| Modified Spectral Solutions for Solving Some Types of Two-Dimensional Integral Equations |
| Paper ID : 1008-ISCH |
| Authors |
|
Ahmed M. Abbas *1, Youssri Hassan Youssri2, Mamdouh Metwally Elkady3, Mohamed Ahmed Abdelhakem4 1T. A., Mathematics Department, Faculty of Science, Helwan University, Helwan 11795, Egypt 2Associate Professor, Mathematics Department, Faculty of Science, Cairo University, Giza 12613, Egypt 3Professor, Mathematics Department, Faculty of Science, Helwan University, Helwan 11795, Egypt 4Associate Professor, Mathematics Department, Faculty of Science, Helwan University, Helwan 11795, Egypt |
| Abstract |
| 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. |
| Keywords |
| Legendre polynomials, spectral methods, pseudo-Galerkin spectral method, integral equations. |
| Status: Abstract Accepted (Poster Presentation) |