Superconvergence of Legendre spectral projection methods for mth order integro-differential equations with weakly singular kernels

In this article, we apply Legendre spectral Galerkin, Legendre spectral multi-projection methods and their iterated versions to find the approximate solution of mth order Fredholm integro-differential equations with weakly singular kernel. Motivated by Mandal et al. (2023), we use Cauchy repeated integral theorem to transform the integro-differential equation to an single integral equation and obtain superconvergence results by iterated Legendre spectral Galerkin method, in spite of the singularity in the kernel function and unbounded differential operator. We have further improved the convergence rate of the approximate solution by using iterated Legendre spectral multi-projection method. Global Legendre polynomials are used as a basis for the approximating space, to reduce the computational cost of our proposed methods. In this article, we have derived theoretical error bounds and obtained the global convergence rates for all the discussed methods in both L2 and infinity norms. Numerical methods are implemented on examples to justify the theoretical results.