# A Computational Method for Two-Point Boundary Value Problems of Fourth-Order Mixed Integrodifferential Equations

Authors: Mohammed Al-Smadi; Omar Abu Arqub; Shaher Momani
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

In this paper, reproducing kernel Hilbert space method is applied to approximate the solution of two-point boundary value problems for fourth-order Fredholm-Volterra integrodifferential equations. The analytical solution was calculated in the form of convergent series in the spaceW25[a,b] with easily computable components. In the proposed method, the n-term approximation is obtained and is proved to
converge to the analytical solution. Meanwhile, the error of the approximate
solution is monotone decreasing in the sense of the norm of W25[a,b]. The proposed technique is applied to several examples to illustrate the
accuracy, efficiency, and applicability of the method.


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## Body

## 1. Introduction

Boundary value problems (BVPs) of fourth-order integrodifferential equations (IDEs) which are a combination of differential and integral equations arise very frequently in many branches of applied mathematics and physics such as fluid dynamics, biological models, chemical kinetics, biomechanics, electromagnetic, elasticity, electrodynamics, heat and mass transfer, and oscillation theory [1–4]. If the BVPs for fourth-order Fredholm-Volterra IDEs cannot be solved analytically, which is the usual case, then recourse must be made to numerical and approximate methods.This paper discusses and investigates the analytical approximate solution using reproducing kernel Hilbert space (RKHS) method for fourth-order Fredholm-Volterra IDE which is as follows:(1)u(4)(x)=f(x)+N(x,u(x))+∫abh1(x,t)G1(u(t))dt+∫axh2(x,t)G2(u(t))dt,a≤t,x≤b,
subject to the boundary conditions
(2)u(a)=α0,u′(a)=α1,u(b)=β0,u′(b)=β1,
where αi, βi, i=0,1 are real finite constants, u∈W25[a,b] is an unknown function to be determined, f∈W21[a,b], h1(x,t), h2(x,t) are continuous functions on [a,b]2, G1(w), G2(y), N(x,z) are continuous terms in W21[a,b] as w=w(x), y=y(x), z=z(x)∈W25[a,b], a≤x≤b, -∞<w, y,z<∞ and are depending on the problem discussed, and W21[a,b], W25[a,b] are two reproducing kernel spaces.Recently, many authors have discussed the numerical solvability of BVPs for fourth-order Volterra IDEs which are special cases of (1) and (2). To mention a few, in [5], the authors have discussed the Adomian decomposition method for solving IDEs (1) and (2) when N(x,u(x))=u(x), G1(u(t))=0, and G2(u(t))=[u(t)]p, where p∈ℕ. The differential transform method has been applied to solve the same equations when N(x,u(x))=u(x), G1(u(t))=0, and G2(u(t))=[u(t)]p, where p∈ℕ as described in [6]. Furthermore, the homotopy perturbation method is carried out in [7] for the aforementioned IDEs in the case that N(x,u(x))=u(x), G1(u(t))=0, and G2(u(t))=[u(t)]p, where p∈ℕ. Recently, the homotopy analysis method for solving (1) and (2) when N(x,u(x))=u(x), G1(u(t))=0, and G2(u(t))=[u(t)]p, where p∈ℕ, is proposed in [8]. But on the other aspects as well, the numerical solvability of differential and integral equations of different type and order can be found in [9–15] and references therein.Investigation about two-point BVPs for fourth-order Fredholm-Volterra IDEs is scarce. Actually, no one has given a method to get approximate solution for this type of equations in the literature. Also, 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 but are usually developed for special types of (1) and (2). The new method is accurate, needs less effort to achieve the results, and is developed especially for nonlinear case. Meanwhile, the proposed technique has an advantage that it is possible to pick any point in the interval of integration, and the approximate solutions and all its derivatives up to order four will be applicable as well.Reproducing kernel theory has important application in numerical analysis, differential equations, integral equations, probability and statistics, and so forth [16–18]. In the last years, extensive work has been done using RKHS method, which provides numerical approximations for linear and nonlinear equations. This method has been implemented in several operator, differential, integral, and integro-differential equations, such as nonlinear operator equations [19], nonlinear system of second-order BVPs [20], nonlinear fourth-order BVPs [21], nonlinear systems of initial value problems [22], nonlinear second-order singular BVPs [23], nonlinear partial differential equations [24], nonlinear Fredholm-Volterra integral equations [25], nonlinear fourth-order Volterra IDEs [26], nonlinear Fredholm-Volterra IDEs [27], and others.The paper is organized in the following form. In the next section, two reproducing kernel spaces are described. In Section3, a linear operator, a complete normal orthogonal system, and some essential results are introduced. Also, a method for the existence of solutions for (1) and (2) based on reproducing kernel space is described. In Section 4, we give an iterative method to solve (1) and (2) numerically in the space W25[a,b]. Numerical examples are presented in Section 5. Section 6 ends this paper with a brief conclusion.


