The Approximate Solution of Fractional Fredholm Integrodifferential Equations by Variational Iteration and Homotopy Perturbation Methods

Abstract

Variational iteration method and homotopy perturbation method are used to solve the fractional Fredholm integrodifferential equations with constant coefficients. The obtained results indicate that the method is efficient and also accurate.

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Abdelouahab Kadem 1 and Adem Kilicman 2

Recommended by Bashir Ahmad

1, L.M.F.N Mathematics Department, University of Setif, Algeria
2, Department of Mathematics, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia

Received 2 February 2012; Revised 20 February 2012; Accepted 23 March 2012

1. Introduction

The topic of fractional calculus has attracted many scientists because of its several applications in many areas, such as physics, chemistry, and engineering. For a detail survey with collections of applications in various fields, see, for example, [1-3].

Further, the fractional derivatives technique has been employed for solving linear fractional differential equations including the fractional integrodifferential equations; in this way, much of the efforts is devoted to searching for methods that generate accurate results, see [4, 5]. In this work, we present two different methods, namely, homotopy perturbation method and variational iteration method [6], for solving a fractional Fredholm integro-differential equations with constant coefficients. There is a vast literature, and we only mention the works of Liao which treat a homotopy method in [7, 8].

For the nonlinear equations with derivatives of integer order, many methods are used to derive approximation solution [9-14]. However, for the fractional differential equation, there are some limited approaches, such as Laplace transform method [3], the Fourier transform method [15], the iteration method [16], and the operational calculus method [17].

Recently, there has been considerable researches in fractional differential equations due to their numerous applications in the area of physics and engineering [18], such as phenomena in electromagnetic theory, acoustics, electrochemistry, and material science [3, 16, 18, 19]. Similarly, there is also growing interest in the integrodifferential equations which are combination of differential and Fredholm-Volterra equations. In this work, we study these kind of equations that have the fractional order usually difficult to solve analytically, thus a numerical method is required, for example, the successive approximations, Adomian decomposition, Chebyshev and Taylor collocation, Haar Wavelet, Tau and Walsh series methods.

This note is devoted to the application of variational iteration method (VIM) and homotopy perturbation method (HPM) for solving fractional Fredholm integrodifferential equations with constant coefficients: [figure omitted; refer to PDF] under the initial-boundary conditions [figure omitted; refer to PDF] [figure omitted; refer to PDF] where a is constant and 1<α<2 and ^D*α is the fractional derivative operator given in the Caputo sense. For the physical understanding of the fractional integrodifferential equations, see [20]. Further, we also note that fractional integrodifferential equations were associated with a certain class of phase angles and suggested a new way for understanding of Riemann's conjecture, see [21].

Outline of this paper is as follows. Section 2 contains preliminaries on fractional calculus. Section 3 is a short review of the homotopy method and Section 4 variational iteration method. Sections 5 and 6 are devoted to VIM and HPM analysis, respectively. Concluding remarks with suggestions for future work are listed in Section 7.

2. Description of the Fractional Calculus

In the following, we give the necessary notations and basic definitions and properties of fractional calculus theory; for more details, see [3, 13, 16, 22].

Definition 2.1.

A real function f(x), x>0 , is said to be in the space _Cα , α∈R if there exists a real number (p>α) , such that f(x)=^xp_f1 (x) , where _f1 (x)=C([0,∞)) . Clearly, _Cα ⊂_Cβ , if β...4;α .

Definition 2.2.

A function f(x), x>0 , is said to be in space ^Cαm , m∈N , if ^f(m) ∈_Cα .

Definition 2.3.

The Riemann-Liouville fractional integral of order μ...5;0 for a function f∈_Cα , (α...5;1) is defined as [figure omitted; refer to PDF] in particular ^I0 f(t)=f(t) .

Definition 2.4.

The Caputo fractional derivative of f ∈ ^C-1m , m∈N , is defined as [figure omitted; refer to PDF] Note that

(i) ^Iμtγ =(Γ(γ+1)/Γ(γ+μ+1))^tγ+μ , μ>0 , γ>-1 , t>0 ,

(ii) ^{Iμ CD0+ μ} f(t)=f(t)-^{∑k=0m-1f(k)} (₀₊ )(^tk /k!) , m-1<μ...4;m , m∈N ,

(iii): ^{D0+ μ} C f(t)=^Dμ (f(t)-^{∑k=0m-1f(k)} (₀₊ )(^tk /k!)) , m-1<μ...4;m , m∈N ,

(iv) [figure omitted; refer to PDF]

(v) ^{CD0+ βDm} f(t)=^Dβ+m f(t) , m=0,1,2,...,n-1<β<n .

Definition 2.5 (see [3, 16]).

The Riemann-Liouville fractional integral operator of order ρ...5;0 for a function f∈_Cμ , (μ...5;-1) is defined as [figure omitted; refer to PDF] having the properties [figure omitted; refer to PDF] According to the Caputo's derivatives, we obtain the following expressions: [figure omitted; refer to PDF]

Lemma 2.6.

If m-1<α...4;m , m∈N , f∈^Cμm , μ...5;-1 , then the following two properties hold: [figure omitted; refer to PDF]

In fact, Kiliçman and Zhour introduced the Kronecker convolution product and expanded to the Riemann-Liouville fractional integrals of matrices by using the Block Pulse operational matrix as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] see [23].

