Wavelet Collocation Method for Solving Multiorder Fractional Differential Equations

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 [1–6]. 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 [9–17]. However, few papers have reported applications of wavelets in solving fractional differential equations [8, 18–21]. In view of successful application of wavelet operational matrices in numerical solution of integral and differential equations [22–27], 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. 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 𝐶_𝜇, 𝜇∈ℝ if there exists a real number 𝑝(>𝜇) such that 𝑓(𝑡)=𝑡^𝑝𝑓₁(𝑡), where 𝑓₁(𝑡)∈𝐶[0,∞], and it is said to be in the space 𝐶^𝑛𝜇 if 𝑓^(𝑛)∈𝐶_𝜇, 𝑛∈ℕ.

Definition 2.2. The Riemann-Liouville fractional integration operator of order 𝛼≥0, of a function 𝑓∈𝐶_𝜇, 𝜇≥−1, is defined as follows [4]: 𝐼^𝛼𝑓(𝑡)=1Γ(𝛼)^𝑡0𝐼(𝑡−𝜏)^𝛼−1𝑓(𝜏)𝑑𝜏,₀𝑓(𝑡)=𝑓(𝑡),(2.1) and according to [4], we have 𝐼𝐼^𝛼𝐼^𝛽𝑓(𝑡)=𝐼^𝛼+𝛽𝑓(𝑡),_𝛼𝐼𝐼^𝛽𝑓(𝑡)=𝐼^𝛽𝐼^𝛼𝑓(𝑡),_𝛼𝑡^𝜗=Γ(𝜗+1)Γ(𝛼+𝜗+1)𝑡^𝛼+𝜗,(2.2) where 𝛼,𝛽≥0, 𝑡>0 and 𝜗>−1.

Definition 2.3. The fractional derivative of order 𝛼>0 in the Riemann-Liouville sense is defined as [4] 𝐷^𝛼𝑓(𝑡)=𝑑𝑑𝑡^𝑛𝐼^𝑛−𝛼𝑓(𝑡),(𝑛−1<𝛼≤𝑛),(2.3) where 𝑛 is an integer and 𝑓∈𝐶^𝑛1.
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 𝐷^𝛼∗ proposed by Caputo [5].

Definition 2.4. The fractional derivative of order 𝛼>0 in the Caputo sense is defined as [5] 𝐷^𝛼∗𝑓(𝑡)=1Γ(𝑛−𝛼)^𝑡0(𝑡−𝜏)^{𝑛−𝛼−1}𝑓^(𝑛)(𝜏)𝑑𝜏,(𝑛−1<𝛼≤𝑛),(2.4) where 𝑛 is an integer 𝑡>0 and 𝑓∈𝐶^𝑛1. Caputos integral operator has a useful property: 𝐼^𝛼𝐷^𝛼∗𝑓(𝑡)=𝑓(𝑡)−^𝑛−1_𝑘=0𝑓^(𝑘)0⁺𝑡^𝑘𝑘!,(𝑛−1<𝛼≤𝑛),(2.5) where 𝑛 is an integer 𝑡>0 and 𝑓∈𝐶^𝑛1.

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]: 𝜓_𝑎,𝑏(𝑡)=|𝑎|^−1/2𝜓𝑡−𝑏𝑎,𝑎,𝑏∈ℝ,𝑎≠0.(2.6)

If we restrict the parameters 𝑎 and 𝑏 to discrete values as 𝑎=𝑎^0−𝑘, 𝑏=𝑛𝑏₀𝑎^0−𝑘, 𝑎₀>1, 𝑏₀>0, for 𝑛 and 𝑘 positive integers, we have the following family of discrete wavelets: 𝜓_𝑘,𝑛(𝑡)=||𝑎₀||^𝑘/2𝜓𝑎^𝑘0𝑡−𝑛𝑏₀,(2.7) where 𝜓_𝑘,𝑛(𝑡) forms a wavelet basis for 𝐿²(ℝ). In particular, when 𝑎₀=2 and 𝑏₀=1, 𝜓_𝑘,𝑛(𝑡) forms an orthonormal basis. That is (𝜓_𝑘,𝑛(𝑡),𝜓_𝑙,𝑚(𝑡))=𝛿_𝑘𝑙𝛿_𝑛𝑚.

