Discrete Legendre spectral projection-based methods for Tikhonov regularization of first kind Fredholm integral equations

Abstract

In this paper, we apply the discrete Legendre Galerkin and multi-Galerkin methods to find the approximate solution of the Tikhonov regularized equation of the Fredholm integral equations of the first kind. We evaluate the error bounds for the approximate solutions with the exact solution in the infinity norm. We provide an a priori parameter choice strategy to find the convergence rates under the infinity norm. Since smoothness of the solution is not known in applied problems, we discuss an adaptive parameter choice rule to choose the regularization parameter, and then using this regularization parameter, we obtain the order of convergence in infinity norm. We give test examples to justify the theoretical estimates.