# Application of Reproducing Kernel Method for Solving Nonlinear Fredholm-Volterra Integrodifferential Equations

Authors: Omar Abu Arqub; Mohammed Al-Smadi; Shaher Momani
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper investigates the numerical solution of nonlinear Fredholm-Volterra integro-differential equations using reproducing kernel Hilbert space method. The solutionux is represented in the form of series in the reproducing kernel space. In the mean time, the n-term approximate solution un(x) is obtained and it is proved to converge to the exact solution ux. Furthermore, the proposed method has an advantage that it is possible to pick any point in the interval of integration and as well the approximate solution and its derivative will be applicable. Numerical examples are included to demonstrate the accuracy and applicability of the presented technique. The results reveal that the method is very effective and simple.


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## Body

## 1. Introduction

In recent years, there has been a growing interest in the integrodifferential equations (IDEs) which are a combination of differential and Fredholm-Volterra integral equations. IDEs are often involved in the mathematical formulation of physical phenomena. IDEs can be encountered in various fields of science such as physics, biology, and engineering. These kinds of equations can also be found in numerous applications, such as biomechanics, electromagnetic, elasticity, electrodynamics, fluid dynamics, heat and mass transfer, and oscillation theory [1–4].The purpose of this paper is to extend the application of the reproducing kernel Hilbert space (RKHS) method to solve the nonlinear Fredholm-Volterra IDE which is as follows:(1.1)u′(x)+f(x)N(x,u(x))+∫ab‍k1(x,t)G1(u(t))dt+∫ax‍k2(x,t)G2(u(t))dt=g(x),a≤x,t≤b,
subject to the initial condition
(1.2)u(a)=α,
where a,b,α are real finite constants, u∈W22[a,b] is an unknown function to be determined, f,g∈W21[a,b], k1(x,t), k2(x,t) are continuous functions on [a,b]×[a,b], G1(w), G2(y), N(x,z) are continuous terms in W21[a,b] as w=w(x), y=y(x), z=z(x)∈W22[a,b], a≤x≤b, -∞<w,y,z<∞ and are depending on the problem discussed, and W21[a,b],W22[a,b] are reproducing kernel spaces.In general, nonlinear Fredholm-Volterra IDEs do not always have solutions which we can obtain using analytical methods. In fact, many of real physical phenomena encountered are almost impossible to solve by this technique. Due to this, some authors have proposed numerical methods to approximate the solutions of nonlinear Fredholm-Volterra IDEs. To mention a few, in [5] the authors have discussed the Taylor polynomial method for solving IDEs (1.1) and (1.2) when N(x,u(x))=1, G1(u(t))=u(t), and G2(u(t))=[u(t)]q, where q∈ℕ. The triangular functions method has been applied to solve the same equations when N(x,u(x))=u(x), G1(u(t))=[u(t)]p, and G2(u(t))=[u(t)]q, where p, q∈ℕ as described in [6]. Furthermore, the operational matrix with block-pulse functions method is carried out in [7] for the aforementioned IDEs in the case N(x,u(x))=1, G1(u(t))=[u(t)]p, and G2(u(t))=[u(t)]q, where p,q∈ℕ. Recently, the Hybrid Legendre polynomials and Block-Pulse functions approach for solving IDEs (1.1) and (1.2) when N(x,u(x))=1, G1(u(t))=[u(t)]p, and G2(u(t))=[u(t)]q, where p,q∈ℕ are proposed in [8]. The numerical solvability of Fredholm and Volterra IDEs and other related equations can be found in [9–11] and references therein. However, none of previous studies propose a methodical way to solve these equations. Moreover, previous studies require more effort to achieve the results, they are not accurate, and usually they are developed for special types of IDEs (1.1) and (1.2).Reproducing kernel theory has important application in numerical analysis, differential equations, integral equations, probability and statistics, and so on [12–14]. Recently, using the RKHS method, the authors in [15–29] have discussed singular linear two-point boundary value problems, singular nonlinear two-point periodic boundary value problems, nonlinear system of boundary value problems, initial value problems, singular integral equations, nonlinear partial differential equations, operator equations, and fourth-order IDEs.The outline of the paper is as follows: several reproducing kernel spaces are described in Section2. In Section 3, a linear operator, a complete normal orthogonal system, and some essential results are introduced. Also, a method for the existence of solutions for (1.1) and (1.2) based on reproducing kernel space is described. In Section 4, we give an iterative method to solve (1.1) and (1.2) numerically in RKHS. Various numerical examples are presented in Section 5. This paper ends in Section 6 with some concluding remarks.


