# Application of Homotopy Perturbation and Variational Iteration Methods for Fredholm Integrodifferential Equation of Fractional Order

Authors: Asma Ali Elbeleze; Adem Kiliçman; Bachok M. Taib
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper presents the application of homotopy perturbation and variational iteration methods as numerical methods for Fredholm integrodifferential equation of fractional order with initial-boundary conditions. The fractional derivatives are described in Caputo sense. Some illustrative examples are presented.


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## Body

## 1. Introduction

Fractional differential equations have attracted much attention, recently, see for instance [1–4]. This is mostly due to the fact that fractional calculus provides an efficient and excellent instrument for the description of many practical dynamical phenomena arising in engineering and scientific disciplines such as, physics, chemistry, biology, economy, viscoelasticity, electrochemistry, electromagnetic, control, porous media and many more, see for example, [5, 6].During the past decades, the topic of fractional calculus has attracted many scientists and researchers due to its applications in many areas, see [4, 7–9]. Thus several researchers have investigated existence results for solutions to fractional differential equations, see [10, 11]. Further, many mathematical formulation of physical phenomena lead to integrodifferential equations, for example, mostly these type of equations arise in fluid dynamics, biological models and chemical kinetics, and continuum and statistical mechanics, for more details see [12–16]. Integrodifferential equations are usually difficult to solve analytically, so it is required to obtain an efficient approximate solution. The homotopy perturbation method and variational iteration method which are proposed by He [17–26] are of the methods which have received much concern. These methods have been successfully applied by many authors, such as the works in [19, 27, 28].In this work, we study the Integrodifferential equations which are combination of differential and Fredholm-Volterra equations that have the fractional order. In particular, we applied the HPM and VIM for fractional Fredholm Integrodifferential equations with constant coefficients(1.1)∑k=0∞‍PkD*αy(t)=g(t)+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 in the Caputo sense.For the geometrical applications and physical understanding of the fractional Integrodifferential equations, see [14, 26]. Further, we also note that fractional integro-differential equations were associated with a certain class of phase angles and suggested a new way for understanding of Riemann's conjecture, see [29].In present paper, we apply the HPM and VIM to solve the linear and nonlinear fractional Fredholm Integrodifferential equations of the form (1.1). The paper is organized as follows. In Section 2, some basic definitions and properties of fractional calculus theory are given. In Section 3, the basic idea of HPM exists. In Section 4, also is the basic idea of VIM. In Sections 5 and 6, analysis of HPM and VIM exsists, respectively. some examples are given in Section 7. Concluding remarks are listed in Section 8.


