Recursive generation of the Galerkin-Chebyshev matrix for convolution kernels (Q1284732)

Today integral equations play a very vital part in formulating, analyzing and solving many practical problems in science, engineering and technology. The computational aspects of solutions are often quite complicated. By careful handling of inputs in explicit situations, however, we can arrive at satisfactory results. The present paper is concerned with integral equations of the first kind having convolution forms in a finite interval. A Galerkin-Chebyshev (G-C) procedure applying to Galerkin's method to the convolution integral \[ \int^1_{-1} K(x-y) g(y)dy= f(x)\quad (-1<x<+1) \] is discussed. Through one basic theorem a general recursion relation for the G-C-matrix components is obtained. Result for the fast generation of the G-C-matrix \(K_{m,n}\) for special convolution kernels are presented and extended to additional recursion relations. A lemma and two more theorems are given on similar lines with proofs. In a numerical examples, the author considers the calculation of G-C-matrix entries \(L^{(0,\sigma)}_{m,n}\) for \(0\leq m\leq M\), \(0\leq n\leq N\) for the modified Bessel function kernel \(K_0(\sigma lx-yl)\). Eleven references are provided.