## 2. Several Reproducing Kernel Spaces

In this section, several reproducing kernels needed are constructed in order to solve (1.1) and (1.2) using RKHS method. Before the construction, we utilize the reproducing kernel concept. Throughout this paper ℂ is the set of complex numbers, L2[a,b]={u∣∫ab‍u2(x)dx<∞}, l2={A∣∑i=1∞(Ai)2<∞}, and the superscript (n) in u(n)(t) denotes the n-th derivative of u(t).Definition 2.1 (see [18]).
LetE be a nonempty abstract set. A function K:E×E→ℂ is a reproducing kernel of the Hilbert space H if(1)
for eacht∈E, K(·,t)∈H,(2)
for eacht∈E and φ∈H, 〈φ(·),K(·,t)〉=φ(t).The last condition is called “the reproducing property”: the value of the functionφ at the point t is reproducing by the inner product of φ with K(·,t). A Hilbert space which possesses a reproducing kernel is called a RKHS [18].Next, we first construct the spaceW22[a,b] in which every function satisfies the initial condition (1.2) and then utilize the space W21[a,b].Definition 2.2 (see [30]).
W
2
2
[
a
,
b
]
=
{
u
:
u
,
u
′ are absolutely continuous on [a,b], u,u′,u′′∈L2[a,b], and u(a)=0}. The inner product and the norm in W22[a,b] are defined, respectively, by
(2.1)〈u,v〉W22=u(a)v(a)+u′(a)v′(a)+∫ab‍u′′(y)v′′(y)dy
and ∥u∥W22=〈u,u〉W22, where u,v∈W22[a,b].Definition 2.3 (see [23]).
W
2
1
[
a
,
b
]
=
{
u
:
u is absolutely continuous on [a,b] and u,u′∈L2[a,b]}. The inner product and the norm in W21[a,b] are defined, respectively, by 〈u,v〉W21=∫ab‍u(t)v(t)+u′(t)v′(t)dt and ∥u∥W21=〈u,u〉W21, where u,v∈W21[a,b].In [23], the authors have proved that the spaceW21[a,b] is a complete reproducing kernel space and its reproducing kernel function is given by
(2.2)Tx(y)=12sinh(b-a)[cosh(x+y-b-a)+cosh(|x-y|-b+a)].
From the definition of the reproducing kernel spaces W21[a,b] and W22[a,b], we get W21[a,b]⊃W22[a,b].The Hilbert spaceW22[a,b] is called a reproducing kernel if for each fixed x∈[a,b] and any u(y)∈W22[a,b], there exist K(x,y)∈W22[a,b] (simply Kx(y)) and y∈[a,b] such that 〈u(y),Kx(y)〉W22=u(x). The next theorem formulates the reproducing kernel function Kx(y).Theorem 2.4.
The Hilbert spaceW22[a,b] is a reproducing kernel and its reproducing kernel function Kx(y) can be written as
(2.3)Kx(y)={∑i=14pi(x)yi-1,y≤x,∑i=14qi(x)yi-1,y>x,
where pi(x) and qi(x) are unknown coefficients of Kx(y).Proof.
Through several integrations by parts for (2.1), we obtain 〈u(y),Kx(y)〉W22=∑i=01u(i)(a)(Kx(i)(a)+(-1)iKx(3-i)(a))+∑i=01(-1)1-iu(i)(b)Kx(3-i)(b)+∫ab‍u(y)Kx(4)(y)dy. Since Kx(y)∈W22[a,b], it follows that Kx(a)=0. Also, since u∈W22[a,b], one obtains u(a)=0. Thus, if Kx(i)(b)=0, i=2,3, and Kx′(a)-Kx′′(a)=0, then 〈u(y),Kx(y)〉W22=∫ab‍u(y)Kx(4)(y)dy. Now, for each x∈[a,b], if Kx(y) also satisfies Kx(4)(y)=δ(x-y), where δ is the dirac-delta function, then 〈u(y),Kx(y)〉W22=u(x). Obviously, Kx(y) is the reproducing kernel function of the space W22[a,b].
Next, we give the expression of the reproducing kernel functionKx(y). The characteristic equation of Kx(4)(y)=δ(x-y) is λ4=0, and their characteristic values are λ=0 with 4 multiple roots. So, let the kernel Kx(y) be as defined in (2.3).
On the other hand, letKx(y) satisfy Kx(m)(x+0)=Kx(m)(x-0), m=0,1,2. Integrating Kx(4)(y)=δ(x-y) from x-ε to x+ε with respect to y and letting ε→0, we have the jump degree of Kx(3)(y) at y=x given by Kx(3)(x-0)-Kx(3)(x+0)=-1. Through the last descriptions the unknown coefficients of (2.3) can be obtained. This completes the proof.By using Mathematica7.0 software package, the representation of the reproducing kernel function Kx(y) is provided by
(2.4)Kx(y)={16(y-a)(2a2-y2+3x(2+y)-a(6+3x+y)),y≤x,16(x-a)(2a2-x2+3y(2+x)-a(6+x+3y)),y>x.The following corollary summarizes some important properties of the reproducing kernel functionKx(y).Corollary 2.5.
The reproducing kernel functionKx(y) is symmetric, unique, and Kx(x)≥0 for any fixed x∈[a,b].Proof.
By the reproducing property, we haveKx(y)=〈Kx(ξ),Ky(ξ)〉=〈Ky(ξ),Kx(ξ)〉=Ky(x) for each x and y. Now, let Kx1(y) and Kx2(y) be all the reproducing kernels of the space W22[a,b]; then Kx1(y)=〈Kx1(ξ),Ky2(ξ)〉=〈Ky2(ξ),Kx1(ξ)〉=Ky2(x)=Kx2(y). Finally, we note that Kx(x)=〈Kx(ξ),Kx(ξ)〉=∥Kx(ξ)∥2≥0.


