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In this article, we propose shifted Jacobi spectral Galerkin method (SJSGM) and iterated SJSGM to solve nonlinear Fredholm integral equations of Hammerstein type with weakly singular kernel. We have rigorously studied convergence analysis of the proposed methods. Even though the exact solution exhibits non-smooth behaviour, we manage to achieve superconvergence order for the iterated SJSGM. Further, using smoothing transformation, we improve the regularity of the exact solution, which enhances the convergence order of the SJSGM and iterated SJSGM. We have also shown the applicability of our proposed methods to high-order nonlinear weakly singular integro-differential equations and achieved superconvergence. Several numerical examples have been implemented to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this article, we propose Chebyshev spectral projection methods to solve the hypersingular integral equation of first kind. The presence of strong singularity in Hadamard sense in the first part of the integral equation makes it challenging to get superconvergence results. To overcome this, we transform the first kind hypersingular integral equation into a second kind integral equation. This is achieved by defining a bounded inverse of the hypersingular integral operator in some suitable Hilbert space. Using iterated Chebyshev spectral Galerkin method on the equivalent second kind integral equation, we obtain improved convergence of O (N − 2 r) , where N is the highest degree of Chebyshev polynomials employed in the approximation space and r is the smoothness of the solution. Further, using commutativity of projection operator and inverse of the hypersingular integral operator, we are able to obtain superconvergence of O (N − 3 r) and O (N − 4 r) , by Chebyshev spectral multi-Galerkin method (CSMGM) and iterated CSMGM, respectively. Finally, numerical examples are presented to verify our theoretical results. • Developed Chebyshev spectral projection methods for hypersingular integral equations. • Established error bounds and convergence rates for all the proposed methods. • Obtained superconvergence of O (N − 4 r) using iterated spectral multi-Galerkin method. • Verified theoretical superconvergence results through numerical examples. [ABSTRACT FROM AUTHOR]