𝑠1,(4.8)𝑌(𝑡)=𝐾𝑇𝑚𝑃𝛼𝑚×𝑚Ψ𝑚(𝑡).(4.9)

Since Ψ𝑚=Φ𝑚×𝑚𝐵𝑚(𝑡), (4.9) can be rewritten as follows: 𝑌(𝑡)=𝐾𝑇𝑚𝑃𝛼𝑚×𝑚Φ𝑚×𝑚𝐵𝑚𝑎(𝑡)=1,𝑎2,,𝑎𝑚𝐵𝑚(𝑡).(4.10)

Now by using (3.4), we obtain [𝑌](𝑡)𝑖𝑠=𝑎1𝑖𝑠,𝑎2𝑖𝑠,,𝑎𝑚𝑖𝑠𝐵𝑚(𝑡),𝑖=1,2,,𝑠2,𝑠=0,1,,𝑖.(4.11)

Moreover, we expand functions ̂𝛽𝑖(𝑡), ̂𝛾𝑖,𝑠(𝑡), and 𝑓(𝑡) by wavelets as follows: ̂𝛽𝑖(𝑡)=Ψ𝑚(𝑡)𝑇̃𝛽𝑖,𝑚,𝑖=0,1,,𝑠1,̂𝛾𝑖,𝑠(𝑡)=Ψ𝑚(𝑡)𝑇̃𝛾𝑖,𝑠,𝑚,𝑖=1,2,,𝑠2,𝑠=0,1,,𝑖,𝑓(𝑡)=𝑓𝑇𝑚Ψ𝑚(𝑡),(4.12) where ̃𝛽𝑖,𝑚, ̃𝛾𝑖,𝑚, and 𝑓𝑇𝑚 are known vectors. By substituting (4.7), (4.8), and (4.11)-(4.12) into (4.5), we obtain Ψ𝑚(𝑡)𝑇̃𝛽0,𝑚𝐾𝑇𝑚Ψ𝑚(𝑡)+𝑠1𝑖=1Ψ𝑚(𝑡)𝑇̃𝛽𝑖,𝑚𝐾𝑇𝑚𝑃𝛼𝛼𝑖𝑚×𝑚Ψ𝑚(𝑡)+𝑠2𝑖𝑖=1𝑠=0Ψ𝑚(𝑡)𝑇̃𝛾𝑖,𝑠,𝑚𝑎1𝑖𝑠,𝑎2𝑖𝑠,,𝑎𝑚𝑖𝑠𝐵𝑚(𝑡)=𝐵𝑚(𝑡)𝑇[]1,1,,1𝑇𝑓𝑇𝑚Ψ𝑚(𝑡).(4.13)

Now, from (3.12), (4.10), and (4.13), we have 𝐵𝑚(𝑡)𝑇Φ𝑇𝑚×𝑚̃𝛽0,𝑚𝑎1,𝑎2,,𝑎𝑚𝐹𝛼1+𝑠1𝑖=1Φ𝑇𝑚×𝑚̃𝛽𝑖,𝑚𝑎1,𝑎2,,𝑎𝑚𝐹𝛼1𝐹𝛼𝑖𝛼𝛼+𝑠2𝑖𝑖=1𝑠=0Φ𝑇𝑚×𝑚̃𝛾𝑖,𝑠,𝑚𝑎1𝑖𝑠,𝑎2𝑖𝑠,,𝑎𝑚𝑖𝑠𝐵𝑚(𝑡)=𝐵𝑚(𝑡)𝑇[]1,1,,1𝑇𝑓𝑇𝑚Φ𝑚×𝑚𝐵𝑚(𝑡).(4.14)

This is a nonlinear algebraic equation for unknown vector [𝑎1,𝑎2,,𝑎𝑚]. Here, by taking collocation points, expressed in (2.17), we transform (4.14) into a nonlinear system of algebraic equations. This nonlinear system can be solved by the Newton iteration method for unknown vector [𝑎1,𝑎2,,𝑎𝑚]. Therefore, 𝑌(𝑡) as the solution of (4.5) is 𝑌𝑎(𝑡)=1,𝑎2,,𝑎𝑚𝐵𝑚(𝑡),(4.15) and finally, 𝑦(𝑡) as the solution of (4.1) will be 𝑦(𝑡)=𝑛1𝑗=0𝑐𝑗𝑡𝑗+𝑎𝑗!1,𝑎2,,𝑎𝑚𝐵𝑚(𝑡).(4.16)

