A new approach of shifted Jacobi spectral Galerkin methods (SJSGM) for weakly singular Fredholm integral equation with non-smooth solution - Kayal, Arnab; Mandal, Moumita (Numerical algorithms, 2024)
This article presents a new approach of shifted Jacobi spectral Galerkin methods to solve weakly singular Fredholm integral equations with non-smooth solutions. We have incorporated the singular part of the kernel into a single Jacobi weight function, by dividing the integration into two parts and using a simple variable transformation. Taking advantage of orthogonal projection operator and weighted inner product with respect to that same Jacobi weight function, we are able to obtain improved convergence rate for iterated shifted Jacobi spectral Galerkin method (SJSGM) and iterated shifted Jacobi spectral multi-Galerkin method (SJSMGM) in both weighted and infinity norms. Further, we obtain improved superconvergence rate for iterated SJSGM and iterated SJSMGM, by improving the regularity of exact solution, using smoothing transformation. Increasing the value of the smoothing parameter we can improve the regularity of the exact solution upto the desired degree. Numerical results with a comparative study of pre and post smoothing transformation are given to illustrate the theoretical results and efficiency of our proposed methods.