# Reproducing Kernel Method for Solving Nonlinear Fractional Fredholm Integrodifferential Equation

Authors: Bothayna S. H. Kashkari; Muhammed I. Syam
Journal: Complexity
 (2018)

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## Abstract

This article is devoted to both theoretical and numerical studies of nonlinear fractional Fredholm integrodifferential equations. In this paper, we implement the reproducing kernel method (RKM) to approximate the solution of nonlinear fractional Fredholm integrodifferential equations. Numerical results demonstrate the accuracy of the present algorithm. In addition, we prove the existence of the solution of the nonlinear fractional Fredholm integrodifferential equation. Uniformly convergence of the approximate solution produced by the RKM to the exact solution is proven.


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## Body

## 1. Introduction

Fractional Fredholm integrodifferential equations have various applications in sciences and engineering. Most of these problems cannot be solved analytically, and hence finding accurate numerical solution for these problems will be very useful. Wazwaz [1, 2] studied the Fredholm integral equations of the form (1)fx=rx+λ∫abHx,tftdt,where a and b are constants, λ is a parameter, r(x) is the data function, H(x,t) is the kernel of the integral equation, and f(x) is the unknown function that will determined. In this paper, we study the generalization of the above problem of the form(2)Dαfx=rx+λ∫abHx,tfmtdt,0<α≤1subject to(3)fa=a0.Note that Dα in (2) is in the Caputo derivative. Equation (2) is called the nonlinear fractional Fredholm integrodifferential equations of the second kind characterized by the occurrence of the unknown function u(x) inside and outside the integral sign. To homogenize the initial condition, we assume q(x)=f(x)-a0. Then,(4)Dαqx=rx+λ∫abHx,tqt+a0mdt,0<α≤1subject to(5)qa=0.In the following definition and theorem, we write the definition of Caputo derivative as well as the power rule which we are used in this paper. For more details on the geometric and physical interpretation for Caputo fractional derivatives, see [3].Definition 1.
Form to be the smallest integer that exceeds α, the Caputo fractional derivatives of order α>0 is defined as (6)
D
α
q
x
=
1
Γ
m
-
α
∫
0
x
x
-
τ
m
-
α
-
1
d
m
q
τ
d
τ
m
d
τ
,
m
-
1
<
α
<
m
,
d
m
q
x
d
x
m
,
m
i
n
t
e
g
e
r
.Theorem 2.
The Caputo fractional derivative of the power function satisfies(7)Dαts=Γs+1Γs-α+1tp-s,n-1<s<n,s>n-1,0,s<α.The reproducing kernel Hilbert space method is a useful numerical technique to solve nonlinear problems [4–6]. The reproducing kernel is given by this definition.Definition 3.
LetM≠ϕ. A function K:A×A→C is a kernel of Z i if(i)
H(.,x)∈Z for all x∈A,(ii)
(ϕ(.),H(.,x))=ϕ(x) for all x∈A and ϕ∈Z.The second condition is called the reproducing property and a Hilbert space which possesses a reproducing kernel is called a reproducing kernel Hilbert space (RKHS). More details can be in [7–14]. A description of the RKM for discretization of the linear fractional Fredholm integrodifferential equations problem (4)-(5) is presented in Section 2. In Section 3, we study the nonlinear fractional Fredholm integrodifferential equations. Several numerical examples and conclusions are discussed in Section 4. Conclusions and closing remarks are given in Section 5.


