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

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 space 𝑊⁵₂[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 𝑊⁵₂[𝑎,𝑏]. The proposed technique is applied to several examples to illustrate the accuracy, efficiency, and applicability of the method.

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: 𝑢⁽⁴⁾(𝑥)=𝑓(𝑥)+𝑁(𝑥,𝑢(𝑥))+^𝑏_𝑎ℎ₁(𝑥,𝑡)𝐺₁(𝑢(𝑡+))𝑑𝑡_𝑥𝑎ℎ₂(𝑥,𝑡)𝐺₂(𝑢(𝑡))𝑑𝑡,𝑎≤𝑡,𝑥≤𝑏, (1) subject to the boundary conditions 𝑢(𝑎)=𝛼₀,𝑢^′(𝑎)=𝛼₁,𝑢(𝑏)=𝛽₀,𝑢^′(𝑏)=𝛽₁, (2) where 𝛼_𝑖, 𝛽_𝑖, 𝑖=0,1 are real finite constants, 𝑢∈𝑊⁵₂[𝑎,𝑏] is an unknown function to be determined, 𝑓∈𝑊¹₂[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 𝑊¹₂[𝑎,𝑏] as 𝑤=𝑤(𝑥), 𝑦=𝑦(𝑥), 𝑧=𝑧(𝑥)∈𝑊⁵₂[𝑎,𝑏], 𝑎≤𝑥≤𝑏, −∞<𝑤, 𝑦,𝑧<∞ and are depending on the problem discussed, and 𝑊¹₂[𝑎,𝑏], 𝑊⁵₂[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 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) 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 𝑊⁵₂[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∣∫^𝑏_𝑎𝑢²(𝑥)𝑑𝑥<∞}, and 𝑙²={𝐴∣∑^∞_𝑖=1(𝐴_𝑖)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]). Let 𝐸 be a nonempty abstract set. A function 𝐾∶𝐸×𝐸→ℂ is a reproducing kernel of the Hilbert space 𝐻 if(1) for each 𝑥∈𝐸, 𝐾(⋅,𝑥)∈𝐻;(2) for each 𝑥∈𝐸 and 𝜑∈𝐻, ⟨𝜑(⋅),𝐾(⋅,𝑥)⟩=𝜑(𝑥).

The second condition in Definition 1 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 space 𝑊⁵₂[𝑎,𝑏] in which every function satisfies the boundary conditions (2) and then utilize the space 𝑊¹₂[𝑎,𝑏].

Definition 2 (see [21]). 𝑊⁵₂[𝑎,𝑏]={𝑢(𝑥)∣𝑢^(𝑖),  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 𝑊⁵₂[𝑎,𝑏] are defined as ⟨𝑢,𝑣⟩_𝑊⁵₂=²_𝑖=0𝑢^(𝑖)(𝑎)𝑣^(𝑖)(𝑎)+¹_𝑖=0𝑢^(𝑖)(𝑏)𝑣^(𝑖)(𝑏)+^𝑏_𝑎𝑢⁽⁵⁾𝑦𝑣⁽⁵⁾𝑦𝑑𝑦 (3) and ‖𝑢‖_𝑊⁵₂=⟨𝑢,𝑢⟩_𝑊⁵₂, respectively, where 𝑢,𝑣∈𝑊⁵₂[𝑎,𝑏].

Definition 3 (see [19]). 𝑊¹₂[𝑎,𝑏]={𝑢(𝑥)∣𝑢 is absolutely continuous real-valued function on [𝑎,𝑏] and 𝑢^′∈𝐿²[𝑎,𝑏]}. The inner product and the norm in 𝑊¹₂[𝑎,𝑏] are defined as ⟨𝑢,𝑣⟩_𝑊¹₂=∫^𝑏_𝑎(𝑢^′𝑣^′+𝑢𝑣)𝑑𝑦 and ‖𝑢‖_𝑊¹₂=⟨𝑢,𝑢⟩_𝑊¹₂, respectively, where 𝑢,𝑣∈𝑊¹₂[𝑎,𝑏].

Remark 4. In [19], the authors have proved that the space 𝑊¹₂[𝑎,𝑏] is a complete reproducing kernel space and its reproducing kernel is 𝑅_𝑥𝑦=1().2sinh𝑏−𝑎cosh𝑥+𝑦−𝑏−𝑎+cosh𝑥+𝑦−𝑏+𝑎 (4)
The Hilbert space 𝑊⁵₂[𝑎,𝑏] is called a reproducing kernel if for each fixed 𝑥∈[𝑎,𝑏] and any 𝑢(𝑦)∈𝑊⁵₂[𝑎,𝑏], there exist 𝐾(𝑥,𝑦)∈𝑊⁵₂[𝑎,𝑏] (simply 𝐾_𝑥(𝑦)) and 𝑦∈[𝑎,𝑏] such that ⟨𝑢(𝑦),𝐾_𝑥(𝑦)⟩_𝑊52=𝑢(𝑥). The next theorem formulates the reproducing kernel function 𝐾_𝑥(𝑦) on the space 𝑊⁵₂[𝑎,𝑏].

Theorem 5. The Hilbert space 𝑊⁵₂[𝑎,𝑏] is a reproducing kernel and its reproducing kernel function 𝐾_𝑥(𝑦) is given by𝐾_𝑥𝑦=⁹_𝑖∑₌₀𝑝_𝑖(𝑥)𝑦^𝑖,𝑦≤𝑥,_9𝑖∑₌₀𝑞_𝑖(𝑥)𝑦^𝑖,𝑦>𝑥, (5) where 𝑝_𝑖(𝑥) and 𝑞_𝑖(𝑥) are unknown coefficients of 𝐾_𝑥(𝑦).