## 2. Two Reproducing Kernel Spaces

In this section, two reproducing kernels needed are constructed in order to solve (1) and (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<∞}, and l2={A∣∑i=1∞(Ai)2<∞}.An abstract set is supposed to have elements, each of which has no structure, and is itself supposed to have no internal structure, except that the elements can be distinguished as equal or unequal, and to have no external structure except for the number of elements.Definition 1 (see [23]).
LetE be a nonempty abstract set. A function K:E×E→ℂ is a reproducing kernel of the Hilbert space H if(1)
for eachx∈E, K(·,x)∈H;(2)
for eachx∈E and φ∈H, 〈φ(·),K(·,x)〉=φ(x).The second condition in Definition1 is called “the reproducing property”; the value of the function φ at the point x is reproducing by the inner product of φ(·) with K(·,x). A Hilbert space which possesses a reproducing kernel is called an RKHS [23].Next, we first construct the spaceW25[a,b] in which every function satisfies the boundary conditions (2) and then utilize the space W21[a,b].Definition 2 (see [21]).
W
2
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[
a
,
b
]
=
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u
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∣
u
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),  i=0,1,2,3,4 are absolutely continuous real-valued functions on [a,b], u(5)∈L2[a,b], and u(a)=u′(a)=u(b)=u′(b)=0}. The inner product and the norm in W25[a,b] are defined as
(3)〈u,v〉W25=∑i=02u(i)(a)v(i)(a)+∑i=01u(i)(b)v(i)(b)+∫abu(5)(y)v(5)(y)dy
and ∥u∥W25=〈u,u〉W25, respectively, where u,v∈W25[a,b].Definition 3 (see [19]).
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2
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a
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=
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u
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u is absolutely continuous real-valued function on [a,b] and u′∈L2[a,b]}. The inner product and the norm in W21[a,b] are defined as 〈u,v〉W21=∫ab‍(u′v′+uv)dy and ∥u∥W21=〈u,u〉W21, respectively, where u,v∈W21[a,b].Remark 4.
In [19], the authors have proved that the space W21[a,b] is a complete reproducing kernel space and its reproducing kernel is
(4)Rx(y)=12sinh(b-a)[cosh(x+y-b-a)+cosh(|x+y|-b+a)].
The Hilbert spaceW25[a,b] is called a reproducing kernel if for each fixed x∈[a,b] and any u(y)∈W25[a,b], there exist K(x,y)∈W25[a,b] (simply Kx(y)) and y∈[a,b] such that 〈u(y),Kx(y)〉W25=u(x). The next theorem formulates the reproducing kernel function Kx(y) on the space W25[a,b].Theorem 5.
The Hilbert spaceW25[a,b] is a reproducing kernel and its reproducing kernel function Kx(y) is given by(5)Kx(y)={∑i=09pi(x)yi,y≤x,∑i=09qi(x)yi,y>x,
where pi(x) and qi(x) are unknown coefficients of Kx(y).Proof.
Through several integrations by parts for (3), one can obtain 〈u(y),Kx(y)〉W25=∑i=02u(i)(a)Kx(i)(a)+∑i=01u(i)(b)Kx(i)(b)+∑i=04(-1)4-iu(i)(y)Kx(9-i)(y)|y=ay=b-∫ab‍u(y)Kx(10)(y)dy. Since Kx(y)∈W25[a,b], it follows that Kx(i)(a)=Kx(i)(b)=0, i=0,1. Again, since u∈W25[a,b], one obtains u(i)(a)=u(i)(b)=0, i=0,1. Thus, if Kx(2)(a)-Kx(7)(a)=0, Kx(i)(a)=0, i=5,6, and Kx(i)(b)=0, i=5,6,7, then 〈u(y),Kx(y)〉W25=∫ab‍u(y)(-Kx(10)(y))dy. Now, for each x∈[a,b], if Kx(y) also satisfies the formula -Kx(10)(y)=δ(x-y), where δ is the dirac-delta function, then 〈u(y),Kx(y)〉W25=u(x). Obviously, Kx(y) is the reproducing kernel of the space W25[a,b].
Next, we give the expression of the reproducing kernel functionKx(y). The characteristic equation of -Kx(10)(y)=δ(x-y) is λ10=0, and their characteristic values are λ=0 with 10 multiple roots. So, let the kernel Kx(y) be as defined in (5).
On the other hand, letKx(y) satisfy Kx(m)(x+0)=Kx(m)(x-0), m=0,1,…,8. Integrating -Kx(10)(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(9)(y) at y=x given by Kx(9)(x-0)-Kx(9)(x+0)=1. Through the last descriptions, the unknown coefficients of (5) can be obtained. This completes the proof.Remark 6.
Without the loss of generality and by using Mathematica 7.0 software package, the two rulesK1x(y)=∑i=09pi(x)yi and K2x(y)=∑i=09qi(x)yi of the reproducing kernel function Kx(y) in (5) are obtained at a=0 and b=1 in (1) and (2) and are given, respectively, as(6)
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.The following corollary summarizes some important properties of the reproducing kernel functionKx(y).Corollary 7.
The reproducing kernel functionKx(y) is symmetric and 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 W25[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. Main Results and the Structure of Solution

