Parallel solution of Fredholm integral equations of the second kind by orthogonal polynomial expansions

Abstract

Given a Fredholm integral equation of the second kind, the Nyström method based on an n-point Gaussian quadrature rule and the perturbed Galerkin method in which inner products and integral transforms are approximated using the Gaussian rule, are equivalent in the sense that the Nyström interpolant and the orthogonal polynomial expansion of degree n − 1> agree at the n quadrature nodes. The perturbed collocation method, using the n quadrature nodes as collocation points, in which integral transforms are approximated using the Gaussian rule, is also equivalent. A practical implication is that one can set up and solve the simple n × n linear system corresponding to the Nyström method, and then easily transform the result to that of the perturbed Galerkin method, in order to economically gain the advantages of the orthogonal expansion. This approach appears particularly useful for the numerical solution on multiprocessor systems of the integral formulation of certain two-point boundary value problems. The paper analyzes the relationship among numerical methods, it establishes the weighted least-squares convergence of the orthogonal expansions, and it describes corresponding parallel algorithms.