2.2.1. The Legendre Wavelets

The Legendre wavelets 𝜓_𝑛,𝑚(𝑡)=𝜓(𝑘,𝑛,̂𝑚,𝑡) have four arguments; 𝑘∈ℕ, 𝑛=1,2,…,2^𝑘−1, and 𝑛̂=2𝑛−1; 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 𝜓_𝑛,𝑚(𝑡)=⎧⎪⎨⎪⎩0,𝑚+212^𝑘/2𝑝_𝑚2^𝑘𝑡−𝑛̂,o𝑛t̂2h−_𝑘e1rw≤ise𝑡,<2𝑛_𝑘̂,(2.8) where 𝑚=0,1,…,𝑀−1 and 𝑀 is a fixed positive integer. The coefficient √𝑚+1/2 in (2.8) is for orthonormality, the dilation parameter is 𝑎=2^−𝑘, and the translation parameter is 𝑏=𝑛2̂^−𝑘. Here, 𝑃_𝑚(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,𝑝_𝑚+1(𝑡)=2𝑚+1𝑚+1𝑡𝑝^𝑚(𝑡)−𝑚𝑚+1𝑝^𝑚−1(𝑡),𝑚=1,2,3,….(2.9)

2.2.2. The Chebyshev Wavelets

The Chebyshev wavelets 𝜓_𝑛,𝑚(𝑡)=𝜓(𝑘,𝑛,̂𝑚,𝑡) have four arguments; 𝑘∈ℕ, 𝑛=1,2,…,2^𝑘−1, and 𝑛̂=2𝑛−1; 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 𝜓_𝑛,𝑚(𝑡)=2^𝑘/2𝑇_𝑚2^𝑘𝑡−𝑛̂,𝑛̂−12^𝑘≤𝑡<𝑛̂2^𝑘0,otherwise,,(2.10) where 𝑇_𝑚(𝑡)=⎧⎪⎨⎪⎩√1𝜋,𝑚=0,2𝜋𝑇^𝑚(𝑡),𝑚>0,(2.11) where 𝑚=0,1,…,𝑀−1 and 𝑀 is a fixed positive integer. The coefficients in (2.11) are used for orthonormality. Here, 𝑇_𝑚(𝑡) are the well-known Chebyshev polynomials of order 𝑚 which are orthogonal with respect to the weight function 𝑤(𝑡)=1/√1−𝑡² on the interval [−1,1] and satisfy the following recursive formula: 𝑇₀(𝑡)=1,𝑇₁(𝑡)=𝑡,𝑇_𝑚+1(𝑡)=2𝑡𝑇_𝑚(𝑡)−𝑇_𝑚−1,𝑚=1,2,3….(2.12) We should note that in dealing with the Chebyshev polynomials the weight function 𝑤=𝑤(2𝑡−1) has to be dilated and translated as follows: 𝑤_𝑛(𝑡)=𝑤2^𝑘𝑡−𝑛̂,(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=1_𝑚=0𝑐_𝑛,𝑚𝜓_𝑛,𝑚(𝑡),(2.14) by the Legendre or Chebyshev wavelets, where 𝑐_𝑛,𝑚=(𝑓(𝑡),𝜓_𝑛,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)≈^2𝑘−1_{𝑛=1𝑀−1𝑚=0}𝑐_𝑛,𝑚𝜓_𝑛,𝑚(𝑡)=𝐶^𝑇Ψ(𝑡),(2.15) where 𝐶 and Ψ(𝑡) are 2^𝑘−1𝑀×1 matrices given by Ψ=𝜓₁₀𝐶=𝑐₁₀,𝑐₁₁,…,𝑐_1𝑀−1,𝑐₂₀,…,𝑐_2𝑀−1,…,𝑐₂^𝑘−1₀,…,𝑐₂^𝑘−1_𝑀−1^𝑇,(𝑡),𝜓₁₁(𝑡),…,𝜓_1𝑀−1(𝑡),𝜓₂₀(𝑡),…,𝜓_2𝑀−1(𝑡),…,𝜓₂^𝑘−1₀(𝑡),…,𝜓₂^𝑘−1_𝑀−1(𝑡)^𝑇.(2.16)

