Application of Hilbert–Schmidt SVD approach to solve linear two-dimensional Fredholm integral equations of the second kind

Abstract

Meshfree techniques based on infinitely smooth radial kernels have the great potential to provide spectrally accurate function approximations with irregular domain in high dimensions. The maximum accuracy can mostly be found when the RBF shape parameter is small, i.e., when the radial kernel is relatively smooth. However, as the shape parameter goes to zero, the standard RBF interpolant matrix will be very ill-conditioned. The ill-conditioning can be alleviated using alternate bases. One of these alternative bases is the Hilbert–Schmidt SVD basis. The Hilbert–Schmidt SVD approach suggests a stable mechanism for replacing a set of near-flat kernels with scattered centres to a well-conditioned base for exactly the same space. In this work, the Gaussian Hilbert–Schmidt SVD basis functions method is presented to numerically solve the linear two-dimensional Fredholm integral equations of the second kind. The method estimates the solution by the discrete collocation method based on Gaussian Hilbert–Schmidt SVD basis functions constructed on a set of scattered points. The emerged integrals in the scheme are approximately computed by the Gauss–Legendre quadrature rule. This approach reduces the problem under study to a linear system of algebraic equations which can be solved easily via applying an appropriate numerical technique. Also, the convergence of the proposed approach is established. Finally, numerical results are compared with standard RBF method to indicate the accuracy and efficiency of the suggested approach.