## 2.  Preliminaries

In order to modeling the real world application the fractional differential equations are considered by using the fractional derivatives. Thus, in this section, we give some basic definitions and properties of fractional calculus theory which is used in this paper. There are many different starting points for the discussion of classical fractional calculus, see for example, [30]. One can begin with a generalization of repeated integration. If f(t) is absolutely integrable on [0,b), as in [31] then
(2.1)∫0tdtn∫0tndtn-1⋯∫0t3dt2∫0t2f(t1)dt1=1(n+1)!∫0t(t-t1)n-1f(t1)dt1=1(n+1)!tn-1*f(t),
where n=1,2,…, and 0≤t≤b. On writing Γ(n)=(n-1)!, an immediate generalization in the form of the operation Iα defined for α>0 is
(2.2)(Iαf)(t)=1Γ(α)∫0t(t-t1)α-1f(t1)dt1=1Γ(α)tα-1*f(t),0≤t<b,
where Γ(α) is the Gamma function and tα-1*f(t)=∫0tf(t-t1)α-1(t1)dt1 is called the convolution product of tα-1 and f(t). Equation (2.2) is called the Riemann-Liouville fractional integral of order α for the function f(t). Then, we have the following definitions.Definition 2.1.
A real functionf(x),x>0 is said to be in space Cμ,μ∈ℝ if there exists a real number p>μ, such that f(x)=xpf1(x), where f1(x)∈C(0,∞), and it is said to be in the space Cμn if fn∈ℝμ, n∈ℕ.Definition 2.2.
The Riemann-Liouville fractional integral operator of orderα≥0 of a function f∈Cμ, μ≥-1 is defined as
(2.3)Jαf(x)=1Γ(α)∫0x(x-t)α-1f(t)dt,α>0,t>0.
In particular, J0f(x)=f(x).
Forβ≥0 and γ≥-1, some properties of the operator Jα:  (1)
JαJβf(x)=Jα+βf(x),(2)
JαJβf(x)=JβJαf(x),(3)
Jαxγ=(Γ(γ+1)/Γ(α+γ+1))xα+γ.Definition 2.3.
The Caputo fractional derivative off∈C-1m, m∈ℕ is defined as
(2.4)Dαf(x)=1Γ(m-α)∫0xfm(t)dt,m-1<α≤m.Lemma 2.4.
Ifm-1<α≤m,m∈ℕ,f∈Cμm,μ>-1 then the following two properties hold  (1)
Dα[Jαf(x)]=f(x),(2)
Jα[Dαf(x)]=f(x)-∑k=1m-1fk(0)(xk/k!).Now, iff(x) is expanded to the block pulse functions, then the Riemann-Liouville fractional integral becomes
(2.5)(Iαf)(x)=1Γ(α)xα-1*f(x)≃ξT1Γ(α){xα-1*ϕm(x)}.
Thus, if xα-1*ϕm(x) can be integrated, then expanded in block pulse functions, the Riemann-Liouville fractional integral is solved via the block pulse functions. Thus, one notes on that Kronecker convolution product can be expanded in order to define the Riemann-Liouville fractional integrals for matrices by using the Block Pulse operational matrix as follows:
(2.6)1Γ(α)∫0t(t-t1)α-1ϕm(t1)dt1≃Fαϕm(t),
where
(2.7)Fα=(bm)α1Γ(α+2)[1ξ2ξ3⋯ξm01ξ2⋯ξm-1001⋯ξm-2000⋱⋮00001],
see [32].


## 3. Homotopy Perturbation Method

To illustrate the basic idea 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 analytic function, and Γ is the boundary of the domain Ω.In general, the operatorA can be divided into two parts L and N, where L is linear, while N is nonlinear. Equation (3.1) therefor, can be rewritten as follows:
(3.3)L(u)+N(u)-f(r)=0.
By the homotopy technique [33–35], we construct a homotopy 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 of (3.1) which satisfies the boundary conditions. From (3.2) and  (3.3) 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 of p from zero to unity is just that of v(r,p) from u0(r) to u(r). In topology, this called deformation, and L(v)-L(u0) and A(v)-f(r) are called homotopic. Now, assume that the solution of (3.2) and (3.3) can be expressed as
(3.7)v=v0+pv1+p2v2+⋯.
Setting p=1 results in the approximate solution of (3.1).Therefore,(3.8)u=limvp→1=v0+v1+v2+⋯.


## 4.  The Variational Iteration Method

To illustrate the basic concepts of VIM, we consider the following differential equation(4.1)L(u)+N(u)=g(x),
where L is a linear operator; N is nonlinear operator, and g(x) is an nonhomogeneous term. According to VIM, one constructs a correction functional as follows:
(4.2)yn+1=yn+∫0xλ[Lyn(s)-Nyn~(s)]ds,
where λ is a general Lagrange multiplier, and yn~ denotes restricted variation that is δyn~=0.