Taking the collocation points 𝑡_𝑖=(2𝑖−1)2^𝑘𝑀,𝑖=1,2,…,𝑚,(2.17) where 𝑚=2^𝑘−1𝑀, we define the wavelet matrix Φ_𝑚×𝑚 as Φ_𝑚×𝑚=Ψ12𝑚,Ψ23𝑚,…,Ψ2𝑚2𝑚−1.(2.18)

Indeed Φ_𝑚×𝑚 has the following form: Φ_𝑚×𝑚=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣⋮𝐴000⋮𝐴000⋱⋯⋯𝐴0⋱⋯⋯00⋮𝐴000⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,(2.19) where 𝐴 is a 𝑀×𝑀 matrix given by 𝐴=⎡⎢⎢⎢⎢⎢⎢⎢⎣𝜓¹⁰𝜓21𝑚𝜓¹⁰23𝑚⋯𝜓¹⁰2𝑀2𝑚₁₁−1𝜓⋮21𝑚𝜓¹¹⋮23𝑚⋯⋮𝜓¹¹2⋮_1𝑀−1𝑀2𝑚−121𝑚𝜓^1𝑀−123𝑚⋯𝜓^1𝑀−12𝑀2𝑚−1⎤⎥⎥⎥⎥⎥⎥⎥⎦.(2.20)

For example, for 𝑀=4 and 𝑘=2, the Legendre and Chebyshev matrices can be expressed as Φ_8×8=⎡⎢⎢⎣𝐴0𝐴0⎤⎥⎥⎦,(2.21) where for the Legendre matrix we have 𝐴=⎡⎢⎢⎢⎢⎢⎢⎣−011.1..240.681833470720811352−−111...642.211582464329671782−−01.116.4..1221258436427961278−111....240861833477020181325⎤⎥⎥⎥⎥⎥⎥⎦,(2.22) and for the Chebyshev matrix we have 𝐴=⎡⎢⎢⎢⎢⎢⎢⎣−001..1.181.9912979846637288113−−111...310.392998876930348902−−−111..31..3092998867933048092−011...1.811992997864638728311⎤⎥⎥⎥⎥⎥⎥⎦.(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)𝑑𝜏≈𝑃Ψ(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 (𝐼^𝛼𝑓)(𝑡)=1Γ(𝛼)^𝑡0(𝑡−𝜏)^𝛼−1𝑓(𝜏)𝑑𝜏=1Γ(𝛼)𝑡^𝛼−1∗𝑓(𝑡),(3.2) where 𝛼∈ℝ is the order of the integration and 𝑡^𝛼−1∗𝑓(𝑡) denotes the convolution product of 𝑡^𝛼−1 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 (𝐼^𝛼𝑓)(𝑡)=1Γ(𝛼)𝑡^𝛼−1∗𝑓(𝑡)≈𝐶^𝑇1Γ(𝛼)𝑡^𝛼−1∗Ψ(𝑡).(3.3)

So that, if 𝑡^𝛼−1∗𝑓(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 𝑏_𝑖(𝑡)=10,,o𝑚𝑖th≤erw𝑡<ise(,𝑖+𝑚1),(3.4) where 𝑖=0,1,2,…,(𝑚−1).

The functions 𝑏_𝑖(𝑡) are disjoint and orthogonal, that is, 𝑏_𝑖(𝑡)𝑏_𝑙(𝑡)=𝑏0,𝑖≠𝑙,_𝑖(𝑡),𝑖=𝑙,₁₀𝑏_𝑖(𝜏)𝑏_𝑙(𝜏)𝑑𝜏=01,𝑖≠𝑙,𝑚,𝑖=𝑙.(3.5)

Similarly, the Legendre and Chebyshev wavelets may be expanded into 𝑚-term block pulse functions (BPFs) as follows: Ψ_𝑚(𝑡)=Φ_𝑚×𝑚𝐵_𝑚(𝑡),(3.6) where 𝐵_𝑚(𝑡)=[𝑏₀(𝑡),𝑏₁(𝑡),…,𝑏_𝑖(𝑡),…,𝑏_𝑚−1(𝑡)]^𝑇.