Proof. Through several integrations by parts for (3), one can obtain ⟨𝑢(𝑦),𝐾_𝑥(𝑦)⟩_𝑊52=∑²_𝑖=0𝑢^(𝑖)(𝑎)𝐾_𝑥^(𝑖)(𝑎)+∑¹_𝑖=0𝑢^(𝑖)(𝑏)𝐾_𝑥^(𝑖)(𝑏)+∑⁴_𝑖=0(−1)^4−𝑖𝑢^(𝑖)(𝑦)𝐾_𝑥^(9−𝑖)(𝑦)|^𝑦=𝑏_𝑦=𝑎−∫^𝑏_𝑎𝑢(𝑦)𝐾_𝑥⁽¹⁰⁾(𝑦)𝑑𝑦. Since 𝐾_𝑥(𝑦)∈𝑊⁵₂[𝑎,𝑏], it follows that 𝐾_𝑥^(𝑖)(𝑎)=𝐾_𝑥^(𝑖)(𝑏)=0, 𝑖=0,1. Again, since 𝑢∈𝑊⁵₂[𝑎,𝑏], one obtains 𝑢^(𝑖)(𝑎)=𝑢^(𝑖)(𝑏)=0, 𝑖=0,1. Thus, if 𝐾_𝑥⁽²⁾(𝑎)−𝐾_𝑥⁽⁷⁾(𝑎)=0, 𝐾_𝑥^(𝑖)(𝑎)=0, 𝑖=5,6, and 𝐾_𝑥^(𝑖)(𝑏)=0, 𝑖=5,6,7, then ⟨𝑢(𝑦),𝐾_𝑥(𝑦)⟩_𝑊52=∫^𝑏_𝑎𝑢(𝑦)(−𝐾_𝑥⁽¹⁰⁾(𝑦))𝑑𝑦. Now, for each 𝑥∈[𝑎,𝑏], if 𝐾_𝑥(𝑦) also satisfies the formula −𝐾_𝑥⁽¹⁰⁾(𝑦)=𝛿(𝑥−𝑦), where 𝛿 is the dirac-delta function, then ⟨𝑢(𝑦),𝐾_𝑥(𝑦)⟩_𝑊52=𝑢(𝑥). Obviously, 𝐾_𝑥(𝑦) is the reproducing kernel of the space 𝑊⁵₂[𝑎,𝑏].
Next, we give the expression of the reproducing kernel function 𝐾_𝑥(𝑦). The characteristic equation of −𝐾_𝑥⁽¹⁰⁾(𝑦)=𝛿(𝑥−𝑦) is 𝜆¹⁰=0, and their characteristic values are 𝜆=0 with 10 multiple roots. So, let the kernel 𝐾_𝑥(𝑦) be as defined in (5).
On the other hand, let 𝐾_𝑥(𝑦) satisfy 𝐾_𝑥^(𝑚)(𝑥+0)=𝐾_𝑥^(𝑚)(𝑥−0), 𝑚=0,1,…,8. Integrating −𝐾_𝑥⁽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 𝐾_𝑥⁽⁹⁾(𝑦) at 𝑦=𝑥 given by 𝐾_𝑥⁽⁹⁾(𝑥−0)−𝐾_𝑥⁽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 rules 𝐾_1𝑥(𝑦)=∑⁹_𝑖=0𝑝_𝑖(𝑥)𝑦^𝑖 and 𝐾_2𝑥(𝑦)=∑⁹_𝑖=0𝑞_𝑖(𝑥)𝑦^𝑖 of the reproducing kernel function 𝐾_𝑥(𝑦) in (5) are obtained at 𝑎=0 and 𝑏=1 in (1) and (2) and are given, respectively, as

𝐾_1𝑥𝑦=1()362880𝑥−1²𝑦²×5𝑥³8−15𝑦𝑦+𝑥⁶19−12𝑦𝑦+𝑦⁷+𝑥𝑦⁶−9+2𝑦+3𝑥⁵𝑦−10+3𝑦+𝑥⁷𝑦−4+3𝑦+5𝑥⁴𝑦1+6𝑦+3𝑥²30240−60480𝑦+30240𝑦²+12𝑦⁵−6𝑦⁶+𝑦⁷,𝐾_2𝑥𝑦=1𝑥362880²𝑦−1²×90720𝑦²+36𝑥⁵𝑦²−9𝑥⁶𝑦1+2𝑦+𝑥⁷1+2𝑦+3𝑦²+𝑥𝑦²−181440+40𝑦+5𝑦²−30𝑦³+19𝑦⁴−4𝑦⁵+3𝑥²𝑦²30240−25𝑦+10𝑦²+3𝑦³−4𝑦⁴+𝑦⁵. (6)

The following corollary summarizes some important properties of the reproducing kernel function 𝐾_𝑥(𝑦).

Corollary 7. The reproducing kernel function 𝐾_𝑥(𝑦) is symmetric and unique, and 𝐾_𝑥(𝑥)≥0 for any fixed 𝑥∈[𝑎,𝑏].

Proof. By the reproducing property, we have 𝐾_𝑥(𝑦)=⟨𝐾_𝑥(𝜉),𝐾_𝑦(𝜉)⟩=⟨𝐾_𝑦(𝜉),𝐾_𝑥(𝜉)⟩=𝐾_𝑦(𝑥) for each 𝑥 and 𝑦. Now, let 𝐾¹_𝑥(𝑦) and 𝐾²_𝑥(𝑦) be all the reproducing kernels of the space 𝑊⁵₂[𝑎,𝑏]; then, 𝐾¹_𝑥(𝑦)=⟨𝐾¹_𝑥(𝜉),𝐾²_𝑦(𝜉)⟩=⟨𝐾²_𝑦(𝜉),𝐾¹_𝑥(𝜉)⟩=𝐾²_𝑦(𝑥)=𝐾²_𝑥(𝑦). Finally, we note that 𝐾_𝑥(𝑥)=⟨𝐾_𝑥(𝜉),𝐾_𝑥(𝜉)⟩=‖𝐾_𝑥(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 𝑊⁵₂[𝑎,𝑏]. After that, we construct an orthogonal function system of 𝑊⁵₂[𝑎,𝑏] based on the Gram-Schmidt orthogonalization process.