## 5.  Analysis of Homotopy Perturbation Method

To illustrate the basic concepts of HPM for Fredholm Integrodifferential equation (1.1) with boundary conditions (1.2) and (1.3). We use the view of He in [19, 20], where the following homotopy was constructed for (1.1) as the following:
(5.1)(1-p)∑k=0∞‍PkD*αy(x)+p[∑k=0∞‍PkD*αy(x)-g(t)-λ∫abH(x,t)y(x)dx]=0
or
(5.2)∑k=0∞‍PkD*αy(x)=p[g(t)+λ∫abH(x,t)y(x)dx],
where p∈[0,1] is an embedding parameter. If p=0, (5.2) becomes linear fractional differential equation
(5.3)∑k=0∞‍PkD*αy(x)=0,
and when p=1, the (5.2) turn out to be the original equation. In view of basic assumption of HPM, solution of (1.1) can be expressed as a power series in p(5.4)y(x)=y0(x)+p1y1(x)+p2y2(x)+⋯,
when p=1, we get the approximate solution of (5.4)
(5.5)y(x)=y0(x)+y1(x)+y2(x)+⋯.
The convergence of series (5.5) has been proved in [21]. Substitution (5.4) into (5.2), and equating the terms with having identical power of p, we obtain the following series of equations:
(5.6)p0:∑k=0∞‍PkD*αy0=0,p1:∑k=0∞‍PkD*αy1=g(t)-λ∫abH(x,t)y0(x)dx,p2:∑k=0∞‍PkD*αy2=-λ∫abH(x,t)y1(x)dx,p3:∑k=0∞‍PkD*αy3=-λ∫abH(x,t)y2(x)dx,⋮
with the initial-boundary conditions
(5.7)D*αy(a)=y(0),D*αy(0)=y′(a).
The initial approximation can be chosen in the following manner.
(5.8)y0=∑j=01‍γjxjj!=γ0+γ1x,whereγ0=D*αy(a)γ1=D*αy(0).
Note that the (5.6) can be solved by applying the operator J*α and by some computation, we approximate the series solution of HPM by the following N-term truncated series
(5.9)χn(x)=y0(x)+y1(x)+⋯+yN-1(x),
which is the approximate solution of (1.1)–(1.3).


## 6. Analysis of VIM

To solve the fractional Integrodifferential equation by using the variational iteration method, with boundary conditions (1.2) and (1.3) we construct the following correction functional:
(6.1)yk+1(x)=yk(x)+Jα[μ(∑k=0∞‍PkD*αy(x)-g~(x)-λ∫0aH(x,s)y~k(s)ds)]
or
(6.2)yk+1(x)=yk(x)+1Γ(α)∫0x(x-s)α-1μ(s)[∑k=0∞‍PkD*αy(s)-g~k(s)-λ∫0aH(x,p)y~k(p)dp],
where μ is a general Lagrange multiplier, and g~k(x) and y~k(x) are considered as restricted variation, that is, δg~k(x)=0 and δy~k(x)=0.Making the above correction functional stationary, the following condition can be obtained(6.3)δyk+1(x)=δyk(x)+∫0x(x-s)α-1μ(s)[∑k=0∞‍PkδD*αy(s)-δg~k(s)-λ∫0aH(x,p)δy~k(p)dp].
It's boundary condition can be obtained as follows:
(6.4)1-μ′(s)|x=s=0,μ(s)|x=s=1.
The Lagrange multipliers can be identified as follows:
(6.5)μ(s)=12(x-s).
We obtain the following iteration formula by substitution of (6.5) in (6.2):
(6.6)yk+1(x)=yk(x)+12Γ(α-1)∫0x(x-s)α-2(s-x)[∑k=0∞‍PkD*αy(s)-g~k(s)-λ∫0aH(x,p)y~k(p)dp]ds.
That is,
(6.7)yk+1(x)=yk(x)-(α-1)2Γ(α)∫0x(x-s)α-1[∑k=0∞‍PkD*αy(s)-g~k(s)-λ∫0aH(x,p)y~k(p)dp]ds.
This yields the following iteration formula:
(6.8)yk+1(x)=yk(x)-(α-1)2Jα[∑k=0∞‍PkD*αy(x)-gk(x)-λ∫0aH(x,s)y~k(s)ds].
The initial approximation y0 can be chosen by the following manner which satisfies initial-boundary conditions (1.2)-(1.3)
(6.9)y0=γ0+γ1x,whereγ0=D*αy(a)γ1=D*αy(0).
We can obtain the following first-order approximation by substitution of (6.9) in (6.8)
(6.10)y1(x)=y0(x)-(α-1)2Jα[∑k=0∞‍PkD*αy(x)-g0(x)-λ∫0aH(x,s)y~0(s)ds].
Finally, by substituting the constant values of γ0 and γ1 in (6.10) we have the results as the approximate solutions of (1.1)–(1.3), see the further details in [36–40].


