A Comprehensive Comparison between Galerkin Method Using Gegenbauer Wavelets and Modified Galerkin Algorithm Using Shifted Jacobi Polynomials for Solving Special Fredholm Integral Equations

Abstract

This paper considers two advanced spectral Galerkin methods for special Fredholm integral equations, namely, the Gegenbauer Wavelet Galerkin Method (GWGM) and the Modified Jacobi Galerkin Algorithm (MJGA),         in a rigorous comparative study. The GWGM, based on Gegenbauer wavelets, is well adapted to catching the localized characters of the solutions, while the MJGA, which is based on shifted Jacobi polynomials, ensures
 a fast convergence rate when the kernels are smooth. We develop theoretical underpinnings of both schemes, including convergence analysis along with computational complexity, and illustrate performance by extensive numerical experiments. It turns out that GWGM is more accurate for problems with localized singularities, whereas MJGA exponentially converges in the case of functions with global smoothness. The study carries out the computational trade-offs of both approaches and derives practical challenges where each is superior. This work considerably extends the numerical analysis by providing an overall performance test that will guide researchers in choosing the best spectral method for solving integral equations arising in mathematical physics and engineering applications.