To do this, we define a differential operator 𝐿 as 𝐿∶𝑊⁵₂[𝑎,𝑏]⟶𝑊¹₂[𝑎,𝑏], (7) such that 𝐿𝑢(𝑥)=𝑢⁽⁴⁾(𝑥). (8) After homogenization of the boundary conditions (2), (1) and (2) can be converted into the equivalent form as follows: (𝑥)((𝑥)(𝑥)(𝑥𝑢(𝑎)𝐿𝑢=𝐹𝑥,𝑢,𝑇𝑢,𝑆𝑢)),𝑎≤𝑥≤𝑏,=0,𝑢^′(𝑎)𝑢(𝑏)=0,=0,𝑢^′(𝑏)=0, (9) such that 𝐹(𝑥,𝑢(𝑥),𝑇𝑢(𝑥),𝑆𝑢(𝑥))=𝑓(𝑥)+𝑁(𝑥,𝑢(𝑥))+𝑇𝑢(𝑥)+𝑆𝑢(𝑥), 𝑇𝑢(𝑥)=∫^𝑏_𝑎ℎ₁(𝑥,𝑡)𝐺₁(𝑢(𝑡))𝑑𝑡, and 𝑆𝑢(𝑥)=∫^𝑥_𝑎ℎ₂(𝑥,𝑡)𝐺₂(𝑢(𝑡))𝑑𝑡, where 𝑢(𝑥)∈𝑊⁵₂[𝑎,𝑏] and 𝐹(𝑥,𝑤,𝑦,𝑧)∈𝑊¹₂[𝑎,𝑏] for 𝑤=𝑤(𝑥), 𝑦=𝑦(𝑥), 𝑧=𝑧(𝑥)∈𝑊⁵₂[𝑎,𝑏],  𝑎≤𝑥≤𝑏,  −∞<𝑤,𝑦,𝑧<∞. It is easy to show that 𝐿 is a bounded linear operator from 𝑊⁵₂[𝑎,𝑏] into 𝑊¹₂[𝑎,𝑏].

Next, we construct an orthogonal function system of 𝑊⁵₂[𝑎,𝑏]. Put 𝜑_𝑖(𝑥)=𝑅_𝑥𝑖(𝑥) and 𝜓_𝑖(𝑥)=𝐿^∗_𝑖𝜑(𝑥), where {𝑥_𝑖}^∞_𝑖=1 is dense on [𝑎,𝑏] and 𝐿^∗ is the adjoint operator of 𝐿. In terms of the properties of reproducing kernel function 𝑅_𝑥(𝑦), one obtains ⟨𝑢(𝑥),𝜓_𝑖(𝑥)⟩_𝑊52=⟨𝑢(𝑥),𝐿^∗𝜑_𝑖(𝑥)⟩_𝑊52=⟨𝐿𝑢(𝑥),𝜑_𝑖(𝑥)⟩_𝑊12=𝐿𝑢(𝑥_𝑖), 𝑖=1,2,….

Remark 8. The orthonormal function system {𝜓_𝑖(𝑥)}^∞_𝑖=1 of the space 𝑊⁵₂[𝑎,𝑏] can be derived from Gram-Schmidt orthogonalization process of {𝜓_𝑖(𝑥)}^∞_𝑖=1 as follows: 𝜓_𝑖(𝑥)=^𝑖_𝑘=1𝛽_𝑖𝑘𝜓_𝑘(𝑥), (10) where 𝛽_𝑖𝑘 are orthogonalization coefficients and are given by the following subroutine: 𝛽_𝑖𝑗=1𝜓₁𝛽,for𝑖=𝑗=1,_𝑖𝑗=1𝑑_𝑖𝑘𝛽,for𝑖=𝑗≠1,_𝑖𝑗1=−𝑑_𝑖𝑘^𝑖−1_𝑘=𝑗𝑐_𝑖𝑘𝛽_𝑘𝑗,for𝑖>𝑗, (11) such that 𝑑_𝑖𝑘=‖𝜓_𝑖‖²−∑^𝑖−1_𝑘=1𝑐²_𝑖𝑘, 𝑐_𝑖𝑘=⟨𝜓_𝑖,𝜓_𝑘⟩_𝑊52, and {𝜓_𝑖(𝑥)}^∞_𝑖=1 is the orthonormal system in the space 𝑊⁵₂[𝑎,𝑏].

Through the next theorem, the subscript 𝑦 by the operator 𝐿 (𝐿_𝑦) indicates that the operator 𝐿 applies to the function of 𝑦.

Theorem 9. If {𝑥_𝑖}^∞_𝑖=1 is dense on [𝑎,𝑏], then {𝜓_𝑖(𝑥)}^∞_𝑖=1 is a complete function system of 𝑊⁵₂[𝑎,𝑏] and 𝜓_𝑖(𝑥)=𝐿_𝑦𝐾_𝑥(𝑦)|_{𝑦=𝑥𝑖}.