In this section, the representation of the analytical solution of (1) and (2) and the implementation method are given in the reproducing kernel space W25[a,b]. After that, we construct an orthogonal function system of W25[a,b] based on the Gram-Schmidt orthogonalization process.To do this, we define a differential operatorL as
(7)L:W25[a,b]⟶W21[a,b],
such that
(8)Lu(x)=u(4)(x).
After homogenization of the boundary conditions (2), (1) and (2) can be converted into the equivalent form as follows:
(9)Lu(x)=F(x,u(x),Tu(x),Su(x)),a≤x≤b,u(a)=0,u′(a)=0,u(b)=0,u′(b)=0,
such that F(x,u(x),Tu(x),Su(x))=f(x)+N(x,u(x))+Tu(x)+Su(x), Tu(x)=∫ab‍h1(x,t)G1(u(t))dt, and Su(x)=∫ax‍h2(x,t)G2(u(t))dt, where u(x)∈W25[a,b] and F(x,w,y,z)∈W21[a,b] for w=w(x), y=y(x), z=z(x)∈W25[a,b],  a≤x≤b,  -∞<w,y,z<∞. It is easy to show that L is a bounded linear operator from W25[a,b] into W21[a,b].Next, we construct an orthogonal function system ofW25[a,b]. Put φi(x)=Rxi(x) and ψi(x)=Li*φ(x), where {xi}i=1∞ is dense on [a,b] and L* is the adjoint operator of L. In terms of the properties of reproducing kernel function Rx(y), one obtains 〈u(x),ψi(x)〉W25=〈u(x),L*φi(x)〉W25=〈Lu(x),φi(x)〉W21=Lu(xi), i=1,2,….Remark 8.
The orthonormal function system{ψ¯i(x)}i=1∞ of the space W25[a,b] can be derived from Gram-Schmidt orthogonalization process of {ψi(x)}i=1∞ as follows:
(10)ψ-i(x)=∑k=1iβikψk(x),
where βik are orthogonalization coefficients and are given by the following subroutine:
(11)βij=1∥ψ1∥,fori=j=1,βij=1dik,fori=j≠1,βij=-1dik∑k=ji-1cikβkj,fori>j,
such that dik=∥ψi∥2-∑k=1i-1cik2, cik=〈ψi,ψ¯k〉W25, and {ψi(x)}i=1∞ is the orthonormal system in the space W25[a,b].Through the next theorem, the subscripty by the operator L (Ly) indicates that the operator L applies to the function of y.Theorem 9.
If{xi}i=1∞ is dense on [a,b], then {ψi(x)}i=1∞ is a complete function system of W25[a,b] and ψi(x)=LyKx(y)|y=xi.Proof.
Clearly,ψi(x)=Li*φ(x)=〈Li*φ(y),Kx(y)〉W25=〈φi(y),LyKx(y)〉W21=LyKx(y)|y=xi∈W25[a,b]. Now, for each fixed u(x)∈W25[a,b], let 〈u(x),ψi(x)〉W25=0, i=1,2,…. In other word, 〈u(x),ψi(x)〉W25=〈u(x),L*φi(x)〉W25=〈Lu(x),φi(x)〉W21=Lu(xi)=0. Note that {xi}i=1∞ is dense on [a,b], and therefore Lu(x)=0. It follows that u(x)=0 from the existence of L-1. So, the proof of the theorem is complete.Lemma 10.
