A composite collocation method for the nonlinear mixed Volterra–Fredholm–Hammerstein integral equations

Abstract

This paper presents a computational technique for the solution of the nonlinear mixed Volterra–Fredholm–Hammerstein integral equations. The method is based on the composite collocation method. The properties of hybrid of block-pulse functions and Lagrange polynomials are discussed and utilized to define the composite interpolation operator. The estimates for the errors are given. The composite interpolation operator together with the Gaussian integration formula are then used to transform the nonlinear mixed Volterra–Fredholm–Hammerstein integral equations into a system of nonlinear equations. The efficiency and accuracy of the proposed method is illustrated by four numerical examples.

Introduction

There is considerable literature that discusses approximating the solution of linear and nonlinear Hammerstein integral equations [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]. For Fredholm–Hammerstein integral equations, the classical method of successive approximations was introduced in [1]. A variation of the Nystrom method was presented in [2], and a collocation-type method was developed in [3]. In [4], Brunner applied a collocation-type method to nonlinear Volterra–Hammerstein integral equations and integro-differential equations and discussed its connection with the iterated collocation method. Han [5] introduced and discussed the asymptotic error expansion of a collocation-type method for Volterra–Hammerstein integral equations. The existence of the solutions to nonlinear Hammerstein integral equations was discussed in [6]. Several authors consider the nonlinear mixed Volterra–Fredholm integral equation of the form

$$y(x)=f(x)+{\lambda }_1{\int }_0^x{K}_1(x\text{,}t)G_1(t\text{,}y(t))\mathit{dt}+{\lambda }_2{\int }_0^1{K}_2(x\text{,}t)G_2(t\text{,}y(t))\mathit{dt}\text{,}0⩽x⩽1\text{,}$$

where λ₁ and λ₂ are constants and f(x) and the kernels K₁(x, t) and K₂(x, t) are given functions assumed to have nth derivatives on the interval 0 ⩽ x, t ⩽ 1. For the case G₁(t, y(t)) = y^p(t) and G₂(t, y(t)) = y^q(t), where p and q are nonnegative integers, Yalcinbas [7] applied Taylor series and Bildik and Inc [8] used the modified decomposition method to find the solution. For the case G₁(t, y(t)) = F₁(y(t)) and G₂(t, y(t)) = F₂(y(t)), where F₁(y(t)) and F₂(y(t)) are given continuous functions which are nonlinear with respect to y(t), Yousefi and Razzaghi [9] applied Legendre wavelets to obtain the solution, and for the general case, where G₁(t, y(t)) and G₂(t, y(t)) are given continuous functions which are nonlinear with respect to t and y(t), Ordokhani [10] applied the rationalized Haar functions to get the solution.

It is well known that the collocation-type methods provide highly-accurate approximations for the solutions of nonlinear operator equations in function spaces, provided that these solutions are sufficiently smooth [11], [12], [13]. In this paper, we are concerned with the extension of the composite collocation method based on the Legendre–Gauss-type points to the numerical solution of the nonlinear mixed Volterra–Fredholm–Hammerstein integral equations of the form of Eq. (1.1). The method consists of reducing the calculation of the solution of this equation to a set of nonlinear equations by first expanding the candidate function as a hybrid function with unknown coefficients. These hybrid functions, which consist of block-pulse functions and Lagrange polynomials, are first introduced. The properties of hybrid functions and composite collocation method based on the Legendre–Gauss-type points are then utilized to evaluate the unknown coefficients and find an approximate solution to Eq. (1.1). This paper is organized as follows: in Section 2, we introduce the composite interpolation operator using the hybrid functions and discuss some approximation errors. In Section 3, we introduce our method, and in Section 4, we report our numerical findings and demonstrate the efficiency and accuracy of the proposed method.