In [28], Kilicman and Al Zhour have given the block pulse operational matrix of the fractional integration 𝐹^𝛼 as follows: 𝐼^𝛼𝐵_𝑚(𝑡)≈𝐹^𝛼𝐵_𝑚(𝑡),(3.7) where 𝐹^𝛼=1𝑚^𝛼1Γ(𝛼+2)⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣1𝜉¹𝜉₂⋯𝜉_𝑚−101𝜉₁⋯𝜉_𝑚−2001⋯𝜉_𝑚−3000000⋱0⋮1⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,(3.8) and 𝜉_𝑖=(𝑖+1)^𝛼+1−2𝑖^𝛼+1+(𝑖−1)^𝛼+1. Next, we derive the Legendre and Chebyshev wavelet operational matrices of the fractional integration. Let 𝐼^𝛼Ψ_𝑚(𝑡)≈𝑃^{𝛼𝑚×𝑚}Ψ_𝑚(𝑡),(3.9) where the matrix 𝑃^{𝛼𝑚×𝑚} is called the wavelet operational matrix of the fractional integration. Using (3.6) and (3.7), we have 𝐼^𝛼Ψ_𝑚(𝑡)≈𝐼^𝛼Φ_𝑚×𝑚𝐵_𝑚×𝑚(𝑡)=Φ_𝑚×𝑚𝐼^𝛼𝐵_𝑚(𝑡)≈Φ_𝑚×𝑚𝐹^𝛼𝐵_𝑚(𝑡),(3.10) and from (3.9) and (3.10), we get 𝑃^{𝛼𝑚×𝑚}Ψ_𝑚(𝑡)≈𝑃^{𝛼𝑚×𝑚}Φ_𝑚×𝑚𝐵_𝑚(𝑡)≈Φ_𝑚×𝑚𝐹^𝛼𝐵_𝑚(𝑡).(3.11)

Thus, the Legendre and Chebyshev wavelet operational matrices of the fractional integration 𝑃^{𝛼𝑚×𝑚} can be approximately expressed by 𝑃^{𝛼𝑚×𝑚}=Φ_𝑚×𝑚𝐹^𝛼Φ^{−1𝑚×𝑚}.(3.12) Also, from (3.3) and (3.12), we obtain (𝐼^𝛼𝑓)(𝑡)≈𝐶^𝑇Φ_𝑚×𝑚𝐹^𝛼𝐵_𝑚(𝑡).(3.13) Here, for the Legendre and Chebyshev wavelets, we can obtain Φ_𝑚×𝑚𝐹^𝛼 as follows: Φ_𝑚×𝑚𝐹^𝛼=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝐵¹0𝐵𝐵₂₁𝐵₃⋯𝐵𝑘̂𝐵₂⋯𝐵𝑘̂₋₁00𝐵₁⋯𝐵𝑘̂₋₂000000⋱0𝐵⋮₁⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,(3.14) where 𝐵_𝑟, 𝑟=1,2,…,𝑘̂(𝑘̂=2^𝑘−1), are 𝑀×𝑀 matrices given by 𝐵_𝑟=𝑏^{(𝑟)𝑖𝑗}_𝑀×𝑀,(3.15) such that for 𝑟=1, 𝑏^(1)𝑖𝑗=^𝑗_𝑙=1𝑎_𝑖𝑙𝜉^′_𝑗−𝑙,𝜉^0=1,(3.16) and for 𝑟=2,3,…,𝑀,𝐵_𝑟=𝑏𝑏^{(𝑟)𝑖𝑗}_𝑀×𝑀,𝑙=2,3,…,𝑘̂,_{(𝑟)𝑖𝑗}=^𝑀_𝑙=1𝑎_𝑖𝑙𝜉^{(𝑟−1)𝑀−𝑙+𝑗},(3.17) where 𝑎_𝑖𝑗=Ψ_1(𝑖−1)2𝑗2−𝑚1,(3.18) and 𝜉^𝑗=(1/𝑚^𝛼Γ(𝛼+2))𝜉_𝑗, 𝑗=0,1,…𝑚−1.