In cases where all of coefficients ̂𝛽𝑖(𝑡) and ̂𝛾𝑖,𝑠(𝑡), are constants, that is, ̂𝛽𝑖̂𝛽(𝑡)=𝑖 and ̂𝛾𝑖,𝑠(𝑡)=̂𝛾𝑖,𝑠 we can reduce (4.13) and (4.14), respectively, as follows: ̂𝛽0𝐾𝑇𝑚Ψ𝑚(𝑡)+𝑠1𝑖=1̂𝛽𝑖𝐾𝑇𝑚𝑃𝛼𝛼𝑖𝑚×𝑚Ψ𝑚(𝑡)+𝑠2𝑖𝑖=1𝑠=0̂𝛾𝑖,𝑠𝑎1𝑖𝑠,𝑎2𝑖𝑠,,𝑎𝑚𝑖𝑠𝐵𝑚(𝑡)=𝑓𝑇𝑚Ψ𝑚𝑎(𝑡),1,𝑎2,,𝑎𝑚̂𝛽0𝐹𝛼1+𝐹𝛼1𝑠1𝑖=1̂𝛽𝑖𝐹𝛼𝛼𝑖+𝑠2𝑖𝑖=1𝑠=0̂𝛾𝑖,𝑠𝑎1𝑖𝑠,𝑎2𝑖𝑠,,𝑎𝑚𝑖𝑠𝐵𝑚(𝑡)=𝑓𝑇𝑚Φ𝑚×𝑚𝐵𝑚(𝑡).(4.17)

This is a nonlinear system of algebraic equations for unknown vector [𝑎1,𝑎2,,𝑎𝑚] and can be solved directly without use of collocation points by the Newton iteration method. Here, as before we obtain 𝑌𝑎(𝑡)=1,𝑎2,,𝑎𝑚𝐵𝑚(𝑡),(4.18) as the solution of (4.5), and finally, 𝑦(𝑡) as the solution of (4.1) will be 𝑦(𝑡)=𝑛1𝑗=0𝑐𝑗𝑡𝑗+𝑎𝑗!1,𝑎2,,𝑎𝑚𝐵𝑚(𝑡).(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 equation [13, 16]: 𝐷2𝑦(𝑡)+𝐷0.5𝑦(𝑡)+𝑦(𝑡)=0,(5.1) subject to the following initial conditions: 𝑦(0)=1,𝑦(0)=0.(5.2) Here, we apply the method of Section 4 for solving this problem. By setting 𝑦(𝑡)=1+𝑌(𝑡).(5.3) Equation (5.1) can be rewritten as 𝐷2𝑌(𝑡)+𝐷0.5𝑌(𝑡)+𝑌(𝑡)+1=0,(5.4) subject to the initial condition 𝑌(0)=𝑌(0)=0.(5.5) By using the proposed method, we get a system of algebraic equations for both the Legendre and the Chebyshev wavelets for (5.4) as follows: 𝑎1,𝑎2,,𝑎𝑚𝐹21+𝐹21𝐹1.5+𝑎1,𝑎2,,𝑎𝑚+[]1,1,,1=0.(5.6) Since the linear system of algebraic equations (5.6) is 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 for (5.4) as follows: 𝑌𝑎(𝑡)=1,𝑎2,,𝑎𝑚𝐵𝑚(𝑡).(5.7) Finally the solution of the original problem (5.1) is 𝑦𝑎(𝑡)=1+1,𝑎2,,𝑎𝑚𝐵𝑚(𝑡).(5.8) Figure 1 shows the behavior of the numerical solution for 𝑚=160(𝑀=5,𝑘=6). As Figure 1 shows, this method is very efficient for numerical solution of this problem and the solution can be derived in a large interval [0,40].

542401.fig.001
Figure 1: Numerical solution by the Legendre and Chebyshev wavelets.

