Journal of Computational Applied Mechanics

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

Document Type : Research Paper

Authors

1 Mathematics Department, Faculty of Science, Helwan University, Cairo 11795, Egypt

2 Basic Science Department, School of Engineering, Canadian International College, New Cairo, Egypt

3 Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt

4 Associate Fellow (AFHEA) of the Higher Education Academy (Advance HE), UK

5 Helwan School of Numerical Analysis in Egypt (HSNAE)

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

Main Subjects

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Journal of Computational Applied Mechanics
Volume 56, Issue 2
April 2025
Pages 364-379
  • Receive Date: 17 March 2025
  • Revise Date: 29 March 2025
  • Accept Date: 30 March 2025