On the Application of a Generalization of Toeplitz Matrices to the Numerical Solution of Integral Equations with Weakly Singular Convolution Kernels
Simon N. Chandler‐Wilde, M. J. C. Gover · IMA Journal of Numerical Analysis · 1989
This paper discusses the numerical solution by product integration of weakly singular Fredholm integral equations of the second kind with symmetric difference kernels. The product integration method uses a piecewise polynomial, in general, at most, continuous at the knots. A main result of the paper is to show that, owing to the difference kernel, a highly patterned linear system of equations arises if the knots are equally spaced. Specifically the order-N coefficient matrix is block-Toeplitz or a generalization, and centrosymmetric. An algorithm to solve this linear system in O(N2) operations is presented. Unfortunately the asymptotic rate of convergence of the product integration solution is limited if a uniform mesh is used. A method to improve the rate of convergence, while retaining a patterned coefficient matrix, is described. This method involves subtraction of the singularities in the solution induced by the weakly singular kernel. A higher rate of convergence for the modified product integration method is shown. These results are illustrated by application to an integral equation arising in outdoor sound propagation.