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
