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arXiv · 1906.03907

A sparse spectral method for Volterra integral equations using orthogonal polynomials on the triangle

Timon S. Gutleb·Sheehan Olver
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Abstract

We introduce and analyse a sparse spectral method for the solution of Volterra integral equations using bivariate orthogonal polynomials on a triangle domain. The sparsity of the Volterra operator on a weighted Jacobi basis is used to achieve high efficiency and exponential convergence. The discussion is followed by a demonstration of the method on example Volterra integral equations of the first and second kind with known analytic solutions as well as an application-oriented numerical experiment. We prove convergence for both first and second kind problems, where the former builds on connections with Toeplitz operators.

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Timon S. Gutleb, Sheehan Olver. 2019-06-10. A sparse spectral method for Volterra integral equations using orthogonal polynomials on the triangle. https://doi.org/10.1137/19m1267441

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