## 3. Introduction to a Linear Operator and a Normal Orthogonal System inW22[a,b]

In this section, we construct an orthogonal function system ofW22[a,b]. Also, representation of the solution of (1.1) and (1.2) is given in the reproducing kernel space W22[a,b].To do this, we define a differential operatorL:W22[a,b]→W21[a,b] such that Lu(x)=u′(x). After homogenization of the initial condition (1.2), IDEs (1.1) and (1.2) can be converted into the equivalent form as follows:
(3.1)Lu(x)=F(x,u(x),Tu(x),Su(x)),a≤x≤b,u(a)=0,
such that F(x,u(x),Tu(x),Su(x))=g(x)-f(x)N(x,u(x))-Tu(x)-Su(x), Tu(x)=∫ab‍k1(x,t)G1(u(t))dt, and Su(x)=∫ax‍k2(x,t)G2(u(t))dt, where u(x)∈W22[a,b] and F(x,w,y,z)∈W21[a,b] for x∈[a,b] and w=w(x),y=y(x),z=z(x)∈W22[a,b], -∞<w,y,z<∞. It is easy to show that L is a bounded linear operator from W22[a,b] to W21[a,b].Now, we construct an orthogonal function system ofW22[a,b]. Let φi(x)=Txi(x) and ψi(x)=L*φi(x), where {xi}i=1∞ is dense on [a,b] and L* is the adjoint operator of L. From the properties of the reproducing kernel Tx(y), we have 〈u(x),φi(x)〉W21=〈u(x),Txi(x)〉W21=u(xi) for every u(x)∈W21[a,b]. In terms of the properties of Kx(y), one obtains 〈u(x),ψi(x)〉W22=〈u(x),L*φi(x)〉W22=〈Lu(x),φi(x)〉W21=Lu(xi), i=1,2,⋯.It is easy to see thatψi(x)=L*φi(x)=〈L*φi(x),Kx(y)〉W22=〈φi(x),LyKx(y)〉W21=LyKx(y)|y=xi. Thus, ψi(x) can be expressed in the form ψi(x)=LyKx(y)|y=xi, where Ly indicates that the operator L applies to the function of y.Theorem 3.1.
For (3.1), if {xi}i=1∞ is dense on [a,b], then {ψi(x)}i=1∞ is the complete function system of the space W22[a,b].Proof.
Clearly,ψi(x)∈W22[a,b]. For each fixed u(x)∈W22[a,b], let 〈u(x),ψi(x)〉W22=0, i=1,2,…, which means that 〈u(x),ψi(x)〉W22=〈u(x),L*φi(x)〉W22=〈Lu(x),φi(x)〉W21=Lu(xi)=0. Note that {xi}i=1∞ is dense on [a,b]; therefore, Lu(x)=0. It follows that u(x)=0 from the existence of L-1. So, the proof of the theorem is complete.The orthonormal function system{ψ¯i(x)}i=1∞ of the space W22[a,b] can be derived from Gram-Schmidt orthogonalization process of {ψi(x)}i=1∞ as follows:
(3.2)ψ-i(x)=∑k=1iβikψk(x),
where βik are orthogonalization coefficients given as β11=1/∥ψ1∥, βii=1/dik, and βij=-(1/dik)∑k=ji-1cikβkj for j<i in which dik=∥ψi∥2-∑k=1i-1cik2, cik=〈ψi,ψ¯k〉W22, and {ψi(x)}i=1∞ is the orthonormal system in the space W22[a,b].Theorem 3.2.
For eachu(x)∈W22[a,b], the series ∑i=1∞〈u(x),ψ-i(x)〉ψ-i(x) is convergent in the norm of W22[a,b]. On the other hand, if {xi}i=1∞ is dense on [a,b] and u(x) is the exact solution of (3.1), then
(3.3)u(x)=∑i=1∞∑k=1iβikF(xk,u(xk),Tu(xk),Su(xk))ψ-i(x).Proof.
Sinceu(x)∈W22[a,b], ∑i=1∞〈u(x),ψ-i(x)〉ψ-i(x) is the Fourier series expansion about normal orthogonal system {ψ-i(x)}i=1∞, and W22[a,b] is the Hilbert space, then the series ∑i=1∞〈u(x),ψ-i(x)〉ψ-i(x) is convergent in the sense of ∥·∥W22. On the other hand, using (3.2), we have
(3.4)u(x)=∑i=1∞〈u(x),ψ-i(x)〉W22ψ-i(x)=∑i=1∞〈u(x),∑k=1iβikψk(x)〉W22ψ-i(x)=∑i=1∞∑k=1iβik〈u(x),ψk(x)〉W22ψ-i(x)=∑i=1∞∑k=1iβik〈u(x),L*φk(x)〉W22ψ-i(x)=∑i=1∞∑k=1iβik〈Lu(x),φk(x)〉W21ψ-i(x)=∑i=1∞∑k=1iβikLu(xk)ψ-i(x).
But since u(x) is the exact solution of (3.1), then Lu(x)=F(x,u(x),Tu(x),Su(x)) and
(3.5)u(x)=∑i=1∞∑k=1iβikF(xk,u(xk),Tu(xk),Su(xk))ψ-i(x).
So, the proof of the theorem is complete.Note that we denote to the approximate solution ofu(x) by
(3.6)un(x)=∑i=1n∑k=1iβikF(xk,u(xk),Tu(xk),Su(xk))ψ-i(x).Theorem 3.3.
Ifu(x)∈W22[a,b], then there exists M>0 such that ||u(i)(x)||C≤M||u(x)||W22, i=0,1, where ||u(x)||C=maxa≤x≤b|u(x)|.Proof.
For anyx,y∈[a,b], we have u(i)(x)=〈u(y),Kx(i)(y)〉W22,i=0,1. By the expression of Kx(y), it follows that ∥Kx(i)(y)∥W22≤Mi, i=0,1. Thus, |u(i)(x)|=|〈u(x),Kx(i)(x)〉W22|≤∥u(x)∥W22∥Kx(i)(x)∥W22≤Mi∥u(x)∥W22, i=0,1. Hence, ||u(i)(x)||C≤M||u(x)||W22,  i=0,1, where M=max{M0,M1}. The proof is complete.Corollary 3.4.
The approximate solutionun(x) and its derivative un′(x) are uniformly convergent.Proof.
By Theorems3.2 and 3.3, for any x∈[a,b], we get
(3.7)|un(x)-u(x)|=|〈un(x)-u(x),Kx(x)〉W22|≤∥Kx(x)∥W22∥un(x)-u(x)∥W22≤M0∥un(x)-u(x)∥W22.
On the other hand,(3.8)|un′(x)-u′(x)|=|〈un(x)-u(x),Kx′(x)〉W22|≤∥Kx′(x)∥W22∥un(x)-u(x)∥W22≤M1∥un(x)-u(x)∥W22.
Hence,|un(i)(x)-u(i)(x)|≤∥un(i)(x)-u(i)(x)∥C≤Mi∥un(x)-u(x)∥W22, where M0 and M1 are positive constants. Hence, if ∥un(x)-u(x)∥W22→0 as n→∞, the approximate solutions un(x) and un′(x) converge uniformly to the exact solution u(x) and its derivative, respectively.


