Original articles
Discrete superconvergent degenerate kernel method for Fredholm integral equations

https://doi.org/10.1016/j.matcom.2018.08.014Get rights and content

Abstract

Approximate solutions of integral equations using methods related to an interpolatory projection involve many integrals which need to be evaluated using a numerical quadrature formula. In this paper, we propose the discrete version of the superconvergent degenerate kernel method for solving Fredholm integral equation of the second kind with a smooth kernel. Using sufficiently accurate numerical quadrature rule, we obtain optimal convergence rates for both approximated solution and iterated discrete solution. Numerical results are presented to illustrate the theoretical estimates for the error of this method.

Introduction

Let us consider the Fredholm integral equation of the second kind defined by uKu=f,where K is the compact linear operator defined on the space C[0,1] by (Ku)(s)01κ(s,t)u(t)dt,s[0,1],with κC[0,1]2 and fC[0,1]. Assume that the homogeneous integral equation uKu=0 has, in C[0,1], only the trivial solution, then the operator (IK) is invertible and therefore, Eq. (1.1) has a unique solution.

A standard technique to numerically solving (1.1) is to replace K by an operator Kn of finite rank qn. The approximate solution is then obtained by solving the system of linear equations of size qn (IKn)un=f.

The Galerkin/collocation, Nyström and degenerate kernel methods are commonly used methods for this purpose. They have been extensively studied in the literature (see [5], [10]). Sloan [17] introduced iterated Galerkin/iterated collocation solutions which are obtained by one step of iteration. It is known that under suitable conditions both the iterated Galerkin and iterated collocation methods exhibit (global) superconvergence, that is they can both converge faster than the rate achieved by the Galerkin or collocation methods themselves. More recently, Kulkarni introduced in [15] an efficient method, based on projections, that consists of approximating K by the finite rank operator πnK+KπnπnKπn,where πn is a sequence of projectors onto a space of discontinuous piecewise polynomials. The solution obtained is shown to converge faster than Sloan’s solution. Recently, the following two new approximating operators based on the interpolatory projection πn are introduced in [3] πnK+Kn,ιπnKn,ι,ι=1,2,where Kn,1 is the degenerate kernel operator obtained by interpolating the kernel with respect to the second variable by πn and Kn,2 is the Nyström operator based on πn. The corresponding approximation schemes (1.2) are called superconvergent Nyström and degenerate kernel methods for approximating the solution of (1.1). Some other methods based on spline quasi-interpolation for the solution of (1.1), using operators (projectors and not projectors) onto spaces of piecewise polynomials of degree d and smoothness Cd1 instead of spaces of discontinuous piecewise polynomials have been also studied in [1], [2], [9], [13].

In [4], the discrete superconvergent Nyström method is developed. It is useful in practice for solving Eq. (1.1) and the corresponding eigenvalue problem. That method is obtained from using a numerical quadrature formula to evaluate the integral in the definition of K. This formula is chosen appropriately so as to preserve the orders of convergence of the regular method. The aim of this paper is to define and analyse the discrete version of the superconvergent degenerate kernel method for solving Fredholm integral equations with a smooth kernel and to establish superconvergence results. The techniques in this paper can be used to give an analyse of the discrete superconvergent degenerate kernel method for solving eigenvalue problems. Several discrete numerical methods for solving (1.1) have also been analysed previously (see [5], [14]). The discrete Galerkin method has been considered in Atkinson–Bogomolny [6] and the discrete version of the iterated collocation method appears in Atkinson–Flores [7]. Important early papers are [12] and [16] which analyse the discrete version of Kulkarni’s method called discrete multi-projection methods.

Here is an outline of the paper. In Section 2, the proposed method for the solution of (1.1) is defined along with relevant notations and the system of linear equations which need to be solved to obtain approximations to the solution is discussed. The orders of convergence for the discrete superconvergent degenerate kernel solution and the discrete iterated solution are obtained in Section 3. Numerical results are given in Section 4.

Section snippets

Method and notations

For any positive integer n, let Δn:0=s0<s1<s2<<sn1<sn=1be the uniform partition of [0,1], with meshlength h=1n and knots si=in,i=0,,n. For a fixed r1, we denote by Πr the space of polynomials of degree r1. Let Xn{υ:[0,1]R:υ|[si1,si]Πr,1in}be the set of piecewise polynomial functions of degree r1, on Δn. The functions in Xn need not be continuous at the node points si. Let r={τ1,,τr} be the set of r Gauss points, i.e. the zeros of the Legendre polynomials pr(t)=drdtr(t21)r in

Orders of convergence

Note that in the remainder of this paper, c denotes generic constant, which may take different values at their different occurrences, but will be independent of n. We state below two estimates for the solution unM and the discrete iterated solution u˜nM.

Proposition 3.1

Assume that (IK)1 exists on C[0,1]. For sufficiently large n, we have uunMcmax{(KKn,q)u,(Iπn)(KKn)u,(Iπn)(KnKnD)u}and uu˜nMcKn,q(KnMKn,q)u+Kn,q(KnMKn,q)(KnMKn,q)u+(Kn,qK)u.

Proof

We write KKnM=πn(KKn,q)+(Iπn

Numerical results

We present a numerical example to illustrate the theoretical estimates obtained in the previous section. We consider the following Fredholm integral equation quoted from [4], [12] u(s)01κ(s,t)u(t)dt=f(s)s[0,1],where κ(s,t)=12est. The exact solution is given by u(s)=escos(s). Let Xn be the space of piecewise constant functions (r=1) with respect to the uniform partition of [0,1] on n subintervals with meshlength h=1n given by 0=1n<2n<<n1n<nn=1.Let πn:C[0,1]Xn be the interpolatory

References (17)

There are more references available in the full text version of this article.

Cited by (3)

  • Generalized spline quasi-interpolants and applications to numerical analysis

    2022, Journal of Computational and Applied Mathematics

Research supported by URAC-05.

View full text