Adv. Appl. Math. Mech., 13 (2021), pp. 1227-1260.
Published online: 2021-06
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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 $0<\kappa<1$, $\beta>0$, 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.
}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2020-0163}, url = {http://global-sci.org/intro/article_detail/aamm/19260.html} }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 $0<\kappa<1$, $\beta>0$, 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.