Stabilité et convergence des méthodes spectrales Application aux équations intégrales

Abstract

In recent years, there has been a growing interest in the formulation of many problems in terms of integral equations, and this has fostered a parallel rapid growth of the literature on their numerical solution. In this sense, our focus will be on spectral methods for solving integral equations. One of the purposes of this research is to provide the mathematical foundations of spectral methods and to analyze their basic theoretical properties (stability, accuracy, computational complexity, and convergence). Furthermore, we have applied the spectral collocation method to find numerical solutions to quadratic Urysohn integral equations. This method reduces the nonlinear integral equation to a system of nonlinear algebraic equations and that algebraic system has been solved by the iterative method. We have derived an error analysis for the current method, which proves that it has exponential convergence order. Finally, several numerical examples are given to show the effectiveness and stability of our approach