Ifu(x)∈W25[a,b], then there exists M>0 such that ∥u(i)(x)∥C≤M∥u(x)∥W25, i=0,1,2,3,4, 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)〉W25, i=0,1,2,3,4. By the expression of Kx(y), it follows that ∥Kx(i)(y)∥W25≤Mi, i=0,1,2,3,4. Thus, |u(i)(x)|=|〈u(x),Kx(i)(x)〉W25|≤∥Kx(i)(x)∥W25∥u(x)∥W25≤Mi∥u(x)∥W25, i=0,1,2,3,4. Hence, ||u(i)(x)||C≤M||u(x)||W25, i=0,1,2,3,4, where M=maxi=0,1,2,3,4{Mi}.The structure of the next theorem is as follows. Firstly, we will give the representation of the exact solution of (1) and (2) in the space W25[a,b]. After that, the convergence of approximate solution un(x) to the analytic solution will be proved.Theorem 11.
For eachu(x) in the space W25[a,b], the series ∑i=1∞〈u(x),ψ-i(x)〉ψ-i(x) is convergent in the sense of the norm of W25[a,b]. On the other hand, if {xi}i=1∞ is dense on [a,b] and the solution of (1) and (2) is unique, then(i)
the exact solution of (9) could be represented by
(12)u(x)=∑i=1∞∑k=1iβikF(xk,u(xk),Tu(xk),Su(xk))ψ-i(x);(ii)
the approximate solution of (9)
(13)un(x)=∑i=1n∑k=1iβikF(xk,u(xk),Tu(xk),Su(xk))ψ-i(x),andun(i)(x), i=0,1,2,3,4 are converging uniformly to the exact solution u(x) and all its derivative as n→∞, respectively.Proof.
For the first part, letu(x) be solution of (9) in the space W25[a,b]. Since u(x) belong to Hilbert space W25[a,b] and ∑i=1∞〈u(x),ψ-i(x)〉ψ-i(x) is the Fourier series about normal orthogonal system {ψ-i(x)}i=1∞; then, the series ∑i=1∞〈u(x),ψ-i(x)〉ψ-i(x) is convergent in the sense of ∥·∥W25. On the other hand, using (10), we have
(14)u(x)=∑i=1∞〈u(x),ψ-i(x)〉W25ψ-i(x)=∑i=1∞∑k=1iβik〈u(x),ψk(x)〉W25ψ-i(x)=∑i=1∞∑k=1iβik〈u(x),L*φk(x)〉W25ψ-i(x)=∑i=1∞∑k=1iβik〈Lu(x),φk(x)〉W21ψ-i(x)=∑i=1∞∑k=1iβik〈F(x,u(x),Tu(x),Su(x)),φk(x)〉W21ψ-i(x)=∑i=1∞∑k=1iβikF(xk,u(xk),Tu(xk),Su(xk))ψ-i(x).
Therefore, the form of (12) is the exact solution of (9).
For the second part, it easy to see that by Lemma10, for any x∈[a,b],
(15)|un(x)-u(x)|=|〈un(x)-u(x),Kx(x)〉W25|≤∥Kx(x)∥W25∥un(x)-u(x)∥W25≤M0∥un(x)-u(x)∥W25.
On the other hand,
(16)|un(i)(x)-u(i)(x)|=|〈un(x)-u(x),Kx(i)(x)〉W25|≤∥Kx(i)(x)∥W25∥un(x)-u(x)∥W25≤Mi∥un(x)-u(x)∥W25,i=1,2,3,4,
where Mi, i=0,1,2,3,4 are positive constants. Hence, if ∥un(x)-u(x)∥W25→0 as n→∞, the approximate solution un(x) and un(i)(x), i=0,1,2,3,4 converge uniformly to the exact solution u(x) and all its derivative, respectively. So, the proof of the theorem is complete.Remark 12.
We mention here that the approximate solutionun(x) in (13) can be obtained by taking finitely many terms in the series representation for u(x) of (12).