Proof. Clearly, 𝜓_𝑖(𝑥)=𝐿^∗_𝑖𝜑(𝑥)=⟨𝐿^∗_𝑖𝜑(𝑦),𝐾_𝑥(𝑦)⟩_𝑊52=⟨𝜑_𝑖(𝑦),𝐿_𝑦𝐾_𝑥(𝑦)⟩_𝑊12=𝐿_𝑦𝐾_𝑥(𝑦)|_{𝑦=𝑥𝑖}∈𝑊⁵₂[𝑎,𝑏]. Now, for each fixed 𝑢(𝑥)∈𝑊⁵₂[𝑎,𝑏], let ⟨𝑢(𝑥),𝜓_𝑖(𝑥)⟩_𝑊52=0, 𝑖=1,2,…. In other word, ⟨𝑢(𝑥),𝜓_𝑖(𝑥)⟩_𝑊52=⟨𝑢(𝑥),𝐿^∗𝜑_𝑖(𝑥)⟩_𝑊52=⟨𝐿𝑢(𝑥),𝜑_𝑖(𝑥)⟩_𝑊12=𝐿𝑢(𝑥_𝑖)=0. Note that {𝑥_𝑖}^∞_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. If 𝑢(𝑥)∈𝑊⁵₂[𝑎,𝑏], then there exists 𝑀>0 such that ‖𝑢^(𝑖)(𝑥)‖_𝐶≤𝑀‖𝑢(𝑥)‖_𝑊⁵₂, 𝑖=0,1,2,3,4, where ‖𝑢(𝑥)‖_𝐶=max_{𝑎≤𝑥≤𝑏}|𝑢(𝑥)|.

Proof. For any 𝑥,𝑦∈[𝑎,𝑏], we have 𝑢^(𝑖)(𝑥)=⟨𝑢(𝑦),𝐾_𝑥^(𝑖)(𝑦)⟩_𝑊52, 𝑖=0,1,2,3,4. By the expression of 𝐾_𝑥(𝑦), it follows that ‖𝐾_𝑥^(𝑖)(𝑦)‖_𝑊52≤𝑀_𝑖, 𝑖=0,1,2,3,4. Thus, |𝑢^(𝑖)(𝑥)|=|⟨𝑢(𝑥),𝐾_𝑥^(𝑖)(𝑥)⟩_𝑊52|≤‖𝐾_𝑥^(𝑖)(𝑥)‖_𝑊52‖𝑢(𝑥)‖_𝑊⁵₂≤𝑀_𝑖‖𝑢(𝑥)‖_𝑊⁵₂, 𝑖=0,1,2,3,4. Hence, ||𝑢^(𝑖)(𝑥)||_𝐶≤𝑀||𝑢(𝑥)||_𝑊⁵₂, 𝑖=0,1,2,3,4, where 𝑀=max_{𝑖=0,1,2,3,4}{𝑀_𝑖}.

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 𝑊⁵₂[𝑎,𝑏]. After that, the convergence of approximate solution 𝑢_𝑛(𝑥) to the analytic solution will be proved.