## 7. Applications

In this section, we have applied homotopy perturbation method and variational iteration method to fractional Fredholm Integrodifferential equations with known exact solution.Example 7.1.
Consider the following linear Fredholm Integrodifferential equation:(7.1)Dαy(x)=(32+e2x2)+∫0xety(t)dt0≤x≤1,1<α≤2,
with initial boundary conditions
(7.2)y(0)=1,y′(1)=e
the exact solution is y(x)=ex. Now we construct
(7.3)Dαy(x)=p((32+e2x2)+∫0xety(t)dt).
Substitution of (5.4) in (7.3) and then equating the terms with same powers of p, we get the series
(7.4)p0:Dαy0(x)=0,p1:Dαy1(x)=(32+23e2x)+∫0xety0(t)dt,p2:Dαy2(x)=-∫0xety1(t)dt.⋮
Now applying the operator Jα to the equations (7.4) and using initial-boundary conditions yields
(7.5)y0(x)=1,(7.6)y1(x)=1+Ax+Jα((32+e2x2)+∫0xety0dt),(7.7)y2(x)=Jα(∫0xety1dt),(7.8)yn(x)=Jα(∫0xetyn-1dt),n=2,3,4,….
Then by solving (7.5)–(7.8), we obtain y1,y2,… as
(7.9)y1(x)=1+Ax+5xα2Γ(α+1)+2xα+1Γ(α+2)+3xα+22Γ(α+3)+5xα+36Γ(α+4),(7.10)y2(x)=xα+1Γ(α+2)+(A+1)xα+22Γ(α+3)+(A3+12)xα+3Γ(α+4)+(A8+112)xα+4Γ(α+5)+Axα+515Γ(α+6)+5x2α+12Γ(2α+2)+⋯.
Now, we can form the 2 term approximation as follows:
(7.11)ϕ2(x)=1+Ax+5xα2Γ(α+1)+3xα+1Γ(α+2)+(A+2)xα+22Γ(α+3)+(A+4)xα+33Γ(α+4)+(A8+112)xα+4Γ(α+5)+Axα+515Γ(α+6)+5x2α+12Γ(2α+2)+⋯,
where A can be determined by imposing initial-boundary conditions (7.2) on ϕ2. Table 1 shows the values of A for different values of α.Table 1
Values ofA for different values of α.
α
=
1.25
α
=
1.5
α
=
1.75
α
=
2
A
-
2.33209843875457
-
1.906444021198994
-
0.88898224618462
-
0.098915873901025Now, we solve (7.1)-(7.2) by variational iteration method. According to variational iteration method, the formula (6.8) for (7.1) can be expressed in the following form:
(7.12)yk+1(x)=yk(x)-(α-1)2Jα[Dαy(x)-(32+e2x2)-∫0xety(t)dt].
Then, in order to avoid the complex and difficult fractional integration, we can consider the truncated Taylor expansions for exponential term in (7.6)–(7.8) for example, ex~1+x+x2/2+x3/6 and further, suppose that an initial approximation has the following form which satisfies the inial-boundary conditions
(7.13)y0(x)=1+Ax.
Now by iteration formula (7.12), the first approximation takes the following form:
(7.14)y1(x)=y0(x)-(α-1)2Jα[Dαy0(x)-(32+e2x2)-∫0xety0(t)dt]=1+Ax+(α-1)2xα[5Γ(α+1)+2xΓ(α+2)+(A+3)x22Γ(α+3)+(5/2+A)x33Γ(α+4)+(A/2+1/6)x46Γ(α+5)-Ax530Γ(α+6)].
By imposing initial-boundary conditions (7.2) on y1, we can obtain the values of A for different α which we show in Table 2.Table 2
Value ofA for different values of α using (7.14).