## 4. Procedure of Constructing Iterative Method

In this section, an iterative method of obtaining the solution of (9) is presented in the reproducing kernel space W25[a,b] for both linear and nonlinear cases. Initially, we will mention the following remark about the exact and approximate solutions of (1) and (2).Remark 13.
In order to apply the RKHS technique to solve (1) and (2), we have the following two cases based on the structure of the functions N, G1, and G2.Case 1.
If (1) is linear, then the exact and approximate solutions can be obtained directly from (12) and (13), respectively.Case 2.
If (1) is nonlinear, then in this case the exact and approximate solutions can be obtained by using the following iterative algorithm.Algorithm 14.
According to (12), the representation of the analytical solution of (1) can be denoted by the following series:
(17)u(x)=∑i=1∞Biψ-i(x),
where Bi=∑k=1iβikF(xk,uk-1(xk),Tuk-1(xk),Suk-1(xk)). In fact, Bi, i=1,2,… in (17) are unknown; we will approximate Bi using known Ai. For numerical computations, we define initial function u0(x1)=0, put u0(x1)=u(x1), and define the n-term approximation to u(x) by
(18)un(x)=∑i=1nAiψ-i(x),
where the coefficients Ai of ψ-i(x), i=1,2,…,n are given as follows:
(19)A1=β11F(x1,u0(x1),Tu0(x1),Su0(x1)),u1(x)=A1ψ-1(x),A2=∑k=12β2kF(xk,uk-1(xk),Tuk-1(xk),Suk-1(xk)),u2(x)=∑i=12Aiψ-i(x),···un-1(x)=∑i=1n-1Aiψ-i(x),An=∑k=1nβnkF(xk,uk-1(xk),Tuk-1(xk),Suk-1(xk)).
Here, note that in the iterative process of (18), we can guarantee that the approximation un(x) satisfies the boundary conditions (2). Now, the approximate solution unN(x) can be obtained by taking finitely many terms in the series representation of un(x) and
(20)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 (18) converges to the exact solution u(x) of (1); in fact, this result is fundamental in the RKHS theory and its applications. The next two lemmas are collected for future use in order to prove the recent theorem.Lemma 15.
Ifun(x)→u(x) in the sense of the norm of W25[a,b], 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 F(xn,un-1(xn),Tun-1(xn),Sun-1(xn))→F(y,u(y),Tu(y),Su(y)) as n→∞.Proof.
Firstly, we will prove thatun-1(xn)→u(y) in the sense of ∥·∥W25, since
(21)|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 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)〉W25|≤∥un-1(x)∥W25∥Kxn(x)-Ky(x)∥W25. From the symmetry of Kx(y), it follows that ∥Kxn(x)-Ky(x)∥W25→0 as n→∞. Hence, |un-1(xn)-un-1(y)|→0 as soon as xn→y. On the other hand, by Theorem 11 part (ii), for any y∈[a,b], it holds that |un-1(y)-u(y)|→0 as n→∞. Therefore, un-1(xn)→u(y) in the sense of ∥·∥W25 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 16.
Forj≤n, one has
(22)Lun(xj)=Lu(xj)=F(xj,uj-1(xj),Tuj-1(xj),Suj-1(xj)).Proof.
The proof of(23)Lun(xj)=F(xj,uj-1(xj),Tuj-1(xj),Suj-1(xj))
will be obtained by induction as follows:
Ifj≤n, then(24)Lun(xj)=∑i=1nAiLψ-i(xj)=∑i=1nAi〈Lψ-i(x),φj(x)〉W21=∑i=1nAi〈ψ-i(x),Lj*φ(x)〉W25=∑i=1nAi〈ψ-i(x),ψj(x)〉W25.
Using the orthogonality of{ψ-i(x)}i=1∞ yields that
(25)∑l=1jβjlLun(xl)=∑i=1nAi〈ψ-i(x),∑l=1jβjlψl(x)〉W25=∑i=1nAi〈ψ-i(x),ψ-j(x)〉W25=Aj=∑l=1jβjlF(xl,ul-1(xl),Tul-1(xl),Sul-1(xl)).