Theorem 11. For each 𝑢(𝑥) in the space 𝑊⁵₂[𝑎,𝑏], the series ∑^∞_𝑖=1⟨𝑢(𝑥),𝜓_𝑖(𝑥)⟩𝜓_𝑖(𝑥) is convergent in the sense of the norm of 𝑊⁵₂[𝑎,𝑏]. On the other hand, if {𝑥_𝑖}^∞_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 𝑢(𝑥)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝐹𝑥_𝑘,𝑢𝑥_𝑘,𝑇𝑢𝑥_𝑘,𝑆𝑢𝑥_𝑘𝜓_𝑖(𝑥); (12)(ii) the approximate solution of (9) 𝑢_𝑛(𝑥)=^𝑛𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝐹𝑥_𝑘,𝑢𝑥_𝑘,𝑇𝑢𝑥_𝑘,𝑆𝑢𝑥_𝑘𝜓_𝑖(𝑥), (13) and 𝑢_𝑛^(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, let 𝑢(𝑥) be solution of (9) in the space 𝑊⁵₂[𝑎,𝑏]. Since 𝑢(𝑥) belong to Hilbert space 𝑊⁵₂[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 ‖⋅‖_𝑊⁵₂. On the other hand, using (10), we have 𝑢(𝑥)=^∞_𝑖=1𝑢(𝑥),𝜓_𝑖(𝑥)_𝑊52𝜓_𝑖(𝑥)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑢(𝑥),𝜓_𝑘(𝑥)_𝑊52𝜓_𝑖(𝑥)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑢(𝑥),𝐿^∗𝜑_𝑘(𝑥)_𝑊52𝜓_𝑖(𝑥)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝐿𝑢(𝑥),𝜑_𝑘(𝑥)_𝑊12𝜓_𝑖(𝑥)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝐹(𝑥,𝑢(𝑥),𝑇𝑢(𝑥),𝑆𝑢(𝑥)),𝜑_𝑘(𝑥)_𝑊12𝜓_𝑖(𝑥)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝐹𝑥_𝑘,𝑢𝑥_𝑘,𝑇𝑢𝑥_𝑘,𝑆𝑢𝑥_𝑘𝜓_𝑖(𝑥). (14) Therefore, the form of (12) is the exact solution of (9).
For the second part, it easy to see that by Lemma 10, for any 𝑥∈[𝑎,𝑏], 𝑢_𝑛(𝑥)−𝑢(𝑥)=𝑢_𝑛(𝑥)−𝑢(𝑥),𝐾_𝑥(𝑥)_𝑊52≤𝐾_𝑥(𝑥)_𝑊52𝑢_𝑛(𝑥)−𝑢(𝑥)_𝑊52≤𝑀₀𝑢_𝑛(𝑥)−𝑢(𝑥)_𝑊52. (15) On the other hand, 𝑢_𝑛^(𝑖)(𝑥)−𝑢^(𝑖)(𝑥)=𝑢_𝑛(𝑥)−𝑢(𝑥),𝐾_𝑥^(𝑖)(𝑥)_𝑊52≤𝐾_𝑥^(𝑖)(𝑥)_𝑊52𝑢_𝑛(𝑥)−𝑢(𝑥)_𝑊52≤𝑀_𝑖𝑢_𝑛(𝑥)−𝑢(𝑥)_𝑊52,𝑖=1,2,3,4, (16) where 𝑀_𝑖, 𝑖=0,1,2,3,4 are positive constants. Hence, if ‖𝑢_𝑛(𝑥)−𝑢(𝑥)‖_𝑊52→0 as 𝑛→∞, the approximate solution 𝑢_𝑛(𝑥) and 𝑢_𝑛^(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 solution 𝑢_𝑛(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 𝑊⁵₂[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: 𝑢(𝑥)=^∞_𝑖=1𝐵_𝑖𝜓_𝑖(𝑥), (17) where 𝐵_𝑖=∑^𝑖_𝑘=1𝛽_𝑖𝑘𝐹(𝑥_𝑘,𝑢_𝑘−1(𝑥_𝑘),𝑇𝑢_𝑘−1(𝑥_𝑘),𝑆𝑢_𝑘−1(𝑥_𝑘)). In fact, 𝐵_𝑖, 𝑖=1,2,… in (17) are unknown; we will approximate 𝐵_𝑖 using known 𝐴_𝑖. 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 𝑢_𝑛(𝑥)=^𝑛_𝑖=1𝐴_𝑖𝜓_𝑖(𝑥), (18) where the coefficients 𝐴_𝑖 of 𝜓_𝑖(𝑥), 𝑖=1,2,…,𝑛 are given as follows: 𝐴₁=𝛽₁₁𝐹𝑥₁,𝑢₀𝑥₁,𝑇𝑢₀𝑥₁,𝑆𝑢₀𝑥₁,𝑢₁(𝑥)=𝐴₁𝜓₁(𝑥),𝐴₂=²_𝑘=1𝛽_2𝑘𝐹𝑥_𝑘,𝑢_𝑘−1𝑥_𝑘,𝑇𝑢_𝑘−1𝑥_𝑘,𝑆𝑢_𝑘−1𝑥_𝑘,𝑢₂(𝑥)=²_𝑖=1𝐴_𝑖𝜓_𝑖(𝑥),𝑢⋅⋅⋅_𝑛−1(𝑥)=^𝑛−1_𝑖=1𝐴_𝑖𝜓_𝑖(𝑥),𝐴_𝑛=^𝑛_𝑘=1𝛽_𝑛𝑘𝐹𝑥_𝑘,𝑢_𝑘−1𝑥_𝑘,𝑇𝑢_𝑘−1𝑥_𝑘,𝑆𝑢_𝑘−1𝑥_𝑘. (19)
Here, note that in the iterative process of (18), we can guarantee that the approximation 𝑢_𝑛(𝑥) satisfies the boundary conditions (2). Now, the approximate solution 𝑢^𝑁_𝑛(𝑥) can be obtained by taking finitely many terms in the series representation of 𝑢_𝑛(𝑥) and 𝑢^𝑁_𝑛(𝑥)=^𝑁𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝐹𝑥_𝑘,𝑢_𝑛−1𝑥_𝑘,𝑇𝑢_𝑛−1𝑥_𝑘,𝑆𝑢_𝑛−1𝑥_𝑘𝜓_𝑖(𝑥). (20)
Next, we will prove that 𝑢_𝑛(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. If 𝑢_𝑛(𝑥)→𝑢(𝑥) in the sense of the norm of 𝑊⁵₂[𝑎,𝑏], 𝑥_𝑛→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(𝑥_𝑛,𝑢_𝑛−1(𝑥_𝑛),𝑇𝑢_𝑛−1(𝑥_𝑛),𝑆𝑢_𝑛−1(𝑥_𝑛))→𝐹(𝑦,𝑢(𝑦),𝑇𝑢(𝑦),𝑆𝑢(𝑦)) as 𝑛→∞.

Proof. Firstly, we will prove that 𝑢_𝑛−1(𝑥_𝑛)→𝑢(𝑦) in the sense of ‖⋅‖_𝑊⁵₂, since 𝑢_𝑛−1𝑥_𝑛𝑦=𝑢−𝑢_𝑛−1𝑥_𝑛−𝑢_𝑛−1𝑦+𝑢_𝑛−1𝑦𝑦≤𝑢−𝑢_𝑛−1𝑥_𝑛−𝑢_𝑛−1𝑦+𝑢_𝑛−1𝑦−𝑢𝑦. (21) By reproducing property of 𝐾_𝑥(𝑦), we have 𝑢_𝑛−1(𝑥_𝑛)=⟨𝑢_𝑛−1(𝑥),𝐾_𝑥𝑛(𝑥)⟩ and 𝑢_𝑛−1(𝑦)=⟨𝑢_𝑛−1(𝑥),𝐾_𝑦(𝑥)⟩. Thus, |𝑢_𝑛−1(𝑥_𝑛)−𝑢_𝑛−1(𝑦)|=|⟨𝑢_𝑛−1(𝑥),𝐾_𝑥𝑛(𝑥)−𝐾_𝑦(𝑥)⟩_𝑊52|≤‖𝑢_𝑛−1(𝑥)‖_𝑊52‖𝐾_𝑥𝑛(𝑥)−𝐾_𝑦(𝑥)‖_𝑊52. From the symmetry of 𝐾_𝑥(𝑦), it follows that ‖𝐾_𝑥𝑛(𝑥)−𝐾_𝑦(𝑥)‖_𝑊52→0 as 𝑛→∞. Hence, |𝑢_𝑛−1(𝑥_𝑛)−𝑢_𝑛−1(𝑦)|→0 as soon as 𝑥_𝑛→𝑦. On the other hand, by Theorem 11 part (ii), for any 𝑦∈[𝑎,𝑏], it holds that |𝑢_𝑛−1(𝑦)−𝑢(𝑦)|→0 as 𝑛→∞. Therefore, 𝑢_𝑛−1(𝑥_𝑛)→𝑢(𝑦) in the sense of ‖⋅‖_𝑊⁵₂ as 𝑥_𝑛→𝑦 and 𝑛→∞.
Thus, by means of the continuation of 𝐺₁, 𝐺₂, and 𝑁, it is obtained that 𝐺₁(𝑢_𝑛−1(𝑥_𝑛))→𝐺₁(𝑢(𝑦)), 𝐺₂(𝑢_𝑛−1(𝑥_𝑛))→𝐺₂(𝑢(𝑦)), and 𝑁(𝑥_𝑛,𝑢_𝑛−1(𝑥_𝑛))→𝑁(𝑦,𝑢(𝑦)) as 𝑛→∞. This shows that 𝑇𝑢_𝑛−1(𝑥_𝑛)→𝑇𝑢(𝑦) and 𝑆𝑢_𝑛−1(𝑥_𝑛)→𝑆𝑢(𝑦) as 𝑛→∞. Hence, the continuity of 𝐹 gives the result.