Now from (3.12) and (3.14), we have 𝑃^{𝛼𝑚×𝑚}=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝐷¹0𝐷𝐷₂₁𝐷₃⋯𝐷𝑘̂𝐷₂⋯𝐷𝑘̂₋₁00𝐷₁⋯𝐷𝑘̂₋₂000000⋱0𝐷⋮₁⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,(3.19) where 𝐷_𝑖=𝐵_𝑖𝐴⁻¹,𝑖=1,2,…𝑘̂.

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)𝐷^𝛼∗𝑦(𝑡)+^𝑠1_𝑖=1𝛽_𝑖(𝑡)𝐷^𝛼𝑖_∗𝑦(𝑡)+^𝑠2_𝑖=1𝛾_𝑖(𝑡)𝑦^𝑖(𝑡)=𝑓(𝑡),𝑦^(𝑗)(0)=𝑐_𝑗,𝑗=0,1,…𝑛−1,(4.1) where 0≤𝑡<1, 𝑛−1<𝛼≤𝑛, 0<𝛼₁<𝛼₂<⋯<𝛼_𝑠1<𝛼, 𝑛, 𝑠₁ and 𝑠₂ are fixed positive integers, 𝐷^𝛼∗ denotes the Caputo fractional derivative of order 𝛼, 𝑓 is a known function of 𝑡, 𝑐_𝑗, 𝑗=0,1,…,𝑛−1, 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)=^𝑛−1_𝑗=0𝑐_𝑗𝑡^𝑗𝑗!+𝑌(𝑡).(4.2)

Under this change of variable, we have 𝑦^𝑖(𝑡)=^𝑛−1_𝑗=0𝑐_𝑗𝑡^𝑗𝑗!+𝑌(𝑡)^𝑖=^𝑖_𝑠=0⎛⎜⎜⎝𝑠𝑖⎞⎟⎟⎠^𝑛−1_𝑗=0𝑐_𝑗𝑡^𝑗𝑗!^𝑠𝐷_𝛼∗𝑌(𝑡)^𝑖−𝑠,𝐷𝑦(𝑡)=𝐷^𝛼∗𝑌(𝑡),_𝛼𝑖∗𝐷𝑦(𝑡)=𝐷^𝛼𝑖_∗𝑌(𝑡),𝑛−1<𝛼_𝑖<𝛼,_𝛼𝑖∗𝑦(𝑡)=𝐷^𝛼𝑖_∗𝑌(𝑡)+ℎ_𝑖(𝑡),𝛼_𝑖≤𝑛−1,(4.3) where ℎ_𝑖(𝑡)=𝐷^𝛼𝑖_∗^𝑛−1_𝑗=0𝑐_𝑗𝑡^𝑗𝑗!.(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: 𝛽̂₀(𝑡)𝐷^𝛼∗𝑌(𝑡)+^𝑠1_𝑖=1𝛽̂_𝑖(𝑡)𝐷^𝛼𝑖_∗𝑌(𝑡)+^𝑠2_𝑖𝑖=1_𝑠=0𝛾̂_𝑖,𝑠(𝑡)𝑌^𝑖−𝑠(𝑡)=𝑓(𝑡),𝑌^(𝑗)(0)=0,𝑗=0,1,…𝑛−1,(4.5) where 𝛽̂𝛽̂₀(𝑡)=𝛽₀(𝑡),_𝑖𝛽̂(𝑡)=𝛽_𝑖(𝑡),𝑛−1<𝛼_𝑖<𝛼,_𝑖𝛾̂(_𝑖,𝑠𝑡)=𝛽_𝑖(𝑡)ℎ_𝑖(𝑡),𝛼_𝑖≤𝑛−1,(𝑡)=𝛾_𝑖(𝑡)⎛⎜⎜⎝𝑠𝑖⎞⎟⎟⎠^𝑛−1_𝑗=0𝑐_𝑗𝑡^𝑗𝑗!^𝑠,(4.6) and 𝑓(𝑡) is a known function.

