A computational method for solving periodic boundary value problems for integro-differential equations of Fredholm–Volterra type

Abstract

In this paper, the numerical solution of periodic Fredholm–Volterra integro–differential equations of first-order is discussed in a reproducing kernel Hilbert space. A reproducing kernel Hilbert space is constructed, in which the periodic condition of the problem is satisfied. The exact solution $u\left(x\right)$ is represented in the form of series in the space $W_2^2$. In the mean time, the n-term approximate solution $u_n\left(x\right)$ is obtained and is proved to converge to the exact solution $u\left(x\right)$. Furthermore, we present an iterative method for obtaining the solution in the space $W_2^2$. Some examples are displayed to demonstrate the validity and applicability of the proposed method. The numerical result indicates that the proposed method is straightforward to implement, efficient, and accurate for solving linear and nonlinear equations.

Introduction

Integro–differential equations (IDEs) are often involved in the mathematical formulation of physical and engineering phenomena. IDEs can be encountered in various fields of science such as physics, chemistry, biology, and engineering. These kinds of equations can also be found in numerous applications, such as, theory of elasticity, biomechanics, electromagnetic, electrodynamics, fluid dynamics, heat and mass transfer, oscillating magnetic field, etc. [1], [2], [3], [4], [5]. In recent years, periodic boundary value problems (BVPs) have been widely studied by many authors [6], [7], [8], [9], due to a wide range of applications in applied mathematics, physics, and engineering, particularly in the homogenization of composite materials with a periodic microstructure [10], [11]. In this paper, solution of periodic Fredholm–Volterra IDEs of first-order which are a combination of Fredholm–Volterra IDEs and periodic boundary condition is discussed. This class of equations is sometimes too complicated to be solved exactly because, generally, the solution cannot be exhibited in a closed form even when it exists. Therefore, finding either the analytical approximation or numerical solution of such equations are of great interest.

In this paper, we are concerned with providing the numerical solution based on the use of reproducing kernel Hilbert space (RKHS) method for periodic Fredholm–Volterra IDEs of first-order of the general form

$$u^{\prime }\left(x\right)=F\left(x\text{,}u\left(x\right)\text{,}\mathit{Su}\left(x\right)\text{,}\mathit{Tu}\left(x\right)\right)\text{,}0⩽x⩽1\text{,}$$

subject to the periodic boundary condition

$$u\left(0\right)=u\left(1\right)\text{,}$$

in which the Fredholm operator, Su, and the Volterra operator, Tu, are given, respectively, as

$$\mathit{Su}\left(x\right)={\int }_0^1{k}_1\left(x\text{,}t\right)F_1\left(u\left(t\right)\right)\mathit{dt}\text{,}0⩽x\text{,}t⩽1\text{,}$$

$$\mathit{Tu}\left(x\right)={\int }_0^x{k}_2\left(x\text{,}t\right)F_2\left(u\left(t\right)\right)\mathit{dt}\text{,}0⩽x\text{,}t⩽1\text{,}$$

where $u\in W_2^2\left[0\text{,}1\right]$ is an unknown function to be determined, $F\left(x\text{,}{v}_1\text{,}{v}_2\text{,}{v}_3\right)\text{,}{F}_1\left(v_4\right)$, and $F_2\left(v_5\right)$ are continuous terms in $W_2^1\left[0\text{,}1\right]$ as $v_i=v_i\left(x\right)\in W_2^2\left[0\text{,}1\right]\text{,}0⩽x⩽1\text{,}-\infty <v_i<\infty \text{,}i=1\text{,}2\text{,}3\text{,}4\text{,}5$ and are depending on the problem discussed, and $W_2^1\left[0\text{,}1\right]\text{,}{W}_2^2\left[0\text{,}1\right]$ are two reproducing kernel spaces.

The numerical solvability of periodic BVPs of different orders and types has been pursued by several authors. To mention a few, the existence and multiplicity of positive solutions have been discussed to first-order periodic BVPs as described in [12]. In [13] also, the authors have discussed the existence and uniqueness of periodic solution to first-order periodic Volterra IDEs. In [14] the author has provided the existence and multiplicity of positive solutions to further investigation to second-order periodic BVPs. Furthermore, the existence of solutions is carried out in [15] for third-order periodic BVPs. The existence of positive solution has been investigated to solve fourth-order periodic BVPs as presented in [16]. However, we assume that Eq. (1) subject to the periodic boundary condition (2) has a unique solution on $\left[0\text{,}1\right]$. But on the other aspect as well, the numerical solvability of IDEs of different types and orders can be found in [17], [18], [19], [20], [21] and references therein, and for numerical solvability of different categories of two-point BVPs one can consult the references [22], [23], [24], [25].

Investigation about periodic Fredholm–Volterra IDEs of first-order numerically is scarce. In this paper, we utilize a methodical way to solve these type of equations. The present method is accurate, need less effort to achieve the results, and is developed especially for nonlinear case. Meanwhile, 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.

The theory of reproducing kernel has recently emerged as a powerful framework in numerical analysis, differential and integral equations, and probability and statistics [26], [27], [28]. On the other aspect as well, a RKHS is a useful framework for constructing approximate solutions for linear and nonlinear equations. This method has been implemented in several differential, integral, Integro–differential, operator, and system of equations, such as initial-boundary-value problems [29], partial differential equations [30], singular BVPs [31], [32], Fredholm–Volterra integral equation [33], Fredholm–Volterra IDEs [18], [19], [20], [21], periodic BVPs [34], [35], operator equations [36], system of equations [37].

This paper is organized in six sections including the introduction. In Section 2, two reproducing kernel spaces are presented in order to construct a reproducing kernel function. In Section 3, the analytical solution for Eqs. (1), (2) in the space $W_2^2\left[0\text{,}1\right]$ and some essential results are introduced. Also, an iterative method to solve Eqs. (1), (2) numerically in the same space is described. In Section 4, the n -term approximate solution $u_n\left(x\right)$ is proved to converge to the exact solution $u\left(x\right)$ in the space $W_2^2\left[0\text{,}1\right]$. Numerical experiments are presented in Section 5. Finally, in Section 6 some concluding remarks and future recommendations are presented.