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

Authors: Abdelouahab Kadem; Adem Kilicman
Journal: Abstract and Applied Analysis
 (2012)

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## 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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## Body

## 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:(1.1)∑k=0∞PkD*αy(t)=g(x)+λ∫0aH(x,t)y(t)dt,a≤x,t≤b,
under the initial-boundary conditions
(1.2)D*αy(a)=y(0),(1.3)D*αy(0)=y′(a),
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. Section2 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 functionf(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)=xpf1(x), where f1(x)=C([0,∞)). Clearly, Cα⊂Cβ, if β≤α.Definition 2.2.
A functionf(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 orderμ≥0for a function f∈Cα,(α≥1) is defined as
(2.1)Iμf(t)=1Γ(μ)∫0t(t-τ)μ-1f(τ)dτ,μ>0,t>0,
in particular I0f(t)=f(t).Definition 2.4.
The Caputo fractional derivative off∈C-1m, m∈N, is defined as
(2.2)Dcμf(t)={[Im-μf(m)(t)],m-1<μ≤m,m∈N,dmdtmf(t),μ=m.
Note that(i)
Iμtγ=(Γ(γ+1)/Γ(γ+μ+1))tγ+μ, μ>0, γ>-1, t>0,(ii)
IμCD0+μf(t)=f(t)-∑k=0m-1f(k)(0+)(tk/k!),
m-1<μ≤m,
m∈N,(iii)
D0+μCf(t)=Dμ(f(t)-∑k=0m-1f(k)(0+)(tk/k!)),
m-1<μ≤m,
m∈N,(iv)
(2.3)DβIαf(t)={Iα-βf(t),ifα>β,f(t),ifα=β,Dβ-αf(t),ifα<β,(v)
CD0+βDmf(t)=Dβ+mf(t),
m=0,1,2,…,n-1<β<n.Definition 2.5 (see [3, 16]).
The Riemann-Liouville fractional integral operator of orderρ≥0 for a function f∈Cμ,(μ≥-1) is defined as
(2.4)Kρf(x)=1Γ(ρ)∫0x(x-t)ρ-1f(t)dt,ρ>0,x>0,K0f(x)=f(x),
having the properties
(2.5)KρKβf(x)=Kρ+βf(x),Kρxβ=Γ(β+1)Γ(ρ+β+1)xα+β.
According to the Caputo’s derivatives, we obtain the following expressions:
(2.6)CDμC=0,C=constant,CDμtβ={Γ(β+1)Γ(α+β+1)tβ-α,β>α-1,0,β≤α-1.Lemma 2.6.
Ifm-1<α≤m, m∈N, f∈Cμm, μ≥-1, then the following two properties hold:
(2.7)(1)DαKαf(t)=f(t),(2)(DαKα)f(t)=f(t)-∑k=0m-1f(k)(0+)tkk!.In fact, Kılıçman 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:(2.8)1Γ(α)∫0t(t-t1)α-1ϕm(t1)dt1≃Fαϕm(t),
where(2.9)Fα=(bm)α1Γ(α+2)[1ξ2ξ3⋯ξm01ξ2⋯ξm-1001⋯ξm-2000⋱⋮00001],
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:(3.1)A(u)+f(r)=0,r∈Ω,
with boundary conditions
(3.2)B(u,∂u∂n)=0,r∈Γ,
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 operatorA 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:(3.3)L(u)+N(u)-f(r)=0.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]→ℛ which satisfies(3.4)H(v,p)=(1-p)[L(v)-L(u0)]+p[A(v)+f(r)]=0,p∈[0,1],r∈Ω,
or
(3.5)H(v,p)=L(v)-L(u0)+pL(u0)+p[N(v)-f(r)]=0,
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
(3.6)H(v,0)=L(v)-L(u0)=0,H(v,1)=A(v)-f(r)=0.The changing in the process ofp from zero to unity is just that of v(r,p) from u0 to u(r). In a topology, this is alsoknown deformation, furtherL(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(3.7)v=v+pv1+p2v2+⋯.
The approximate solution of (3.1), therefore, can be readily obtained:(3.8)u=lim⁡p→1v=v0+v1+v2+⋯.
