@Article{AAMM-13-5,
author = {Kapil, Kant and Moumita, Mandal and Nelakanti, Gnaneshwar},
title = {Jacobi Spectral Galerkin Methods for a Class of Nonlinear Weakly Singular Volterra Integral Equations},
journal = {Advances in Applied Mathematics and Mechanics},
year = {2021},
volume = {13},
number = {5},
pages = {1227--1260},
abstract = {<p style="text-align: justify;">We propose the Jacobi spectral Galerkin and Jacobi spectral multi Galerkin methods with their iterated versions for obtaining the superconvergence results of a general class of nonlinear Volterra integral equations with a kernel $x^{\beta}(z-x)^{-\kappa},$ where $0&lt;\kappa&lt;1$, $\beta&gt;0$, which have an Abel-type and an endpoint singularity. The exact solutions for these types of integral equations are singular at the initial point of integration. First, we apply a transformation of independent variables to find a new integral equation with a sufficiently smooth solution. Then we discuss the superconvergence rates for the transformed equation in both uniform and weighted $L^2$-norms. We obtain the order of convergence in Jacobi spectral Galerkin method $\mathcal{O}(N^{\frac{3}{4}-r})$ and $\mathcal{O}(N^{-r})$ in uniform and weighted $L^2$-norms, respectively. Whereas iterated Jacobi spectral Galerkin method converges with the order of convergence $\mathcal{O}(N^{-2r})$ in both uniform and weighted $L^2$-norms. We also show that iterated Jacobi spectral multi Galerkin method converges with the orders $\mathcal{O}(N^{-3r}\log{N})$ and $\mathcal{O}(N^{-3r})$ in uniform and weighted $L^2$-norms, respectively. Theoretical results are verified by numerical illustrations.</p>},
issn = {2075-1354},
doi = {https://doi.org/10.4208/aamm.OA-2020-0163},
url = {https://global-sci.com/article/73008/jacobi-spectral-galerkin-methods-for-a-class-of-nonlinear-weakly-singular-volterra-integral-equations}
}