## 2. Analysis of RKHSM for Linear Fractional Fredholm Integrodifferential Equations

In this section, we discuss how to solve the following linear fractional Fredholm integrodifferential equation using RKHSM:(8)Dζqx=rx+λ∫abHx,tqt+a0dt,0<ζ≤1subject to(9)qa=0.In order to solve problem (8)-(9), we construct the kernel Hilbert spaces W21[a,b] and W22[a,b] in which every function satisfy the boundary conditions (9). Let (10)W22a,b=fs:f,f′areabsolutelycontinuousrealvaluefunctions,f′′∈L2a,b,fa=0.The inner product in W22[a,b] is defined as (11)uy,vyW22a,b=uava+∫abu′yv′ydyand the norm uW22[a,b] is given by (12)uW22a,b=uy,uyW22a,bwhere u,v∈W22[a,b].Theorem 4.
There existsQ(s,y)∈W23[0,1] such that, for any u∈W22[0,1] and each fixed y,x∈[a,b], we have (13)uy,Qx,yW22a,b=ux.In this case, Q(x,y) is given by (14)Qx,y=α0x+α1xy,y≤xβ0x+β1xy,y>xwhere(15)α0x=ab-xa-b,α1x=-b-xa-b,β0x=ba-xa-b,β1x=-a-xa-b.Proof.
Using the integration by parts, one can get(16)uy,Qx,yW22a,b=uaQx,a+ubQyx,b-uaQyx,a-∫abuy∂2Q∂y2x,ydy.Since u(y) and Q(x,y)∈W22[a,b], (17)ua=0and(18)Qx,a=0.Thus, (19)uy,Qx,yW22a,b=ubQyx,b-∫abuy∂2Q∂y2x,ydy.Since Q(x,y) is a reproducing kernel of W22[a,b],(20)uy,Qx,yW22a,b=ux.Thus,(21)∂2Q∂y2x,y=δy-xwhere δ is the Dirac-delta function and(22)Qyx,b=0.Since the characteristic equation of ∂2Q/∂y2(x,y)=δ(y-x) is λ2=0 and its characteristic value is λ=0 with 2 multiplicity roots, we write Q(x,y) as(23)Qx,y=α0x+α1xy,y≤xβ0x+β1xy,y>x.Since ∂2Q/∂y2(x,y)=δ(y-x), we have(24)Qx,x+0=Qx,x-0.On the other hand, integrate ∂2Q/∂y2(x,y)=δ(y-x) from x-ϵ to x+ϵ with respect to y and let ϵ→0 to get(25)∂Q∂yx,x+0-∂Q∂yx,x-0=-1.
Using conditions (18) and (22)-(25), we get the following system of equations: (26)α0x+α1xa=0,β0x+β1xb=0,α0x+α1xx=β0x+β1xxβ1-c1=-1,
We solved the last system using Mathematica to get(27)α0x=ab-xa-b,α1x=-b-xa-b,β0x=ba-xa-b,β1x=-a-xa-b.
Next, we study the spaceW21[a,b]. Let(28)W21a,b=ux:uareabsolutelycontinuousrealvaluefunctions,u′∈L2a,b. The inner product in W21[a,b] is defined as (29)uy,vyW21a,b=uava+∫abu′yv′ydy,and the norm uW21[a,b] is given by (30)uW21a,b=uy,uyW21a,bwhere u,v∈W21[a,b].Theorem 5.
There existsR(s,y)∈W21[a,b] such that, for any u∈W21[a,b] and each fixed y,x∈W21[a,b], we have (31)uy,Rx,yW21a,b=ux.
In this case,R(x,y) is given by(32)Rx,y=α0x+α1xy,y≤xβ0x+β1xy,y>xwhere(33)α0x=1-a,α1x=1,β0x=1-a+x,β1x=0.Proof.