We assume that 𝐷^𝛼∗𝑌(𝑡) is given by 𝐷^𝛼∗𝑌(𝑡)=𝐾^𝑇𝑚Ψ_𝑚(𝑡),(4.7) where 𝐾^𝑇𝑚 is an unknown vector and Ψ_𝑚(𝑡) is the vector which is defined in (3.6). By using initial conditions and (2.4), we have 𝐷^𝛼𝑖_∗𝑌(𝑡)=𝑌𝐾(^𝑇𝑚𝑡)𝑃=𝐾^{𝛼−𝛼𝑖}_𝑚×𝑚Ψ_𝑚𝑇𝑚(𝑡),𝑖=1,2,…,𝑠₁,(4.8)𝑃^{𝛼𝑚×𝑚}Ψ_𝑚(𝑡).(4.9)

Since Ψ_𝑚=Φ_𝑚×𝑚𝐵_𝑚(𝑡), (4.9) can be rewritten as follows: 𝑌(𝑡)=𝐾^𝑇𝑚𝑃^{𝛼𝑚×𝑚}Φ_𝑚×𝑚𝐵_𝑚(𝑡)=𝑎₁,𝑎₂,…,𝑎_𝑚𝐵_𝑚(𝑡).(4.10)

Now by using (3.4), we obtain [𝑌(𝑡)]^𝑖−𝑠=𝑎^1𝑖−𝑠,𝑎^2𝑖−𝑠,…,𝑎^{𝑚𝑖−𝑠}𝐵_𝑚(𝑡),𝑖=1,2,…,𝑠₂,𝑠=0,1,…,𝑖.(4.11)

Moreover, we expand functions 𝛽̂_𝑖(𝑡), 𝛾̂_𝑖,𝑠(𝑡), and 𝑓(𝑡) by wavelets as follows: 𝛾̂_𝑖,𝑠𝛽̂_𝑖(𝑡)=Ψ_𝑚(𝑡)^𝑇𝛽̃_𝑖,𝑚,𝑖=0,1,…,𝑠₁,(𝑡)=Ψ_𝑚(𝑡)^𝑇𝛾̃𝑓_{𝑖,𝑠,𝑚}(,𝑡)=𝑖=𝑓_𝑇𝑚1,2,…,𝑠₂,𝑠=0,1,…,𝑖,Ψ_𝑚(𝑡),(4.12) where 𝛽̃_𝑖,𝑚, 𝛾̃_𝑖,𝑚, and 𝑓^𝑇𝑚 are known vectors. By substituting (4.7), (4.8), and (4.11)-(4.12) into (4.5), we obtain Ψ_𝑚(𝑡)^𝑇𝛽̃_0,𝑚𝐾^𝑇𝑚Ψ_𝑚(𝑡)+^𝑠1_𝑖=1Ψ_𝑚(𝑡)^𝑇𝛽̃_𝑖,𝑚𝐾^𝑇𝑚𝑃^{𝛼−𝛼𝑖}_𝑚×𝑚Ψ_𝑚(𝑡)+^𝑠2_𝑖𝑖=1_𝑠=0=𝐵_𝑚Ψ_𝑚(𝑡)^𝑇𝛾̃_{𝑖,𝑠,𝑚}𝑎^1𝑖−𝑠,𝑎^2𝑖−𝑠,…,𝑎^{𝑚𝑖−𝑠}𝐵_𝑚(𝑡)(𝑡)^𝑇[1,1,…,1]^𝑇𝑓^𝑇𝑚Ψ_𝑚(𝑡).(4.13)

Now, from (3.12), (4.10), and (4.13), we have 𝐵_𝑚(𝑡)^𝑇Φ^{𝑇𝑚×𝑚}𝛽̃_0,𝑚𝑎₁,𝑎₂,…,𝑎_𝑚𝐹^𝛼−1+^𝑠1_𝑖=1Φ^{𝑇𝑚×𝑚}𝛽̃_𝑖,𝑚𝑎₁,𝑎₂,…,𝑎_𝑚𝐹^𝛼−1𝐹^{𝛼𝑖𝛼−𝛼}+^𝑠2_𝑖𝑖=1_𝑠=0Φ^{𝑇𝑚×𝑚}𝛾̃_{𝑖,𝑠,𝑚}𝑎^1𝑖−𝑠,𝑎^2𝑖−𝑠,…,𝑎^{𝑚𝑖−𝑠}𝐵_𝑚(𝑡)=𝐵_𝑚(𝑡)^𝑇[1,1,…,1]^𝑇𝑓^𝑇𝑚Φ_𝑚×𝑚𝐵_𝑚(𝑡).(4.14)

