An efficient spectral method for two-dimensional Fredholm integro-differential equations in complex geometries

Abstract

Classical spectral methods are confined to numerically solving Fredholm integro-differential equations in regular domains, such as rectangles and discs. This paper aims to numerically address two-dimensional Fredholm integro-differential equations in complex geometries by combining spectral methods with mapping techniques. Initially, we transform the computational domain into a rectangular one via coordinate mapping. Subsequently, classical spectral methods are applied within this rectangular domain for numerical simulations. Our analysis primarily discusses the existence, uniqueness and convergence of numerical solutions. Numerical results demonstrate that the proposed method achieves high-order accuracy.