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

Timon S. Gutleb, Sheehan Olver

TL;DR

A sparse spectral method for the solution of Volterra integral equations using bivariate orthogonal polynomials on a triangle domain is introduced and convergence for both first and second kind problems is proved.

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.

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

TL;DR

A sparse spectral method for the solution of Volterra integral equations using bivariate orthogonal polynomials on a triangle domain is introduced and convergence for both first and second kind problems is proved.

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.

Paper Structure

This paper contains 20 sections, 10 theorems, 97 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Function approximation with orthogonal polynomials
  3. Jacobi polynomials on the real interval
  4. Jacobi polynomials on the triangle
  5. Function evaluation using Clenshaw's algorithm
  6. A numerical method for Volterra integral equations
  7. Volterra integrals on the triangle
  8. Kernel computations using Clenshaw's algorithm
  9. Numerical solutions to linear Volterra integral equations
  10. Equations of the first kind
  11. Equations of the second kind
  12. Different limits of integration
  13. Examples and applications
  14. Set 1: Volterra integral equations of the first kind
  15. Set 2: Volterra integral equations of the second kind with oscillatory kernels
  16. ...and 5 more sections

Key Result

Lemma 5.1

The coefficient space Volterra integral operator $\mathrm{V}_K$ is compact, where $\mathrm{V}_K: \ell^2 \rightarrow \ell^2$ for a given kernel $K(x,y) \in L^2[T^2]$ with limits of integration $0$ to $x$ acting on the coefficient vector Banach space $\ell^2$ of the Jacobi polynomials $\tilde{\mathbf{ with the respective operators defined as in section sec:volterra.

Figures (5)

  • Figure 1: (A) shows absolute error between (\ref{['eq:set1_analyticintegral']}) and the known analytic solution while (B) compares (\ref{['eq:set1_involvedintegral']}) to a solution computed with $n=5050$.
  • Figure 2: Contour plots of oscillatory kernels for equations (\ref{['eq:set2K1']}--\ref{['eq:set2K3']}) on their natural triangle domains.
  • Figure 3: Absolute errors for equations (\ref{['eq:set2K1']}--\ref{['eq:set2K3']}). $u_1(x)$ is compared to the analytic solution, $u_2(x)$ and $u_3(x)$ are compared to a solution computed with $n=5050$.
  • Figure 4: Numerical and analytic solutions to the problem in (\ref{['eq:conductionexample']}).
  • Figure :

Theorems & Definitions (22)

  • Definition 5.1
  • Definition 5.2
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • proof
  • Lemma 5.3
  • proof
  • Lemma 5.4
  • Corollary 5.5
  • ...and 12 more