Abstract
In this paper, Legendre spectral projection methods are applied for the Volterra integral equations of second kind with a smooth kernel. We prove that the approximate solutions of the Legendre Galerkin and Legendre collocation methods converge to the exact solution with the order \({\mathcal {O}}(n^{-r})\) in \(L^2\)-norm and order \({\mathcal {O}}(n^{-r+\frac{1}{2}})\) in infinity norm, and the iterated Legendre Galerkin solution converges with the order \({\mathcal {O}}(n^{-2r})\) in both \(L^2\)-norm and infinity norm, whereas the iterated Legendre collocation solution converges with the order \({\mathcal {O}}(n^{-r })\) in both \(L^2\)-norm and infinity norm, n being the highest degree of Legendre polynomials employed in the approximation and r being the smoothness of the kernels. We have also considered multi-Galerkin method and its iterated version, and prove that the iterated multi-Galerkin solution converges with the order \({\mathcal {O}}(n^{-3r})\) in both infinity and \(L^2\) norm. Numerical examples are given to illustrate the theoretical results.
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Communicated by Jose Alberto Cuminato.
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Mandal, M., Nelakanti, G. Superconvergence results of Legendre spectral projection methods for Volterra integral equations of second kind. Comp. Appl. Math. 37, 4007–4022 (2018). https://doi.org/10.1007/s40314-017-0563-5
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DOI: https://doi.org/10.1007/s40314-017-0563-5
Keywords
- Volterra integral equations
- Smooth kernels
- Projection methods
- Legendre polynomial
- Superconvergence rates