Lemma 16. For 𝑗≤𝑛, one has 𝐿𝑢_𝑛𝑥_𝑗=𝐿𝑢𝑥_𝑗=𝐹𝑥_𝑗,𝑢_𝑗−1𝑥_𝑗,𝑇𝑢_𝑗−1𝑥_𝑗,𝑆𝑢_𝑗−1𝑥_𝑗. (22)

Proof. The proof of 𝐿𝑢_𝑛𝑥_𝑗=𝐹𝑥_𝑗,𝑢_𝑗−1𝑥_𝑗,𝑇𝑢_𝑗−1𝑥_𝑗,𝑆𝑢_𝑗−1𝑥_𝑗 (23)
will be obtained by induction as follows:
If 𝑗≤𝑛, then𝐿𝑢_𝑛𝑥_𝑗=^𝑛_𝑖=1𝐴_𝑖𝐿𝜓_𝑖𝑥_𝑗=^𝑛_𝑖=1𝐴_𝑖𝐿𝜓_𝑖(𝑥),𝜑_𝑗(𝑥)_𝑊12=^𝑛_𝑖=1𝐴_𝑖𝜓_𝑖(𝑥),𝐿^∗_𝑗𝜑(𝑥)_𝑊52=^𝑛_𝑖=1𝐴_𝑖𝜓_𝑖(𝑥),𝜓_𝑗(𝑥)_𝑊52. (24)
Using the orthogonality of {𝜓_𝑖(𝑥)}^∞_𝑖=1 yields that ^𝑗_𝑙=1𝛽_𝑗𝑙𝐿𝑢_𝑛𝑥_𝑙=^𝑛_𝑖=1𝐴_𝑖𝜓_𝑖(𝑥),^𝑗_𝑙=1𝛽_𝑗𝑙𝜓_𝑙(𝑥)_𝑊52=^𝑛_𝑖=1𝐴_𝑖𝜓_𝑖(𝑥),𝜓_𝑗(𝑥)_𝑊52=𝐴_𝑗=^𝑗_𝑙=1𝛽_𝑗𝑙𝐹𝑥_𝑙,𝑢_𝑙−1𝑥_𝑙,𝑇𝑢_𝑙−1𝑥_𝑙,𝑆𝑢_𝑙−1𝑥_𝑙. (25)
Now, if 𝑗=1, then𝐿𝑢_𝑛𝑥₁=𝐹𝑥₁,𝑢₀𝑥₁,𝑇𝑢₀𝑥₁,𝑆𝑢₀𝑥₁. (26)
Again, if 𝑗=2, then 𝛽₂₁𝐿𝑢_𝑛𝑥₁+𝛽₂₂𝐿𝑢_𝑛𝑥₂=𝛽₂₁𝐹𝑥₁,𝑢₀𝑥₁,𝑇𝑢₀𝑥₁,𝑆𝑢₀𝑥₁+𝛽₂₂𝐹𝑥₂,𝑢₁𝑥₂,𝑇𝑢₁𝑥₂,𝑆𝑢₁𝑥₂. (27)
Thus, 𝐿𝑢_𝑛(𝑥₂)=𝐹(𝑥₂,𝑢₁(𝑥₂),𝑇𝑢₁(𝑥₂),𝑆𝑢₁(𝑥₂)). It is easy to see by using mathematical induction concept that 𝐿𝑢_𝑛(𝑥_𝑗)=𝐹(𝑥_𝑗,𝑢_𝑗−1(𝑥_𝑗),𝑇𝑢_𝑗−1(𝑥_𝑗),𝑆𝑢_𝑗−1(𝑥_𝑗)).
On the other hand, from Theorem 11, 𝑢_𝑛(𝑥) converge uniformly to 𝑢(𝑥). It follows that on taking limits in (18) 𝑢(𝑥)=∑^∞_𝑖=1𝐵_𝑖𝜓_𝑖(𝑥). Therefore, 𝑢_𝑛(𝑥)=𝑃_𝑛𝑢(𝑥), where 𝑃_𝑛 is an orthogonal projector from 𝑊⁵₂[𝑎,𝑏] to Span {𝜓₁,𝜓₂,…,𝜓_𝑛}. Thus, 𝐿𝑢_𝑛(𝑥_𝑗)=⟨𝐿𝑢_𝑛(𝑥),𝜑_𝑗(𝑥)⟩_𝑊12=⟨𝑢_𝑛(𝑥),𝐿^∗_𝑗𝜑(𝑥)⟩_𝑊52 = ⟨𝑃_𝑛𝑢(𝑥),𝜓_𝑗(𝑥)⟩_𝑊52=⟨𝑢(𝑥),𝑃_𝑛𝜓_𝑗(𝑥)⟩_𝑊52=⟨𝑢(𝑥),𝜓_𝑗(𝑥)⟩_𝑊52=⟨𝐿𝑢(𝑥),𝜑_𝑗(𝑥)⟩_𝑊12=𝐿𝑢(𝑥_𝑗).