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:(4.1)Lu+Nu=g(x),
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:(4.2)un+1(x)=un(x)+∫0xλ[Lun(s)+Nũn(s)-g(s)]ds,
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, ũu is considered variation [6, 27], that is, δũu=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:
(5.1)yk+1(x)=yk(x)+∫0tμ∑k=0∞PkD*αy(s)ds-μg̃k(x)-λ∫abμH(x,s)ỹk(s)ds,
where μ is a general Lagrange multiplier and g̃k(x) and ỹk(x) are considered as restricted variations, that is, δg̃k(x)=0 and δỹk(x)=0.Making the above correction functional stationary, the following conditions can be obtained:(5.2)δyk+1(x)=δyk(x)+∫0t[∑k=0∞Pkμ(s)δD*αy(s)-δg̃k(x)-λ∫abH(x,s)μ(s)δỹk(s)ds],
having the boundary conditions as follows:(5.3)1-μ′(s)∣x=s=0,μ(s)∣x=s=1.
The Lagrange multipliers can be identified as follows:(5.4)μ(s)=12(s-x).Substituting the value ofμ from (5.4) into correction functional of (5.1) leads to the following iteration formulae:(5.5)yk+1(x)=yk(x)+μ2Γ(α-1)∫0x(x-s)α-2(s-x)×[∫0t∑k=0∞PkD*αy(s)ds-g̃k(x)-λ∫abH(x,s)ỹk(s)]ds,yk+1(x)=yk(x)-μ(α-1)2Γ(α)∫0x(x-s)α-1×[∫0t∑k=0∞PkD*αy(s)ds-g̃k(x)-λ∫abH(x,s)ỹk(s)]ds,
by applying formulae (2.4), we get(5.6)yk+1(x)=yk(x)-(α-1)Kα2Γ(α)[∫0tμ∑k=0∞PkD*αy(v)dv-μg̃k(x)-λ∫abμH(x,v)ỹk(v)]dv.The initial approximation can be chosen in the following manner which satisfies initial boundary conditions (1.2)-(1.3):(5.7)y0(x)=υ0+υ1x,whereυ1=D*αy(0),υ0=D*αy(a).We can obtain the following first-order approximation by substitution of (5.7) into (5.6)(5.8)y1(x)=y0(x)-(α-1)Kα2Γ(α)[∫0tμ∑k=0NPkD*αy(v)dv-μg̃0(x)-λ∫abμH(x,v)ỹk(v)]dv.
Substituting the constant value of υ0 and υ1 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):(6.1)∑k=0∞PkD*αy(x)=p[∑k=0∞PkD*αy(x)+(g(t)-λ∫abH(x,t)y(x)dx)].In view of basic assumption of HPM, solution of (1.1) can be expressed as a power series in p:(6.2)y(x)=D*αy0(x)+pD*αy1(x)+p2D*αy2(x)+p3D*αy3(x)+⋯.
If we put p→1 in (6.2), we get the approximate solution of (1.1):(6.3)y(x)=D*αy0(x)+D*αy1(x)+D*αy2(x)+D*αy3(x)+⋯.
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:(6.4)p0:∑k=0∞PkD*αy0(t)=0,p1:∑k=0∞PkD*αy1(t)=∑k=0∞PkD*αy0(t)-λ∫abH(x,t)y0(x)dx,p2:∑k=0∞PkD*αy2(t)=∑k=0∞PkD*αy1(t)+g(x)-λ∫abH(x,t)y1(x)dx,p3:∑k=0∞PkD*αy3(t)=∑k=0∞PkD*αy2(t)-λ∫abH(x,t)y2(x)dx,p4:∑k=0∞PkD*αy4(t)=∑k=0∞PkD*αy3(t)-λ∫abH(x,t)y3(x)dx,
with the initial-boundary conditions(6.5)D*αy(a)=y(0),D*αy(0)=y′(a).We can also take the initial approximation in the following manner which satisfies initial-boundary conditions (1.2)-(1.3):(6.6)y0(x)=υ0+υ1x,whereυ1=D*αy(0),υ0=D*αy(a).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]:(6.7)χn(x)=D*αy0(x)+D*αy1(x)+D*αy2(x)+D*αy3(x)+⋯+D*αyn-1(x),
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 parameterp 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.



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