High-Accuracy Modified Spectral Techniques for Two-Dimensional Integral Equations

This research introduces a numerical method for solving two-dimensional integral equations. The exact solution is assumed to be a limit point for the set of all polynomials and is approximated to be a finite series of constant multiples of basis functions for the polynomial functions space. Legendre’s first derivative polynomials have been chosen in this work as the orthogonal basis functions. Some new relations are constructed, such as the linearization formula. Subsequently, applying the pseudo-Galerkin spectral method results in a system of algebraic equations in the constant coefficients of the approximated expansion. Lastly, we solve the algebraic system using the Gauss elimination method for linear systems or Newton’s iteration method with zero initial guesses for nonlinear systems that are most likely to appear out of the presented procedure. This approach yields the desired semi-analytic approximate solution. Convergence and error analyses have been studied. To clarify the efficiency and accuracy of the presented method, we solved some numerical test problems.