Extrapolation method of iterated collocation solution for two-dimensional nonlinear Volterra integral equations

Abstract

In this paper, we study the numerical solution of two-dimensional nonlinear Volterra integral equation by collocation and iterated collocation method. Asymptotic error expansion of iterated collocation solution is obtained. Using Richardson's extrapolation, we show that the accuracy of numerical solution in the iterated collocation method can be improved greatly. Some numerical examples are given to illustrate this theory.

Introduction

Consider the nonlinear Volterra equation of the second kind

$$\text{u(x,y)=g(x,y)+∫}{}_0{}^{\mathrm{x}}\text{∫}{}_0{}^{\mathrm{y}}\text{K(x,y,t,s,u(t,s))}\text{d}\text{t}\text{d}\text{s,}\text{(x,y)∈D,}$$

where g(x,y), K(x, y, t, s, u) are given continuous functions defined, respectively, on D=[0,X]×[0,Y] and $\text{E={(x,y,t,s,u):0⩽t⩽x⩽X,}\text{0⩽s⩽y⩽Y,}\text{−∞<u<+∞}}$, with K(x, y, t, s, u) nonlinear in u.

Let

$$\text{(Ku)(x,y)=∫}{}_0{}^{\mathrm{x}}\text{∫}{}_0{}^{\mathrm{y}}\text{K(x,y,t,s,u(t,s))}\text{d}\text{t}\text{d}\text{s.}$$

Then Eq. (1.1)becomes

$$\text{u=Ku+g.}$$

It follows from the classical theory of Volterra that (1.1) possesses a unique solution $\text{u}{}^{\mathrm{\ast }}\text{(x,y)∈C(D)}$. Especially, when g and K are r times continuously differentiable on D and E, respectively, then u^* is r times continuously differentiable on D.

During the last 10 years, significant progress has been made in numerical analysis of one-dimensional version of (1.1) (see, 2, 4, 6 and in the references cited there). However, the numerical methods for two-dimensional integral equation (1.1) seem to have been discussed in only a few places. Ref. [1] proposed a class of explicit Runge–Kutta-type methods of order 3 (without analyzing their convergence). Bivariate cubic spline functions method of full continuity was obtained by Singh [8]. Brunner and Kauthen [3] introduced collocation and iterated collocation methods for two-dimensional linear Volterra integral equation. They gave an analysis of global and local convergence properties of collocation method and iterated collocation method, and derived results on attainable orders of global convergence and local superconvergence. In [7], asymptotic error expansion of iterated collocation solution for two-dimensional linear Volterra integral equation was obtained. In this paper, we discuss collocation method and iterated collocation method for Eq. (1.1). The asymptotic error expansion for the iterated collocation solution of (1.1) is obtained. We show that when piecewise polynomials of π_p−1,q−1 are used, the iterated collocation solution admits an error expansion in powers of the stepsizes h and k. For a special choice of the collocation points, the leading terms in the error expansion contain only even powers of the stepsizes h and k, beginning with terms in h^2p and k^2q. Thus, Richardson's extrapolation can be performed on the numerical solution; and this will increase the accuracy of the numerical solution greatly. This paper is organized as follows. In Section 2, we describe the approximating spline spaces, give asymptotic error expansions for a class of spline interpolation and derive collocation method of (1.1). Section 3deals with asymptotic error expansion of iterated collocation solution for Eq. (1.1). Finally, some numerical examples are given in illustration of this theory in Section 4.