Using the integration by parts, one can get(34)uy,Rs,yW21a,b=uaRs,a+∫abu′y∂R∂ys,ydy=uaRs,a+ub∂R∂ys,b-ua∂R∂ys,a-∫abuy∂2R∂y2s,ydy.Since R(x,y) is a reproducing kernel of W21[a,b],(35)uy,Rx,yW21a,b=ux.Then,(36)-∂2R∂y2x,y=δy-xand(37)Rx,a-∂R∂yx,a=0,(38)∂R∂yx,b=0.Since the characteristic equation of -∂2R/∂y2(x,y)=δ(y-x) is λ2=0 and its characteristic value is λ=0 with 2 multiplicity roots, we write R(x,y) as(39)Rx,y=α0x+α1xy,y≤xβ0x+β1xy,y>x.Since ∂2R/∂y2(x,y)=-δ(y-x), we have(40)Rx,x+0-Rx,x-0=0(41)∂R∂yx,x+0-∂R∂yx,x-0=-1Hence,(42)α0x+α1xa-α1x=0,β1x=0,α0x+α1xx=β0x+β1xx,d1x-α1x=-1.Then,(43)α0x=1-a,α1x=1,β0x=1-a+x,β1x=0.
Now, we present how to solve problem (8)-(9) using the reproducing kernel method. Let (44)σix=Rxi,.for i∈N. It is clear that L:W22[a,b]→W21[a,b] is bounded. Let (45)ψix=L∗σixwhere L(σi(x))=Dαf(x)-λ∫0b‍K(x,t)(f(t)-a0)dt and L∗is the adjoint operator of L. Using Gram-Schmidt orthonormalization to generate orthonormal set of functions ψ¯i(x)i=1∞ where(46)ψ¯ix=∑j=1iαijψjxand αij are coefficients of Gram-Schmidt orthonormalization. In the next theorem, we show the existence of the solution of Problem (8)-(9).Theorem 6.
Ifxii=1∞ is dense on [a,b], then(47)fx=∑i=1∞∑j=1iαijgxjψ¯ix.Proof.
First, we want to prove thatψi(x)i=1∞ is the complete system of W22[a,b] and ψi(x)=L(Q(x,xi)). It is clear that ψi(x)∈W22[a,b] for i∈N. Simple calculations implies that(48)ψix=L∗σix=L∗σix,Qx,yW230,1=σix,LQx,yW22a,b=LQx,xi.For each fixed f(x)∈W22[a,b], let(49)fx,ψixW22a,b=0,i∈N.Then,(50)fx,ψixW22a,b=fx,L∗σixW22a,b=Lfx,σixW22a,b=Lfxi=0.Since xii=1∞ is dense on [a,b],Lf(x)=0. Since L-1 exists, f(x)=0. Thus, ψi(x)i=1∞ is the complete system of W22[a,b]. Second, we prove (47). Simple calculations imply that (51)qx=∑i=1∞qx,ψ¯ixW22a,bψ¯ix=∑i=1∞∑j=1iαijqx,L∗Qx,xjW22a,bψ¯ix=∑i=1∞∑j=1iαijLqx,Qx,xjW22a,bψ¯ix=∑i=1∞∑j=1iαijrx,Qx,xjW22a,bψ¯ix=∑i=1∞∑j=1iαijrxjψ¯ix.
Let the approximate solution of problem (8)-(9) be given by(52)qNs=∑i=1N∑j=1iαijrsjψ¯is.In the next theorem, we show the uniformly convergent of theqN(x)N=1∞ to q(x).Theorem 7.
Ifq(x) and qN(x) are given as in (47) and (52), then qN(x)N=1∞ converges uniformly to q(x).Proof.
For anyx∈[a,b], (53)qx-qNxW22a,b2=qx-qNx,qx-qNxW22a,b=∑i=N+1∞qx,ψ¯ixW22a,bψ¯ix,qx,ψ¯ixW22a,bψ¯ixW22a,b=∑i=N+1∞qx,ψ¯ixW22a,b2. Hence, (54)supx∈a,bqx-qNxW22a,b2=supx∈a,b∑i=N+1∞qx,ψ¯ixW22a,b2.From Theorem 6, one can see that ∑i=1∞‍(q(x),ψ¯i(x))W22[a,b]ψ¯i(x) converges uniformly to q(x). Hence, (55)supx∈a,bqx-qNxW22a,b=0asN→∞which implies that qN(x)N=1∞ converges uniformly to qN(x).