Example 5.2. Consider the nonhomogeneous Bagley-Torvik equation [10, 1315, 17, 20]: 𝑎𝐷2𝑦(𝑡)+𝑏𝐷1.5𝑦(𝑡)+𝑐𝑦(𝑡)=𝑓(𝑡),(5.9) where 𝑓(𝑡)=𝑐(1+𝑡),(5.10) subject to the initial condition 𝑦(0)=𝑦(0)=1,(5.11) which has the exact solution 𝑦(𝑡)=1+𝑡.(5.12) Here, we solve this problem by applying the method of Section 4. By setting 𝑦(𝑡)=1+𝑡+𝑌(𝑡).(5.13) Equation (5.9) can be rewritten as 𝑎𝐷2𝑌(𝑡)+𝑏𝐷1.5𝑌(𝑡)+𝑐𝑌(𝑡)=0,(5.14) subject to the following initial conditions: 𝑌(0)=𝑌(0)=0.(5.15) The system of algebraic equations for both the Legendre and the Chebyshev wavelets for (5.14) has the following form: 𝑎1,𝑎2,,𝑎𝑚𝑎𝐹21+𝑏𝐹21𝐹0.5+𝑐𝐼𝑚×𝑚=0.(5.16) Here, the linear system of algebraic equations (5.16) is nonsingular and so has only the trivial solution, that is, 𝑌(𝑡)=0. Then the solution of the original problem (5.9) is 𝑦(𝑡)=1+𝑡,(5.17) which is the exact solution. It is mentionable that this equation has been solved by Diethelm and Luchko [10] (for 𝑎=𝑏=𝑐=1) with error 1.60𝐸4, Diethelm and Ford [17] (for 𝑎=1, 𝑏=𝑐=0.5) with error 5.62𝐸3, while in [20] Li and Zhao obtained approximation solution by the Haar wavelet.

Example 5.3. Consider the following nonhomogenous MOFDE [13]: 𝑎𝐷𝛼𝑦(𝑡)+𝑏𝐷𝛼2𝑦(𝑡)+𝑐𝐷𝛼1𝑦(𝑡)+𝑒𝑦(𝑡)=𝑓(𝑡),0<𝛼11,1<𝛼22,3<𝛼<4,(5.18) where 𝑓(𝑡)=2𝑏Γ3𝛼2𝑡2𝛼2+2𝑐Γ3𝛼1𝑡2𝛼1𝑡+𝑒2𝑡,(5.19) subject to the following initial conditions: 𝑦(0)=0,𝑦(0)=1,𝑦(0)=2,𝑦(0)=0,(5.20) which has the exact solution 𝑦(𝑡)=𝑡2𝑡.(5.21) Here, we apply the method of Section 4 for solving this problem. By setting 𝑦(𝑡)=𝑡+𝑡2+𝑌(𝑡).(5.22) Equation (5.18)can be rewritten as follows: 𝑎𝐷𝛼𝑌(𝑡)+𝑏𝐷𝛼2𝑌(𝑡)+𝑐𝐷𝛼1𝑌(𝑡)+𝑒𝑌(𝑡)=0,0<𝛼11,1<𝛼22,3<𝛼<4,(5.23) subject to the initial conditions 𝑌(0)=0,𝑌(0)=0,𝑌(0)=0,𝑌(0)=0.(5.24) The system of algebraic equations corresponding to the Legendre and Chebyshev wavelets for (5.23) has the following form: 𝑎1,𝑎2,,𝑎𝑚𝑎𝐹𝛼1+𝑏𝐹𝛼1𝐹𝛼𝛼2+𝑐𝐹𝛼1𝐹𝛼𝛼1+𝑒𝐼𝑚×𝑚=0.(5.25) Here, the linear system of algebraic (5.25) is nonsingular and so has only the trivial solution, that is, 𝑌(𝑡)=0. Then the solution of the original problem (5.18) is 𝑦(𝑡)=𝑡2𝑡,(5.26) which is the exact solution. This equation has been solved by El-Sayed et al. [13] (for 𝑎=𝑏=𝑐=𝑒=1, 𝛼1=0.77, 𝛼2=1.44, and 𝛼=0.91) with error 9.53𝐸4.