In our work, we consider Caputo fractional derivatives and apply the homotopy method in order to derive an approximate solutions of the fractional integrodifferential equations.

3. Homotopy Method

To illustrate the basic ideas of this method, we consider the following nonlinear differential equation: [figure omitted; refer to PDF] with boundary conditions [figure omitted; refer to PDF] where A is a general differential operator, B is a boundary operator, f(r) is a known analytical function, and Γ is the boundary of the domain Ω , see [24].

In general, the operator A can be divided into two parts L and N , where L is linear, while N is nonlinear. Equation (3.1), therefore, can be rewritten as follows: [figure omitted; refer to PDF]

By using the homotopy technique that was proposed by Liao in [7, 8], we construct a homotopy of (3.1) v(r,p):Ω×[0,1][arrow right]... which satisfies [figure omitted; refer to PDF] or [figure omitted; refer to PDF] where p∈[0,1] is an embedding parameter and _u0 is an initial approximation which satisfies the boundary conditions. By using (3.4) and (3.5), we have [figure omitted; refer to PDF]

The changing in the process of p from zero to unity is just that of v(r,p) from _u0 to u(r) . In a topology, this is also

known deformation, further L(v)-L(_u0 ) and A(v)-f(r) are homotopic.

Now, assume that the solution of (3.4) and (3.5) can be expressed as [figure omitted; refer to PDF] The approximate solution of (3.1), therefore, can be readily obtained: [figure omitted; refer to PDF] The convergence of the series of (3.8) has been proved in the [25, 26].

4. The Variational Iteration Method

To illustrate the basic concepts of the VIM, we consider the following differential equation: [figure omitted; refer to PDF] where L is a linear operator, N is a nonlinear operator, and g(x) is an nonhomogenous term; for more details, see [19].

According to the VIM, one construct a correction functional as follows: [figure omitted; refer to PDF] where λ is a general Lagrange multiplier, which can be identified optimally via the variational theory, and the subscript n denotes the order of approximation, _u...u is considered variation [6, 27], that is, δ_u...u =0 .

5. Analysis of VIM

To solve the fractional integrodifferential equation (1.1) by using the variational iteration method, with boundary conditions (1.2), one can construct the following correction functional: [figure omitted; refer to PDF] where μ is a general Lagrange multiplier and _g...k (x) and _y...k (x) are considered as restricted variations, that is, δ_g...k (x)=0 and δ_y...k (x)=0 .

Making the above correction functional stationary, the following conditions can be obtained: [figure omitted; refer to PDF] having the boundary conditions as follows: [figure omitted; refer to PDF] The Lagrange multipliers can be identified as follows: [figure omitted; refer to PDF]

Substituting the value of μ from (5.4) into correction functional of (5.1) leads to the following iteration formulae: [figure omitted; refer to PDF] by applying formulae (2.4), we get [figure omitted; refer to PDF]

The initial approximation can be chosen in the following manner which satisfies initial boundary conditions (1.2)-(1.3): [figure omitted; refer to PDF]

We can obtain the following first-order approximation by substitution of (5.7) into (5.6) [figure omitted; refer to PDF] Substituting the constant value of _[upsilon]0 and _[upsilon]1 in the expression (5.8) results in the approximation solution of (1.1)-(1.3).

6. Analysis of HPM

This section illustrates the basic of HPM for fractional Fredholm integrodifferential equations with constant coefficients (1.1) with initial-boundary conditions (1.2).

In view of HPM [25, 26], construct the following homotopy for (1.1): [figure omitted; refer to PDF]

In view of basic assumption of HPM, solution of (1.1) can be expressed as a power series in p : [figure omitted; refer to PDF] If we put p[arrow right]1 in (6.2), we get the approximate solution of (1.1): [figure omitted; refer to PDF] The convergence of series (6.3) has been proved in [28].

Now, we substitute (6.2) into (6.1); then equating the terms with identical power of p , we obtain the following series of linear equations: [figure omitted; refer to PDF] with the initial-boundary conditions [figure omitted; refer to PDF]

We can also take the initial approximation in the following manner which satisfies initial-boundary conditions (1.2)-(1.3): [figure omitted; refer to PDF]

Note that (6.4) can be solved by applying the operator ^Kβ , which is the inverse of operator ^Dα we approximate the series solution of HPM by the following n -term truncated series [29]: [figure omitted; refer to PDF] which results, the approximate solutions of (1.2)-(1.3). For further analysis, the variational iteration method, see [30] and the algorithm by the homotopy perturbation method, see [31].

7. Conclusion

The proposed methods are used to solve fractional Fredholm integrodifferential equations with constant coefficients. Comparison of the results obtained by the present method with that obtained by other method reveals that the present method is very effective and convenient. Unfortunately, the disadvantage of the second method is that the embedding parameter p is quite casual, and often enough the approximations obtained by this method will not be uniform. So, in our future work we expect to study this kind of equation by using a combination of the variational iteration method and the homotopy perturbation method which has shown reliable results in supplying analytical approximation that converges very rapidly. However, we note that the papers [32, 33] suggest alternative ways for similar problems.

Acknowledgments

The first author would like to thank Professor Juan J. Trujillo for the very helpful discussion. The authors also acknowledge that this research was partially supported by University Putra Malaysia under the Research University Grant Scheme 05-01-09-0720RU.

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