## 4. Iterative Method and Convergence Theorem

In this section, an iterative method of obtaining the solution of (3.1) is presented in the reproducing kernel space W22[a,b].First of all, we will mention the following remark in order to solve (1.1) and (1.2) numerically. If (1.1) is linear, then the exact and approximate solutions can be obtained directly from (3.5) and (3.6), respectively. On the other hand, if (1.1) is nonlinear, then the exact and approximate solutions can be obtained using the following iterative method.According to (3.5), the representation of the solution of (1.1) can be denoted by
(4.1)u(x)=∑i=1∞Aiψ-i(x),
where Ai=∑k=1iβikF(xk,u(xk),Tu(xk),Su(xk)). In fact, Ai, i=1,2,…, in (4.1) are unknown, and we will approximate Ai using known Bi. For a numerical computation, we define initial function u0(x1)=0 and the n-term approximation to u(x) by
(4.2)un(x)=∑i=1nBiψ-i(x),
where the coefficients Bi are given as
(4.3)B1=β11F(x1,u0(x1),Tu0(x1),Su0(x1)),u1(x)=B1ψ-1(x),B2=∑k=12β2kF(xk,uk-1(xk),Tuk-1(xk),Suk-1(xk)),u2(x)=∑i=12Biψ-i(x),⋮un-1(x)=∑i=1n-1Biψ-i(x),Bn=∑k=1nβnkF(xk,uk-1(xk),Tuk-1(xk),Suk-1(xk)).We mention here the following remark: in the iterative process of (4.2), we can guarantee that the approximation un(x) satisfies the initial condition (1.2).Now, the approximate solutionunN(x) can be obtained by taking finitely many terms in the series representation of un(x) and
(4.4)unN(x)=∑i=1N∑k=1iβikF(xk,un-1(xk),Tun-1(xk),Sun-1(xk))ψ-i(x).Next, we will prove thatun(x) in the iterative formula (4.2) is convergent to the exact solution u(x) of (1.1).Lemma 4.1.
If∥un(x)-u(x)∥W22→0, xn→y as n→∞ and F(x,v,w,z) is continuous in [a,b] with respect to x,v,w,z, for x∈[a,b] and v,w,z∈(-∞,∞), then as n→∞, one has F(xn,un-1(xn),Tun-1(xn),Sun-1(xn))→F(y,u(y),Tu(y),Su(y)).Proof.
Firstly, we will prove thatun-1(xn)→u(y) in the sense of ∥·∥W22. Since
(4.5)|un-1(xn)-u(y)|=|un-1(xn)-un-1(y)+un-1(y)-u(y)|≤|un-1(xn)-un-1(y)|+|un-1(y)-u(y)|,
by reproducing kernel property of Kx(y), we have un-1(xn) = 〈un-1(x),Kxn(x)〉 and un-1(y)=〈un-1(x),Ky(x)〉. Thus, |un-1(xn)-un-1(y)| = |〈un-1(x),Kxn(x)-Ky(x)〉W22|≤∥un-1(x)∥W22∥Kxn(x)-Ky(x)∥W22. From the symmetry of Kx(y), it follows that ∥Kxn(x)-Ky(x)∥W22→0 as n→∞. Hence, |un-1(xn)-un-1(y)|→0 as soon as xn→y.
On the other hand, by Corollary3.4, for any y∈[a,b], it holds that |un-1(y)-u(y)|→0. Therefore, un-1(xn)→u(y) in the sense of ∥·∥W22 as xn→y and n→∞.
Thus, by means of the continuation ofG1, G2, and N, it is obtained that G1(un-1(xn))→G1(u(y)), G2(un-1(xn))→G2(u(y)), and N(xn,un-1(xn))→N(y,u(y)) as n→∞. This shows that Tun-1(xn)→Tu(y) and Sun-1(xn)→Su(y) as n→∞. Hence, the continuity of F gives the result.Lemma 4.2.
{
u
n
}
n
=
1
∞ in (4.2) is monotonically increasing in the sense of the norm of W22[a,b].Proof.
By Theorem3.1, {ψ¯i}i=1∞ is the complete orthonormal system in the space W22[a,b]. Hence, we have ∥un∥W222=〈un(x),un(x)〉W22=〈∑i=1nBiψ-i(x),∑i=1nBiψ-i(x)〉W22=∑i=1n(Bi)2. Therefore, ∥un∥W22 is monotonically increasing.Lemma 4.3.
One hasLun(xj)=F(xj,uj-1(xj),Tuj-1(xj),Suj-1(xj)),j≤n.Proof.
The proof will be obtained by mathematical induction as follows: ifj≤n, then Lun(xj) = ∑i=1nBiLψ-i(xj) = ∑i=1nBi〈Lψ-i(x),φj(x)〉W21 = ∑i=1nBi〈ψ-i(x),Lj*φ(x)〉W22 = ∑i=1nBi〈ψ-i(x),ψj(x)〉W22. Thus,