Now, ifj=1, then(26)Lun(x1)=F(x1,u0(x1),Tu0(x1),Su0(x1)).
Again, ifj=2, then
(27)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 by using mathematical induction concept that Lun(xj)=F(xj,uj-1(xj),Tuj-1(xj),Suj-1(xj)).
On the other hand, from Theorem11, un(x) converge uniformly to u(x). It follows that on taking limits in (18) u(x)=∑i=1∞Biψ-i(x). Therefore, un(x)=Pnu(x), where Pn is an orthogonal projector from W25[a,b] to Span {ψ1,ψ2,…,ψn}. Thus, Lun(xj)=〈Lun(x),φj(x)〉W21=〈un(x),Lj*φ(x)〉W25 = 〈Pnu(x),ψj(x)〉W25=〈u(x),Pnψj(x)〉W25=〈u(x),ψj(x)〉W25=〈Lu(x),φj(x)〉W21=Lu(xj).Theorem 17.
If{xi}i=1∞ is dense on [a,b] and ||un||W25 is bounded, then the n-term approximate solution un(x) in the iterative formula (18) converges to the exact solution u(x) of (9) in the space W25[a,b] and u(x)=∑i=1∞Aiψ-i(x), where Ai are given by (19).Proof.
The proof consists of the following three steps. Firstly, we will prove that the sequence{un}n=1∞ in (18) is monotone increasing in the sense of ∥·∥W25. By Theorem 9, {ψ¯i}i=1∞ is the complete orthonormal system in W25[a,b]. Hence, we have ∥un∥W252=〈un(x),un(x)〉W25 = 〈∑i=1nAiψ-i(x),∑i=1nAiψ-i(x)〉W25=∑i=1n(Ai)2. Therefore, ∥un∥W25 is monotone increasing.
Secondly, we will prove the convergence ofun(x). From (18), we have un+1(x)=un(x)+An+1ψ-n+1(x). From the orthogonality of {ψ-i(x)}i=1∞, it follows that ∥un+1∥W252=∥un∥W252+(An+1)2=∥un-1∥W252+(An)2+(An+1)2=⋯=∥u0∥W252+∑i=1n+1(Ai)2. Since, the sequence ∥un∥W25 is monotone increasing. Due to the condition that ∥un∥W25 is bounded, ∥un∥W25 is convergent as n→∞. Then, there exists a constant c such that ∑i=1∞(Ai)2=c. It implies that Ai=∑k=1iβikF(xk,uk-1(xk),Tuk-1(xk),Suk-1(xk))∈l2,i=1,2,…. On the other hand, since (um-um-1)⊥(um-1-um-2)⊥⋯⊥(un+1-un), it follows for m>n that
(28)∥um(x)-un(x)∥W252=∥um(x)-um-1(x)+um-1(x)-⋯+un+1(x)-un(x)∥W252=∥um(x)-um-1(x)∥W252+⋯+∥un+1(x)-un(x)∥W252.
Furthermore, ∥um(x)-um-1(x)∥W252=(Am)2. Consequently, as n,m→∞, we have ∥um(x)-un(x)∥W252=∑i=n+1m(Ai)2→0. Considering the completeness of W25[a,b], there exists a u(x)∈W25[a,b] such that un(x)→u(x) as n→∞ in the sense of ∥·∥W25.
Thirdly, we will prove thatu(x) is the solution of (9). 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 Lemma 16, it is clear that Lu(xnj)=F(xnj,unj-1(xk),Tunj-1(xk),Sunj-1(xk)). Hence, let j→∞, by Lemma 15 and the continuity of F, we have Lu(x)=F(x,u(x),Tu(x),Su(x)). That is, u(x) satisfies (1). Also, since ψ-i(x)∈W25[a,b], clearly, u(x) satisfies the boundary conditions (2). In other words, u(x) is the solution of (1) and (2), where u(x)=∑i=1∞Aiψ-i(x) and Ai are given by (19). The proof is complete.It obvious that if we letu(x) denote the exact solution of (9), un(x) the approximate solution obtained by the RKHS method as given by (18), and rn(x) the difference between u(x) and un(x), where x∈[a,b], then ∥rn(x)∥W252=∥u(x)-un(x)∥W252=∥∑i=n+1∞Aiψ-i(x)∥W252=∑i=n+1∞(Ai)2 and ∥rn-1(x)∥W252=∑i=n∞(Ai)2 or ∥rn(x)∥W25≤∥rn-1(x)∥W25. In fact, this is just the proof of the following theorem.Theorem 18.
The difference functionrn(x) is monotone decreasing in the sense of the norm of W25[a,b].