## 3. Analysis of RKHSM for Nonlinear Fractional Fredholm Integrodifferential Equations

In this section, we discuss how to solve the following the following problem using RKHSM:(56)Dζqx=rx+λ∫abhx,tqt+a0mdt,0<ζ≤1subject to(57)qa=0.Let (58)Gx,q=λ∫abHx,tqt+a0mdt.We construct a homotopy as follows:(59)Hq,p=Dζqx-rx-pGx,q=0where p∈[0,1] is an embedding parameter. For p=0, we get a linear equation (60)Dζfqx-rx=0which can be solved by using RKHSM as we described in the pervious section. If λ=1, we turn out to be problem (56). Following the Homotopy Perturbation method [15], we expand the solution in terms of the Homotopy parameter λ as(61)q=q0+pq1+p2q2+p3q3+⋯.Substitute (61) into (59) and equate the coefficients of the identical powers of λ to get the following system:(62)λ0:Dζq0x=rx,q0a=0,λ1:Dζq1x=Gx,∑i=0∞piqip=0,q1a=0,λ2:Dζq2x=dGx,∑i=0∞piqidλp=0,q2a=0,λ3:Dζq3x=d2Gx,∑i=0∞piqidλ2p=0,q3a=0,⋮λk:Dζqkx=dk-1Gx,∑i=0∞piqidλk-1p=0,qka=0.From (61), it is easy to see that the solution to problem (56)-(57) is given by (63)qx=∑k=0∞qkx=∑k=0∞∑i=1∞∑j=1iαijrkxjψ¯ix.We approximate the solution of problem (56)-(57) by(64)fN,Mx=∑k=0m∑i=1n∑j=1iαijgksjψ¯ix.


## 4. Numerical Results

In this section, we present three numerical examples to show the efficiency of the proposed method.Example 1.
Consider the following fractional Fredholm integrodifferential equation:(65)D1/2ux=gx+λ∫01xtutdtsubject to(66)u0=0where g(x)=32/3πx3/2+16/πx5/2-2λx. Using N=10, the approximate solution is (67)u10x=4x2+5x3.Thus, u10(x) is the exact solution.Example 2.
Consider the following fractional Fredholm integrodifferential equation:(68)D1/4ux=gx+2∫01ex-tu2tdtsubject to(69)u0=1where g(x)=x1/4E1,7/2(x)-2(e-1)ex and Ea,b(x) is the two-parameter of Mittag-Leffler function. The exact solution is u(x)=ex. Using N=15, the approximate solution is given in Figure 1. The error (70)en=max⁡ux-unx:x=0,0.1,0.2,…,1is given in Table 1.Table 1
The error for Example1.
n e 15  10 2.8 ∗ 1 0 - 10  13 4.1 ∗ 1 0 - 12  16 3.9 ∗ 1 0 - 13  19 1.1 ∗ 1 0 - 14Figure 1
The exact and approximate solution forN=15.Example 3.
Consider the following fractional Fredholm integrodifferential equation:(71)D3/4ux=gx+∫01extu4tdtsubject to (72)u0=2where g(x)=24/Γ(17/4)x13/4-10307/630ex and Γ is the gamma function. The exact solution is u(x)=ex. Using N=12, the approximate solution is (73)u12x=x4+2.Hence, u12(x) is the exact solution.


## 5. Conclusions and Closing Remarks

In this paper, we investigate the nonlinear fractional Fredholm integrodifferential equations where0<α≤1. We implement the reproducing kernel method to approximate the solution of the proposed problem. Numerical results demonstrate the accuracy of the present algorithm. In addition, we prove the existence of the solution of the nonlinear fractional Fredholm integrodifferential equation. Uniformly convergence of the approximate solution produced by the RKM to the exact solution is proven. We noted the following:(i)
The proposed method is very accurate. We get the exact solution in Examples1 and 3.(ii)
Form Table1, we note that the error is very small in Example 2.(iii)
Figure1 shows that the approximate solution and the exact solution are identical.(iv)
The proposed method can be generalized for more models in Physics and Engineering.



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