(4.6)Lun(xj)=∑i=1nBi〈ψ-i(x),ψj(x)〉W22.
Multiplying both sides of (4.6) by βjl, summing for l from 1 to j, and using the orthogonality of {ψ-i(x)}i=1∞ yield that
(4.7)∑l=1jβjlLun(xl)=∑i=1nBi〈ψ-i(x),∑l=1jβjlψl(x)〉W22=∑i=1nBi〈ψ-i(x),ψ-j(x)〉W22=Bj=∑l=1jβjlF(xl,ul-1(xl),Tul-1(xl),Sul-1(xl)).
Now, ifj=1, then Lun(x1)=F(x1,u0(x1),Tu0(x1),Su0(x1)). On the other hand, if j=2, then β21Lun(x1)+β22Lun(x2)=β21F(x1,u0(x1),Tu0(x1),Su0(x1))+β22F(x2,u1(x2),Tu1(x2),Su1(x2)). Thus, Lun(x2)=F(x2,u1(x2),Tu1(x2),Su1(x2)). It is easy to see that Lun(xj)=F(xj,uj-1(xj),Tuj-1(xj),Suj-1(xj)) by using mathematical induction.Lemma 4.4.
One hasLun(xj)=Lu(xj),j≤n.Proof.
It is clear that on taking limits in (4.2) u(x)=∑i=1∞Biψ-i(x). Therefore, un(x)=Pnu(x), where Pn is an orthogonal projector from W22[a,b] to Span{ψ1,ψ2,…,ψn}. Thus, Lun(xj) = 〈Lun(x),φj(x)〉W21 = 〈un(x),Lj*φ(x)〉W22 = 〈Pnu(x),ψj(x)〉W22 = 〈u(x),Pnψj(x)〉W22 = 〈u(x),ψj(x)〉W22 = 〈Lu(x),φj(x)〉W21=Lu(xj).Theorem 4.5.
Suppose that∥un∥W22 is bounded in (4.2). If {xi}i=1∞ is dense on [a,b], then the n-term approximate solution un(x) in the iterative formula (4.2) converges to the exact solution u(x) of (3.1) in the space W22[a,b] and u(x)=∑i=1∞Biψ-i(x), where Bi is given by (4.3).Proof.
First of all, we will prove the convergence ofun(x). From (4.2), we infer that un+1(x)=un(x)+Bn+1ψ-n+1(x). The orthogonality of {ψ-i(x)}i=1∞ yields that
(4.8)||un+1||W222=||un||W222+(Bn+1)2=||un-1||W222+(Bn)2+(Bn+1)2=⋯=||u0||W222+∑i=1n+1(Bi)2.
From Lemma4.2, the sequence ∥un∥W22 is monotonically increasing. Due to the condition that ∥un∥W22 is bounded, ∥un∥W22 is convergent as n→∞. Then, there exists a constant c such that ∑i=1∞(Bi)2=c. This implies that Bi=∑k=1iβikF(xk,uk-1(xk),Tuk-1(xk),Suk-1(xk))∈l2,  i=1,2,….
Ifm>n, using (um-um-1)⊥(um-1-um-2)⊥⋯⊥(un+1-un), then one gets
(4.9)||um(x)-un(x)||W222=||um(x)-um-1(x)+um-1(x)-⋯+un+1(x)-un(x)||W222=||um(x)-um-1(x)||W222+⋯+||un+1(x)-un(x)||W222.
Furthermore, ||um(x)-um-1(x)||W222=(Bm)2. Consequently, ||um(x)-un(x)||W222=∑i=n+1m(Bi)2→0 as n,m→∞. Considering the completeness of the space W22[a,b], there exists a u(x)∈W22[a,b] such that un(x)→u(x) as n→∞ in the sense of ∥·∥W22.
Secondly, we will prove thatu(x) is the solution of (3.1). Since {xi}i=1∞ is dense on [a,b], for any x∈[a,b], there exists subsequence {xnj}j=1∞ such that xnj→x as j→∞. From Lemmas 4.3 and 4.4, it is easy to see that Lu(xnj)=F(xnj,unj-1(xk),Tunj-1(xk),Sunj-1(xk)). Hence, letting j→∞, by Lemma 4.1 and the continuity of F, we have Lu(x)=F(x,u(x),Tu(x),Su(x)). That is, u(x) is the solution of (3.1).
Sinceψ-i(x)∈W22[a,b], clearly, u(x) satisfies the initial condition (1.2). In other words, u(x) is the solution of (1.1) and (1.2), where u(x)=∑i=1∞Biψ-i(x) and Bi is given by (4.3). The proof is complete.Theorem 4.6.
Assume thatu(x) is the solution of (3.1) and rn(x) is the difference between the approximate solution un(x) and the exact solution u(x). Then, rn(x) is monotonically decreasing in the sense of the norm of W22[a,b].Proof.
It obvious that||rn(x)||W222=||u(x)-un(x)||W222=∥∑i=n+1∞Biψ-i(x)∥W222=∑i=n+1∞(Bi)2 and ||rn-1(x)||W222=∑i=n∞(Bi)2. Thus, ||rn(x)||W22≤||rn-1(x)||W22; consequently, the difference rn(x) is monotonically decreasing in the sense of ∥·∥W22. So, the proof of the theorem is complete.