Theorem 17. If {𝑥_𝑖}^∞_𝑖=1 is dense on [𝑎,𝑏] and ||𝑢_𝑛||_𝑊52 is bounded, then the 𝑛-term approximate solution 𝑢_𝑛(𝑥) in the iterative formula (18) converges to the exact solution 𝑢(𝑥) of (9) in the space 𝑊⁵₂[𝑎,𝑏] and 𝑢(𝑥)=∑^∞_𝑖=1𝐴_𝑖𝜓_𝑖(𝑥), where 𝐴_𝑖 are given by (19).

Proof. The proof consists of the following three steps. Firstly, we will prove that the sequence {𝑢_𝑛}^∞_𝑛=1 in (18) is monotone increasing in the sense of ‖⋅‖_𝑊⁵₂. By Theorem 9, {𝜓_𝑖}^∞_𝑖=1 is the complete orthonormal system in 𝑊⁵₂[𝑎,𝑏]. Hence, we have ‖𝑢_𝑛‖²_𝑊52=⟨𝑢_𝑛(𝑥),𝑢_𝑛(𝑥)⟩_𝑊52 = ⟨∑^𝑛_𝑖=1𝐴_𝑖𝜓_𝑖(𝑥),∑^𝑛_𝑖=1𝐴_𝑖𝜓_𝑖(𝑥)⟩_𝑊52=∑^𝑛_𝑖=1(𝐴_𝑖)². Therefore, ‖𝑢_𝑛‖_𝑊52 is monotone increasing.
Secondly, we will prove the convergence of 𝑢_𝑛(𝑥). From (18), we have 𝑢_𝑛+1(𝑥)=𝑢_𝑛(𝑥)+𝐴_𝑛+1𝜓_𝑛+1(𝑥). From the orthogonality of {𝜓_𝑖(𝑥)}^∞_𝑖=1, it follows that ‖𝑢_𝑛+1‖²_𝑊52=‖𝑢_𝑛‖²_𝑊52+(𝐴_𝑛+1)²=‖𝑢_𝑛−1‖²_𝑊52+(𝐴_𝑛)²+(𝐴_𝑛+1)²=⋯=‖𝑢₀‖²_𝑊52+∑^𝑛+1_𝑖=1(𝐴_𝑖)². Since, the sequence ‖𝑢_𝑛‖_𝑊52 is monotone increasing. Due to the condition that ‖𝑢_𝑛‖_𝑊52 is bounded, ‖𝑢_𝑛‖_𝑊52 is convergent as 𝑛→∞. Then, there exists a constant 𝑐 such that ∑^∞_𝑖=1(𝐴_𝑖)²=𝑐. It implies that 𝐴_𝑖=∑^𝑖_𝑘=1𝛽_𝑖𝑘𝐹(𝑥_𝑘,𝑢_𝑘−1(𝑥_𝑘),𝑇𝑢_𝑘−1(𝑥_𝑘),𝑆𝑢_𝑘−1(𝑥_𝑘))∈𝑙²,𝑖=1,2,…. On the other hand, since (𝑢_𝑚−𝑢_𝑚−1)⟂(𝑢_𝑚−1−𝑢_𝑚−2)⟂⋯⟂(𝑢_𝑛+1−𝑢_𝑛), it follows for 𝑚>𝑛 that 𝑢_𝑚(𝑥)−𝑢_𝑛(𝑥)²_𝑊52=𝑢_𝑚(𝑥)−𝑢_𝑚−1(𝑥)+𝑢_𝑚−1(𝑥)−⋯+𝑢_𝑛+1(𝑥)−𝑢_𝑛(𝑥)²_𝑊52=𝑢_𝑚(𝑥)−𝑢_𝑚−1(𝑥)²_𝑊52+⋯+𝑢_𝑛+1(𝑥)−𝑢_𝑛(𝑥)²_𝑊52. (28) Furthermore, ‖𝑢_𝑚(𝑥)−𝑢_𝑚−1(𝑥)‖²_𝑊52=(𝐴_𝑚)². Consequently, as 𝑛,𝑚→∞, we have ‖𝑢_𝑚(𝑥)−𝑢_𝑛(𝑥)‖²_𝑊52=∑^𝑚_𝑖=𝑛+1(𝐴_𝑖)²→0. Considering the completeness of 𝑊⁵₂[𝑎,𝑏], there exists a 𝑢(𝑥)∈𝑊⁵₂[𝑎,𝑏] such that 𝑢_𝑛(𝑥)→𝑢(𝑥) as 𝑛→∞ in the sense of ‖⋅‖_𝑊⁵₂.
Thirdly, we will prove that 𝑢(𝑥) is the solution of (9). Since {𝑥_𝑖}^∞_𝑖=1 is dense on [𝑎,𝑏], for any 𝑥∈[𝑎,𝑏], there exists subsequence {𝑥_𝑛𝑗}^∞_𝑗=1, such that 𝑥_𝑛𝑗→𝑥 as 𝑗→∞. From Lemma 16, it is clear that 𝐿𝑢(𝑥_𝑛𝑗)=𝐹(𝑥_𝑛𝑗,𝑢_𝑛𝑗−1(𝑥_𝑘),𝑇𝑢_𝑛𝑗−1(𝑥_𝑘),𝑆𝑢_𝑛𝑗−1(𝑥_𝑘)). 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)∈𝑊⁵₂[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=1𝐴_𝑖𝜓_𝑖(𝑥) and 𝐴_𝑖 are given by (19). The proof is complete.