α
=
1.25
α
=
1.5
α
=
1.75
α
=
2
A
1.23429062479478
0.73267858113358
0.66218167845861
0.54744784230252Example 7.2.
Consider the following linear Fredholm Integrodifferential equation:(7.15)Dαy(x)=(1-x4)+∫0xxty2(t)dt0≤x≤1,1<α≤2
with initial boundary conditions
(7.16)y(0)=0,y′(1)=1.
then the exact solution is y(x)=x. By applying the HPM, we have
(7.17)Dαy(x)=p((1-x4)+∫0xxty2(t)dt).
Substitution of (5.4) in (7.15) and then equating the terms with same powers of p, we get the following series expressions:
(7.18)p0:Dαy0(x)=0,p1:Dαy1(x)=(1-x4)+∫0xxty02(t)dt,p2:Dαy2(x)=2∫0xxty0(t)y1(t)dt,p3:Dαy3(x)=∫0xxt(y0(t)y2(t)+y12(t))dt,p4:Dαy4(x)=∫0xxt(2y0(t)y4(t)+2y1y3+y22(t))dt,⋮
Applying the operator Jα to (7.18) and using initial-boundary conditions, then we get
(7.19)y0(x)=0,y1(x)=Ax+Jα((1-x4)+∫0xxty02(t)dt),y2(x)=0,y3(x)=Jα(∫0xxt(y0(t)y2(t)+y12(t))dt),y4(x)=Jα(∫0xxt(2y0(t)y4(t)+2y1y3+y22(t))dt),⋮
Thus, by solving (7.19), we obtain y1,y2,y3,…(7.20)y1(x)=Ax+xαΓ(α+1)-xα+14Γ(α+2),y2(x)=0,y3(x)=A2xα+54Γ(α+6)+2Ax2α+4(α+3)Γ(α+1)Γ(2α+5)+x3α+3(2α+2)Γ(α+1)Γ(α+1)Γ(3α+4)+⋯.
Now, we can form the 3 term approximation
(7.21)ϕ2(x)=Ax+xαΓ(α+1)-xα+14Γ(α+2)+A2xα+54Γ(α+6)+2Ax2α+4(α+3)Γ(α+1)Γ(2α+5)+x3α+3(2α+2)Γ(α+1)Γ(α+1)Γ(3α+4)+⋯,
where A can be determined by imposing initial-boundary conditions (7.16) on ϕ2. Thus, we have Table 3.
Similarly, by variational iteration method we have the following form:(7.22)yk+1(x)=yk(x)-(α-1)2Jα[Dαy(x)-(1-x4)+∫0xxty2(t)dt],
where we suppose that an initial approximation has the following form which satisfies the initial-boundary conditions y0(x)=Ax. Now by using the iteration formula, the first approximation takes the following form:
(7.23)y1(x)=y0(x)-(α-1)2Jα[Dαy0(x)-(1-x4)+∫0xxty02(t)dt]=Ax+(α-1)2Jα[-xαΓ(α+1)+xα+14Γ(α+2)].
By imposing initial-boundary conditions, we can obtain the following Table 4.Table 3
Value ofA for different values of α.
α
=
1.25
α
=
1.5
α
=
1.75
α
=
2
A
0.179304
0.153796
0.0673477
0.124989Table 4
Value ofA for different values of α.
α
=
1.25
α
=
1.5
α
=
1.75
α
=
2
A
0.88967375
0.76492075
0.650263
0.5625


## 8. Conclusion

In this work, homotopy perturbation method (HPM) and variational iteration method (VIM) have been applied to linear and nonlinear initial-boundary value problems for fractional Fredholm Integrodifferential equations. Two examples are presented in order to illustrate the accuracy of the present methods. Comparisons of HPM and VIM with exact solution have been given in the Tables1–4.



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