## 5. Numerical Examples

In this section, some numerical examples are studied to demonstrate the accuracy and applicability of the present method. Results obtained are compared with the exact solution of each example and are found to be in good agreement with each other. In the process of computation, all the symbolic and numerical computations were performed by using Mathematica7.0 software package.Example 5.1.
Consider the nonlinear Fredholm-Volterra IDE:(5.1)u′(x)+u(x)-14∫01tu3(t)dt+12∫0xxu2(t)dt=g(x),0≤x,t≤1,u(0)=0,
where g(x)=(1/10)x6+x2+2x-(1/32). The exact solution is u(x)=x2.Using RKHS method, takingxi=(i-1)/(N-1), i=1,2,…,N, with the reproducing kernel function Kx(y) on [0,1], the approximate solution unN(x) is calculated by (4.4). The numerical results at some selected grid points for N=26 and n=5 are given in Table 1.Table 1
Numerical results for Example5.1.
x
Exact solution
Approximate solution
Absolute error
Relative error
0.16
0.0256
0.025600985200558
9.85201
×
10
-
7
3.84844
×
10
-
5
0.32
0.1024
0.102401995676314
1.99568
×
10
-
6
1.94890
×
10
-
5
0.48
0.2304
0.230403057350893
3.05735
×
10
-
6
1.32698
×
10
-
5
0.64
0.4096
0.409604197461425
4.19746
×
10
-
6
1.02477
×
10
-
5
0.80
0.6400
0.640005445257301
5.44526
×
10
-
6
8.50821
×
10
-
6
0.96
0.9216
0.921606832750570
6.83275
×
10
-
6
7.41401
×
10
-
6As we mention, we used the grid nodes mentioned earlier in order to obtain approximate solutions. Moreover, it is possible to pick any point in[a,b] and as well the approximate solution and its derivative will be applicable. Next, the numerical results for Example 5.1 at some selected gird nodes in [0,1] of u′(x) are given in Table 2.Table 2
Numerical results ofu′(x) for Example 5.1.
x
Exact solution
Approximate solution
Absolute error
Relative error
0.16
0.32
0.319999014799438
9.85201
×
10
-
7
3.07875
×
10
-
6
0.32
0.64
0.639998004323680
1.99568
×
10
-
6
3.11824
×
10
-
6
0.48
0.96
0.959996942649100
3.05735
×
10
-
6
3.18474
×
10
-
6
0.64
1.28
1.279995802538568
4.19746
×
10
-
6
3.27927
×
10
-
6
0.80
1.60
1.599994554742679
5.44526
×
10
-
6
3.40329
×
10
-
6
0.96
1.92
1.919993167249420
6.83275
×
10
-
6
3.55872
×
10
-
6Table3 shows, a comparison between the absolute errors of our method together with triangular functions method [6], operational matrix with block-pulse functions method [7], and Hybrid Legendre polynomials and block-pulse functions method [8]. As it is evident from the comparison results, it was found that our method in comparison with the mentioned methods is better with a view to accuracy and utilization.Table 3
Numerical comparison of absolute error for Example5.1.
x
Method in [6]
Method in [7]
Method in [8]
RKHS method
0.0
0
0
0
0
0.1
1.6600
×
10
-
4
2.1840
×
10
-
3
3.1000
×
10
-
5
6.1415
×
10
-
7
0.2
2.5400
×
10
-
4
1.4630
×
10
-
3
7.5000
×
10
-
5
1.2345
×
10
-
6
0.3
2.6200
×
10
-
4
1.6780
×
10
-
3
1.7100
×
10
-
4
1.8671
×
10
-
6
0.4
1.9100
×
10
-
4
7.2390
×
10
-
3
9.4000
×
10
-
5
2.5185
×
10
-
6
0.5
4.1000