It obvious that if we let 𝑢(𝑥) denote the exact solution of (9), 𝑢_𝑛(𝑥) the approximate solution obtained by the RKHS method as given by (18), and 𝑟_𝑛(𝑥) the difference between 𝑢(𝑥) and 𝑢_𝑛(𝑥), where 𝑥∈[𝑎,𝑏], then ‖𝑟_𝑛(𝑥)‖²_𝑊52=‖𝑢(𝑥)−𝑢_𝑛(𝑥)‖²_𝑊52=‖∑^∞_𝑖=𝑛+1𝐴_𝑖𝜓_𝑖(𝑥)‖²_𝑊52=∑^∞_𝑖=𝑛+1(𝐴_𝑖)² and ‖𝑟_𝑛−1(𝑥)‖²_𝑊52=∑^∞_𝑖=𝑛(𝐴_𝑖)² or ‖𝑟_𝑛(𝑥)‖_𝑊52≤‖𝑟_𝑛−1(𝑥)‖_𝑊52. In fact, this is just the proof of the following theorem.

Theorem 18. The difference function 𝑟_𝑛(𝑥) is monotone decreasing in the sense of the norm of 𝑊⁵₂[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: 𝑢⁽⁴⁾(𝑥)+𝑢(𝑥)−¹₀(𝑥𝑡)𝑢(𝑡)𝑑𝑡−^𝑥₀𝑥()𝑢(𝑡)(𝑥)𝑡+1𝑑𝑡=𝑓,0≤𝑥,𝑡≤1, (29) subject to the boundary conditions 𝑢(0)(1)𝑢=−2,𝑢=0,_′(0)=2,𝑢^′(1)=2, (30) where 𝑓(𝑥)=−(2/3)𝑥⁴+2𝑥²+(7/3)𝑥−2. The exact solution is 𝑢(𝑥)=2𝑥−2.

Using RKHS method, taking 𝑥_𝑖=(𝑖−1)/(𝑛−1), 𝑖=1,2,…,𝑛 with the reproducing kernel function 𝐾_𝑥(𝑦) on [0,1], the approximate solution 𝑢_𝑛(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 Example 19.

Example 20. Consider the following nonlinear Fredholm-Volterra IDE: 𝑢⁽⁴⁾(𝑥)(𝑥)−1+𝑢4¹₀(𝑡)𝑢³(𝑡)1𝑑𝑡+2^𝑥₀(𝑥)𝑢²(𝑡)(𝑥)𝑑𝑡=𝑓,0≤𝑥,𝑡≤1, (31) subject to the boundary conditions 𝑢(0)(1)𝑢=0,𝑢=1,_′(0)=0,𝑢^′(1)=2, (32) where 𝑓(𝑥)=(1/10)𝑥⁶+𝑥²−(1/32). The exact solution is 𝑢(𝑥)=𝑥².
Using RKHS method, taking 𝑥_𝑖=(𝑖−1)/(𝑁−1), 𝑖=1,2,…,𝑁 with the reproducing kernel function 𝐾_𝑥(𝑦) on [0,1], the approximate solution 𝑢^𝑁_𝑛(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 Example 20.

Example 21. Consider the following nonlinear Fredholm-Volterra IDE: 𝑢⁽⁴⁾(𝑥)+𝑢²(𝑥)−¹₀(𝑡+1)sinh(𝑢(𝑡))𝑑𝑡−^𝑥₀(𝑥)𝑒^𝑢(𝑡)(𝑥)𝑑𝑡=𝑓,0≤𝑥,𝑡≤1, (33) subject to the boundary conditions 𝑢(0)(1)(2),𝑢=0,𝑢=ln_′(0)=1,𝑢^′(1)=12, (34) 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, taking 𝑥_𝑖=(𝑖−1)/(𝑁−1), 𝑖=1,2,…,𝑁 with the reproducing kernel function 𝐾_𝑥(𝑦) on [0,1], the approximate solution 𝑢^𝑁_𝑛(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 Example 21.

Example 22. Consider the following nonlinear Fredholm-Volterra IDE: 𝑢⁽⁴⁾(𝑥)+cos(𝑢(𝑥))−¹₀(𝑡+𝑥)𝑢(𝑡)𝑑𝑡−^𝑥₀𝑒^𝑡−𝑥𝑒^𝑢(𝑡)(𝑥)𝑑𝑡=𝑓,0≤𝑥,𝑡≤1, (35) subject to the boundary conditions 𝑢(0)(1)𝑢=1,𝑢=0,_′(0)=−1,𝑢^′(1)=−1, (36) where 𝑓(𝑥)=cos(1−𝑥)−𝑥𝑒^1−𝑥−(1/2)𝑥−(1/6). The exact solution is 𝑢(𝑥)=1−𝑥.
Using RKHS method, taking 𝑥_𝑖=(𝑖−1)/(𝑁−1), 𝑖=1,2,…,𝑁 with the reproducing kernel function 𝐾_𝑥(𝑦) on [0,1], the approximate solution 𝑢^𝑁_𝑛(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 Example 22.

Example 23. Consider the following nonlinear Fredholm-Volterra IDE: 𝑢⁽⁴⁾(𝑥)+𝑢(𝑥)−¹₀cos⁻¹(𝑢(𝑡−))𝑑𝑡_𝑥0(𝑥+𝑡)1−𝑢²(𝑡)𝑑𝑡=𝑓(𝑥),0≤𝑥,𝑡≤1, (37) subject to the boundary conditions 𝑢(0)(1)(1),𝑢=1,𝑢=cos_′(0)=0,𝑢^′(1)=−sin(1), (38) 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, taking 𝑥_𝑖=(𝑖−1)/(𝑁−1), 𝑖=1,2,…,𝑁 with the reproducing kernel function 𝐾_𝑥(𝑦) on [0,1], the approximate solution 𝑢^𝑁_𝑛(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 Example 23.

As 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 𝑥_𝑖=(𝑖−1)/(𝑁−1), 𝑖=1,2,…,𝑁 in which 𝑁=26 and 𝑛=5 are given in Table 6.

Table 6 

Absolute error of 𝑢^(𝑚)(𝑥) for Example 23.

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 𝑢_𝑛(𝑥) are represented in the form of series in 𝑊⁵₂[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.