## 5. Numerical Outcomes

In this section, we propose few numerical simulations implemented by Mathematica 7.0 software package for solving some specific examples of (1) and (2). However, we apply the techniques described in the previous sections to some linear and nonlinear test examples in order to demonstrate the efficiency, accuracy, and applicability of the proposed method. Results obtained by the method are compared with the analytical solution of each example by computing the exact and relative errors and are found to be in good agreement with each other.Example 19.
Consider the following linear Fredholm-Volterra IDE:(29)u(4)(x)+u(x)-∫01(xt)u(t)dt-∫0xx(t+1)u(t)dt=f(x),0≤x,t≤1,
subject to the boundary conditions
(30)u(0)=-2,u(1)=0,u′(0)=2,u′(1)=2,
where f(x)=-(2/3)x4+2x2+(7/3)x-2. The exact solution is u(x)=2x-2.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 un(x) is calculated by (13). The numerical results at some selected grid points for n=26 are given in Table 1.Table 1
Numerical results for Example19.
x
Exact solution
Approximate solution
Absolute error
Relative error
0.16
-
1.68
-
1.679999999248148
7.51852
×
1
0
-
10
4.47531
×
1
0
-
10
0.32
-
1.36
-
1.359999998030627
1.96937
×
1
0
-
9
1.44807
×
1
0
-
9
0.48
-
1.04
-
1.039999997409132
2.59087
×
1
0
-
9
2.49122
×
1
0
-
9
0.64
-
0.72
-
0.719999997792201
2.20780
×
1
0
-
9
3.06639
×
1
0
-
9
0.80
-
0.40
-
0.399999998934791
1.06521
×
1
0
-
9
2.66302
×
1
0
-
9
0.96
-
0.08
-
0.079999999937888
6.21123
×
1
0
-
11
7.76404
×
1
0
-
10Example 20.
Consider the following nonlinear Fredholm-Volterra IDE:(31)u(4)(x)+u(x)-14∫01(t)u3(t)dt+12∫0x(x)u2(t)dt=f(x),0≤x,t≤1,
subject to the boundary conditions
(32)u(0)=0,u(1)=1,u′(0)=0,u′(1)=2,
where f(x)=(1/10)x6+x2-(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 (20). The numerical results at some selected grid points for N=11 and n=3 are given in Table 2.Table 2
Numerical results for Example20.
x
Exact solution
Approximate solution
Absolute error
Relative error
0.16
0.0256
0.025600007512175
7.51217
×
1
0
-
9
2.93444
×
1
0
-
7
0.32
0.1024
0.102400019690182
1.96902
×
1
0
-
8
1.92287
×
1
0
-
7
0.48
0.2304
0.230400025906381
2.59064
×
1
0
-
8
1.12441
×
1
0
-
7
0.64
0.4096
0.409600022074531
2.20745
×
1
0
-
8
5.38929
×
1
0
-
8
0.80
0.6400
0.640000010646264
1.06463
×
1
0
-
8
1.66348
×
1
0
-
8
0.96
0.9216
0.921600000613291
6.13291
×
1
0
-
10
6.65463
×
1
0
-
10Example 21.
Consider the following nonlinear Fredholm-Volterra IDE:(33)u(4)(x)+u2(x)-∫01(t+1)sinh(u(t))dt-∫0x(x)eu(t)dt=f(x),0≤x,t≤1,
subject to the boundary conditions
(34)u(0)=0,u(1)=ln(2),u′(0)=1,u′(1)=12,
where f(x)=(ln(x+1))2-(6/(x+1)4)-(1/2)x2(x+2)-(2/3). The exact solution is u(x)=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 (20). The numerical results at some selected grid points for N=26 and n=5 are given in Table 3.Table 3
Numerical results for Example21.
x
Exact solution
Approximate solution
Absolute error
Relative error
0.16
0.148420005118273
0.148419864687877
1.40430
×
1
0
-
7
9.46169
×
1
0
-
7
0.32
0.277631736598280
0.277631416799765
3.19799
×
1
0
-
7
1.15188
×
1
0
-
6
0.48
0.392042087776024
0.392041716651615
3.71124
×
1
0
-
7
9.46644
×
1
0
-
7
0.64
0.494696241836107
0.494695959934014
2.81902
×
1
0
-
7
5.69849
×
1
0
-
7
0.80
0.587786664902119
0.587786542757544
1.22145
×
1
0
-
7
2.07804
×
1
0
-
7
0.96
0.672944473242426
0.672944466882463
6.35996
×
1
0
-
9
9.45095
×
1
0
-
9Example 22.
Consider the following nonlinear Fredholm-Volterra IDE:(35)u(4)(x)+cos(u(x))-∫01(t+x)u(t)dt-∫0xet-xeu(t)dt=f(x),0≤x,t≤1,
subject to the boundary conditions
(36)u(0)=1,u(1)=0,u′(0)=-1,u′(1)=-1,