×
10
-
5
1.6024
×
10
-
2
2.2800
×
10
-
4
3.1950
×
10
-
6
0.6
2.0200
×
10
-
4
1.1480
×
10
-
2
5.0200
×
10
-
4
3.9035
×
10
-
6
0.7
2.8300
×
10
-
4
4.5150
×
10
-
3
5.8300
×
10
-
4
4.6511
×
10
-
6
0.8
2.8300
×
10
-
4
4.8720
×
10
-
3
3.7400
×
10
-
4
5.4453
×
10
-
6
0.9
2.0200
×
10
-
4
1.6682
×
10
-
2
4.7000
×
10
-
5
6.2939
×
10
-
6
1.0
3.7000
×
10
-
5
3.0914
×
10
-
2
1.4000
×
10
-
5
7.2055
×
10
-
6Example 5.2.
Consider the nonlinear Fredholm-Volterra IDE:(5.2)u′(x)+2xu(x)-∫01(x-t)u(t)dt-∫0x(x+t)u3(t)dt=g(x),0≤x,t≤1,u(0)=1,
where g(x)=(-(2/3)x+(1/9))e3x+(2x+1)ex+((4/3)-e)x+(8/9). The exact solution is u(x)=ex.Using RKHS method, takingxi=(i-1)/(N-1), i=1,2,…,N, with the reproducing kernel function Kx(y) on [0,1], the approximate solution unN(x) is calculated by (4.4). The numerical results at some selected grid points for N=26 and n=1 are given in Table 4.Table 4
Numerical results for Example5.2.
x
Exact solution
Approximate solution
Absolute error
Relative error
0.0
1
1
0
0
0.1
1.105170918075648
1.105171575587132
6.57511
×
10
-
7
5.94941
×
10
-
7
0.2
1.221402758160170
1.221404273302208
1.51514
×
10
-
6
1.24049
×
10
-
6
0.3
1.349858807576003
1.349861385895122
2.57832
×
10
-
6
1.91007
×
10
-
6
0.4
1.491824697641270
1.491828549248914
3.85161
×
10
-
6
2.58181
×
10
-
6
0.5
1.648721270700128
1.648726613238682
5.34254
×
10
-
6
3.24041
×
10
-
6
0.6
1.822118800390509
1.822125866868782
7.06648
×
10
-
6
3.87817
×
10
-
6
0.7
2.013752707470476
2.013761760093792
9.05262
×
10
-
6
4.49540
×
10
-
6
0.8
2.225540928492467
2.225552279886237
1.13514
×
10
-
5
5.10051
×
10
-
6
0.9
2.459603111156949
2.459617154977751
1.40438
×
10
-
5
5.70979
×
10
-
6
1.0
2.718281828459045
2.718299082481196
1.72540
×
10
-
5
6.34740
×
10
-
6The comparison among the RKHS solution besides the solutions of triangular functions [6], operational matrix with block-pulse functions solution [7], and exact solutions are shown in Table 5.Table 5
Numerical comparison of approximate solution for Example5.2.
x
Exact solution
RKHS method
Method in [6]
Method in [7]
0.0
1.000000
1.000000
1.000000
1.000000
0.1
1.105171
1.105172
1.105223
1.115708
0.2
1.221403
1.221404
1.221494
1.225369
0.3
1.349859
1.349861
1.349971
1.345821
0.4
1.491825
1.491829
1.491933
1.478128
0.5
1.648721
1.648727
1.648795
1.675019
0.6
1.822119
1.822126
1.822484
1.839752
0.7
2.013753
2.013762
2.014465
2.020740
0.8
2.225541
2.225552
2.226719
2.219630
0.9
2.459603
2.459617
2.461507
2.438278
1.0
2.718280
2.718300
2.721505
2.678832Example 5.3.
Consider the nonlinear Fredholm-Volterra IDE:(5.3)u′(x)+u2(x)-∫01(t+1)sinh(u(t)-1)dt-∫0xxeu(t)dt=g(x),0≤x,t≤1,u(0)=1,
where g(x)=(ln(x+1)+1)2+(x+1)-1-(e/2)(x+2)x2-(2/3). The exact solution is u(x)=1+ln(x+1).Using RKHS method, takingxi=(i-1)/(N-1), i=1,2,…,N, with the reproducing kernel function Kx(y) on [0,1], the approximate solution unN(x) is calculated by (4.4). The numerical results at some selected grid points for N=51 and n=1 are given in Table 6.Table 6
Numerical results for Example5.3.