This is a nonlinear algebraic equation for unknown vector [𝑎₁,𝑎₂,…,𝑎_𝑚]. 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,…,𝑎_𝑚]. Therefore, 𝑌(𝑡) as the solution of (4.5) is 𝑌(𝑡)=𝑎₁,𝑎₂,…,𝑎_𝑚𝐵_𝑚(𝑡),(4.15) and finally, 𝑦(𝑡) as the solution of (4.1) will be 𝑦(𝑡)=^𝑛−1_𝑗=0𝑐_𝑗𝑡^𝑗𝑗!+𝑎¹,𝑎₂,…,𝑎_𝑚𝐵_𝑚(𝑡).(4.16)

In cases where all of coefficients 𝛽̂_𝑖(𝑡) and 𝛾̂_𝑖,𝑠(𝑡), are constants, that is, 𝛽̂_𝑖(𝑡)=𝛽̂_𝑖 and 𝛾̂_𝑖,𝑠(𝑡)=𝛾̂_𝑖,𝑠 we can reduce (4.13) and (4.14), respectively, as follows: 𝛽̂₀𝐾^𝑇𝑚Ψ_𝑚(𝑡)+^𝑠1_𝑖=1𝛽̂_𝑖𝐾^𝑇𝑚𝑃^{𝛼−𝛼𝑖}_𝑚×𝑚Ψ_𝑚(𝑡)+^𝑠2_𝑖𝑖=1_𝑠=0𝑎𝛾̂_𝑖,𝑠𝑎^1𝑖−𝑠,𝑎^2𝑖−𝑠,…,𝑎^{𝑚𝑖−𝑠}𝐵_𝑚(𝑡)=𝑓^𝑇𝑚Ψ_𝑚(𝑡),₁,𝑎₂,…,𝑎_𝑚𝛽̂₀𝐹^𝛼−1+𝐹^{𝛼−1𝑠1}_𝑖=1𝛽̂_𝑖𝐹^{𝛼−𝛼𝑖}+^𝑠2_𝑖𝑖=1_𝑠=0𝛾̂_𝑖,𝑠𝑎^1𝑖−𝑠,𝑎^2𝑖−𝑠,…,𝑎^{𝑚𝑖−𝑠}𝐵_𝑚(𝑡)=𝑓^𝑇𝑚Φ_𝑚×𝑚𝐵_𝑚(𝑡()4..17)

This is a nonlinear system of algebraic equations for unknown vector [𝑎₁,𝑎₂,…,𝑎_𝑚] and can be solved directly without use of collocation points by the Newton iteration method. Here, as before we obtain 𝑌(𝑡)=𝑎₁,𝑎₂,…,𝑎_𝑚𝐵_𝑚(𝑡),(4.18) as the solution of (4.5), and finally, 𝑦(𝑡) as the solution of (4.1) will be 𝑦(𝑡)=^𝑛−1_𝑗=0𝑐_𝑗𝑡^𝑗𝑗!+𝑎¹,𝑎₂,…,𝑎_𝑚𝐵_𝑚(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]: 𝐷^2∗𝑦(𝑡)+𝐷^∗0.5𝑦(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 𝐷^2∗𝑌(𝑡)+𝐷^∗0.5𝑌(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,…,𝑎_𝑚𝐹²⁻¹+𝐹²⁻¹𝐹^1.5+𝑎¹,𝑎₂,…,𝑎_𝑚+[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: 𝑌(𝑡)=𝑎₁,𝑎₂,…,𝑎_𝑚𝐵_𝑚(𝑡).(5.7) Finally the solution of the original problem (5.1) is 𝑦(𝑡)=1+𝑎₁,𝑎₂,…,𝑎_𝑚𝐵_𝑚(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].

Figure 1 

Numerical solution by the Legendre and Chebyshev wavelets.