Example 5.4. Consider the following nonlinear MOFDE [15]: 𝐷3𝑦(𝑡)+𝐷2.5[]𝑦(𝑡)+𝑦(𝑡)2=𝑡4,(5.27) subject to the initial conditions 𝑦(0)=𝑦(0)=0,𝑦(0)=2,(5.28) which has the exact solution 𝑦(𝑡)=𝑡2.(5.29) Here, we apply the method of Section 4 for solving this problem. By setting 𝑦(𝑡)=𝑡2+𝑌(𝑡).(5.30) Equation (5.27) can be rewritten as follows: 𝐷3𝑌(𝑡)+𝐷2.5[]𝑌(𝑡)+𝑌(𝑡)2+2𝑡2𝑌(𝑡)=0,(5.31) subject to the following initial conditions: 𝑌(0)=0,𝑌(0)=0,𝑌(0)=0.(5.32) The nonlinear algebraic equation corresponding to the Legendre and Chebyshev wavelets for (5.31) has the following form: 𝐵𝑚(𝑡)𝑇[]1,1,,1𝑇𝑎1,𝑎2,,𝑎𝑚𝐹31+𝐹31𝐹0.5+[]1,1,,1𝑇𝑎21,𝑎22,,𝑎2𝑚+Φ𝑇𝑚×𝑚̃𝛾𝑚𝑎1,𝑎2,,𝑎𝑚𝐵𝑚(𝑡)=0,(5.33) where ̃𝛾𝑚 is a known constant vector corresponding to the kind of wavelet expansions and Φ𝑚×𝑚 is the wavelet matrix. By taking collocation points expressed in (2.17), 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, 𝑌(𝑡)=0. Then the solution of the original problem (5.27) is 𝑦(𝑡)=𝑡2,(5.34) which is the exact solution.

Example 5.5. Consider the following nonlinear MOFDE [14]: 𝑎𝐷3𝑦(𝑡)+𝑏𝐷𝛼2𝑦(𝑡)+𝑐𝐷𝛼1[]𝑦(𝑡)+𝑒𝑦(𝑡)2=𝑓(𝑡),0<𝛼11,1<𝛼2<2,(5.35) where 𝑓(𝑡)=𝑐𝑡1𝛼1Γ2𝛼1+𝑒𝑡2,(5.36) subject to the initial conditions 𝑦(0)=0,𝑦(0)=1,𝑦(0)=0,(5.37) which has the exact solution 𝑦(𝑡)=𝑡.(5.38) Here, we apply the method of Section 4 for solving this problem. By setting 𝑦(𝑡)=𝑡+𝑌(𝑡).(5.39) Equation (5.35) can be rewritten as follows: 𝑎𝐷3𝑌(𝑡)+𝑏𝐷𝛼2𝑌(𝑡)+𝑐𝐷𝛼1[]𝑌(𝑡)+𝑒𝑌(𝑡)2+2𝑒𝑡𝑌(𝑡)=0,(5.40) subject to the initial conditions 𝑌(0)=0,𝑌(0)=0,𝑌(0)=0.(5.41) The nonlinear algebraic equation corresponding to the Legendre and Chebyshev wavelets for (5.40) has the following form: 𝐵𝑚(𝑡)𝑇[]1,1,,1𝑇𝑎1,𝑎2,,𝑎𝑚𝑎𝐹31+𝑏𝐹31𝐹3𝛼2+𝑐𝐹31𝐹3𝛼1+[]1,1,,1𝑇𝑎21,𝑎22,,𝑎2𝑚+Φ𝑇𝑚×𝑚̃𝛾𝑚𝑎1,𝑎2,,𝑎𝑚𝐵𝑚(𝑡)=0,(5.42) where ̃𝛾𝑚 is a known constant vector corresponding to the kind of wavelet expansion and Φ𝑚×𝑚 is the wavelet matrix. By taking collocation points expressed in (2.17), 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, 𝑌(𝑡)=0. Then the solution of the original problem (5.35) is 𝑦(𝑡)=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 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.

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