where f(x)=cos(1-x)-xe1-x-(1/2)x-(1/6). 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 (20). The numerical results at some selected grid points for N=26 and n=5 are given in Table 4.Table 4
Numerical results for Example22.
x
Exact solution
Approximate solution
Absolute error
Relative error
0.16
0.84
0.840000000735357
7.35357
×
1
0
-
10
8.75425
×
1
0
-
10
0.32
0.68
0.680000002094899
2.09490
×
1
0
-
9
3.08073
×
1
0
-
9
0.48
0.52
0.520000002788682
2.78868
×
1
0
-
9
5.36285
×
1
0
-
9
0.64
0.36
0.360000002335676
2.33568
×
1
0
-
9
6.48799
×
1
0
-
9
0.80
0.20
0.200000001005203
1.00520
×
1
0
-
9
5.02601
×
1
0
-
9
0.96
0.04
0.039999999826935
1.73065
×
1
0
-
10
4.32663
×
1
0
-
9Example 23.
Consider the following nonlinear Fredholm-Volterra IDE:(37)u(4)(x)+u(x)-∫01cos-1(u(t))dt-∫0x(x+t)(1-u2(t))dt=f(x),0≤x,t≤1,
subject to the boundary conditions
(38)u(0)=1,u(1)=cos(1),u′(0)=0,u′(1)=-sin(1),
where f(x)=2cos(x)+(1/2)xsin(2x)-(1/4)sin2(x)-(3/4)x2-(1/2). The exact solution is u(x)=cos(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 (20). The numerical results at some selected grid points for N=26 and n=3 are given in Table 5.Table 5
Numerical results for Example23.
x
Exact solution
Approximate solution
Absolute error
Relative error
0.16
0.987227283375627
0.987227193344231
9.00314
×
1
0
-
8
9.11962
×
1
0
-
8
0.32
0.949235418082441
0.949235185492173
2.32590
×
1
0
-
7
2.45029
×
1
0
-
7
0.48
0.886994922779284
0.886994621805469
3.00974
×
1
0
-
7
3.39319
×
1
0
-
7
0.64
0.802095757884293
0.802095506191996
2.51692
×
1
0
-
7
3.13793
×
1
0
-
7
0.80
0.696706709347165
0.696706590465593
1.18882
×
1
0
-
7
1.70634
×
1
0
-
7
0.96
0.573519986072450
0.573519979379673
6.69278
×
1
0
-
9
1.16697
×
1
0
-
8As we mention earlier, it is possible to pick any point in[a,b], and the approximate solutions and all its derivative up to order four will be applicable as well. Next, new numerical results for Example 23 which include the absolute error at some selected nodes in [0,1] for u(m)(x), m=0,1,2,3,4, where xi=(i-1)/(N-1), i=1,2,…,N in which N=26 and n=5 are given in Table 6.Table 6
Absolute error ofu(m)(x) for Example 23.
m
x
=
0.16
x
=
0.48
x
=
0.64
x
=
0.96
0
9.00314
×
1
0
-
8
3.00974
×
1
0
-
7
2.51692
×
1
0
-
7
6.69278
×
1
0
-
9
1
2.31757
×
1
0
-
7
9.03440
×
1
0
-
7
6.42840
×
1
0
-
7
3.21697
×
1
0
-
8
2
1.77287
×
1
0
-
6
4.84667
×
1
0
-
6
3.48865
×
1
0
-
6
7.08367
×
1
0
-
7
3
5.13644
×
1
0
-
6
2.04982
×
1
0
-
5
1.74082
×
1
0
-
5
4.70439
×
1
0
-
6
4
1.00313
×
1
0
-
5
3.00974
×
1
0
-
5
2.51692
×
1
0
-
5
6.69283
×
1
0
-
6


## 6. Conclusions

The main concern of this work has been to propose an efficient algorithm for the solutions of two-point BVPs for fourth-order Fredholm-Volterra IDEs (1) and (2). The main goal has been achieved by introducing the RKHS method to solve this class of IDEs. We can conclude that the RKHS method is powerful and efficient technique in finding approximate solution for linear and nonlinear problems. In the proposed algorithm, the solution u(x) and the approximate solution un(x) are represented in the form of series in W25[a,b]. Moreover, the approximate solution and all its derivatives converge uniformly to the exact solution and all its derivatives up to order four, respectively. There is an important point to make here; the results obtained by the RKHS method are very effective and convenient in linear and nonlinear cases with less computational, iteration steps, work, and time. This confirms our belief that the efficiency of our technique gives it much wider applicability in the future for general classes of linear and nonlinear problems.



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*Source: 832074-2013-04-22.xml*