A sparse spectral method for Volterra integral equations using orthogonal polynomials on the triangle
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2019
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| _version_ | 1866929508583473152 |
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| author | Gutleb, Timon S. Olver, Sheehan |
| author_facet | Gutleb, Timon S. Olver, Sheehan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1906_03907 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | A sparse spectral method for Volterra integral equations using orthogonal polynomials on the triangle Gutleb, Timon S. Olver, Sheehan Numerical Analysis 65N35, 45D05 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. |
| title | A sparse spectral method for Volterra integral equations using orthogonal polynomials on the triangle |
| topic | Numerical Analysis 65N35, 45D05 |
| url | https://arxiv.org/abs/1906.03907 |