x
Exact solution
Approximate solution
Absolute error
Relative error
0.16
1.148420005118273
1.148420410550930
4.05433
×
10
-
7
3.53035
×
10
-
7
0.32
1.277631736598280
1.277632408078983
6.71481
×
10
-
7
5.25567
×
10
-
7
0.48
1.392042087776024
1.392042943683538
8.55908
×
10
-
7
6.14857
×
10
-
7
0.64
1.494696241836107
1.494697231133553
9.89297
×
10
-
7
6.61872
×
10
-
7
0.80
1.587786664902119
1.587787753995242
1.08909
×
10
-
6
6.85919
×
10
-
7
0.96
1.672944473242426
1.672945638649806
1.16541
×
10
-
6
6.96620
×
10
-
7Example 5.4.
Consider the nonlinear Fredholm-Volterra IDE:(5.4)u′(x)+cos(u(x))-∫01(t+x)u(t)dt-∫0xteu(t)dt=g(x),0≤x,t≤1,u(0)=1,
where g(x)=cos(1-x)+e1-x(1+x)-(1/2)x-(7/6)-e. The exact solution is u(x)=1-x.Using RKHS method, takingxi=(i-1)/(N-1), i=1,2,…,N, with the reproducing kernel function Kx(y) on [0,1], the approximate solution unN(x) is calculated by (4.4). The numerical results at some selected grid points for N=51 and n=5 are given in Table 7.Table 7
Numerical results for Example5.4.
x
Exact solution
Approximate solution
Absolute error
Relative error
0.16
0.84
0.840000573907808
5.73908
×
10
-
7
6.83224
×
10
-
7
0.32
0.68
0.680001166402260
1.16640
×
10
-
6
1.71530
×
10
-
6
0.48
0.52
0.520001763060855
1.76306
×
10
-
6
3.39050
×
10
-
6
0.64
0.36
0.360002349461090
2.34946
×
10
-
6
6.52628
×
10
-
6
0.80
0.20
0.200002911180463
2.91118
×
10
-
6
1.45559
×
10
-
5
0.96
0.04
0.040003433796472
3.43380
×
10
-
6
8.58449
×
10
-
5Example 5.5.
Consider the nonlinear Fredholm-Volterra IDE:(5.5)u′(x)+xu2(x)-∫01(t+x)(1+u2(t))dt-∫0xxcos(u(t))dt=g(x),0≤x,t≤1,u(0)=0,
where g(x)=-xsin(x)+x3-(4/3)x+(1/4). The exact solution is u(x)=x.
Using RKHS method, takingxi=(i-1)/(N-1), i=1,2,…,N with the reproducing kernel function Kx(y) on [0,1], the approximate solution unN(x) is calculated by (4.4). The numerical results at some selected grid points for N=26 and n=5 are given in Table 8.Table 8
Numerical results for Example5.5.
x
Exact solution
Approximate solution
Absolute error
Relative error
0.16
0.16
0.159999994448885
5
.
55112
×
10
-
9
3
.
46945
×
10
-
8
0.32
0.32
0.319999983346655
1
.
66533
×
10
-
8
5
.
20417
×
10
-
8
0.48
0.48
0.479999983346655
1
.
66533
×
10
-
8
3
.
46945
×
10
-
8
0.64
0.64
0.639999777955395
2
.
22045
×
10
-
7
3
.
46945
×
10
-
7
0.80
0.80
0.799999555910790
4
.
44089
×
10
-
7
5
.
55112
×
10
-
7
0.96
0.96
0.959999555910790
4
.
44089
×
10
-
7
4
.
62593
×
10
-
7


## 6. Conclusion

In this paper, the RKHS method was employed to solve the nonlinear Fredholm-Volterra IDEs (1.1) and (1.2). The solution u(x) and the approximate solution un(x) are represented in the form of series in the space W22[a,b]. Moreover, the approximate solution and its derivative converge uniformly to the exact solution and its derivative, respectively. Meanwhile, the error of the approximate solution is monotonically decreasing in the sense of the norm of W22[a,b].



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*Source: 839836-2012-09-09.xml*