Example 5.2. Consider the nonhomogeneous Bagley-Torvik equation [10, 13–15, 17, 20]: 𝑎𝐷^2∗𝑦(𝑡)+𝑏𝐷^∗1.5𝑦(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 𝑎𝐷^2∗𝑌(𝑡)+𝑏𝐷^∗1.5𝑌(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,…,𝑎_𝑚𝑎𝐹²⁻¹+𝑏𝐹²⁻¹𝐹^0.5+𝑐𝐼_𝑚×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 1.60𝐸−4, Diethelm and Ford [17] (for 𝑎=1, 𝑏=𝑐=0.5) with error 5.62𝐸−3, while in [20] Li and Zhao obtained approximation solution by the Haar wavelet.

Example 5.3. Consider the following nonhomogenous MOFDE [13]: 𝑎𝐷^𝛼∗𝑦(𝑡)+𝑏𝐷^𝛼2_∗𝑦(𝑡)+𝑐𝐷^𝛼1_∗𝑦(𝑡)+𝑒𝑦(𝑡)=𝑓(𝑡),0<𝛼₁≤1,1<𝛼₂≤2,3<𝛼<4,(5.18) where 𝑓(𝑡)=2𝑏Γ3−𝛼₂𝑡^2−𝛼2+2𝑐Γ3−𝛼₁𝑡^2−𝛼1+𝑒𝑡²−𝑡,(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)=𝑡²−𝑡.(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: 𝑎𝐷^𝛼∗𝑌(𝑡)+𝑏𝐷^𝛼2_∗𝑌(𝑡)+𝑐𝐷^𝛼1_∗𝑌(𝑡)+𝑒𝑌(𝑡)=0,0<𝛼₁≤1,1<𝛼₂≤2,3<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,…,𝑎_𝑚𝑎𝐹^𝛼−1+𝑏𝐹^𝛼−1𝐹^𝛼−𝛼2+𝑐𝐹^𝛼−1𝐹^𝛼−𝛼1+𝑒𝐼_𝑚×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)=𝑡²−𝑡,(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 9.53𝐸−4.

Example 5.4. Consider the following nonlinear MOFDE [15]: 𝐷^3∗𝑦(𝑡)+𝐷^∗2.5𝑦(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: 𝐷^3∗𝑌(𝑡)+𝐷^∗2.5𝑌(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: 𝐵_𝑚(𝑡)^𝑇[+1Φ,1,…_{𝑇𝑚×𝑚},1]^𝑇𝑎₁,𝑎₂,…,𝑎_𝑚𝐹³⁻¹+𝐹³⁻¹𝐹^0.5+[1,1,…,1]^𝑇𝑎²¹,𝑎²²,…,𝑎^2𝑚𝛾̃_𝑚𝑎₁,𝑎₂,…,𝑎_𝑚𝐵_𝑚(𝑡)=0,(5.33) where 𝛾̃_𝑚 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 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]: 𝑎𝐷^3∗𝑦(𝑡)+𝑏𝐷^𝛼2_∗𝑦(𝑡)+𝑐𝐷^𝛼1_∗𝑦(𝑡)+𝑒[𝑦(𝑡)]²=𝑓(𝑡),0<𝛼₁≤1,1<𝛼₂<2,(5.35) where 𝑓(𝑡)=𝑐𝑡^1−𝛼1Γ2−𝛼₁+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: 𝑎𝐷^3∗𝑌(𝑡)+𝑏𝐷^𝛼2_∗𝑌(𝑡)+𝑐𝐷^𝛼1_∗𝑌(𝑡)+𝑒[𝑌(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: 𝐵_𝑚(𝑡)^𝑇[+1[,11,,1…,…,1,]1^𝑇]_𝑇𝑎₁,𝑎₂,…,𝑎_𝑚𝑎𝐹³⁻¹+𝑏𝐹³⁻¹𝐹^3−𝛼2+𝑐𝐹³⁻¹𝐹^3−𝛼1𝑎²¹,𝑎²²,…,𝑎^2𝑚+Φ^{𝑇𝑚×𝑚}𝛾̃_𝑚𝑎₁,𝑎₂,…,𝑎_𝑚𝐵_𝑚(𝑡)=0,(5.42) where 𝛾̃_𝑚 is a known constant vector corresponding to the kind of wavelet expansion and Φ_𝑚×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.