We introduce and analyze 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 kinds with or without 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.


  1. Volterra integral equations
  2. spectral methods
  3. sparse operators
  4. orthogonal polynomials

MSC codes

  1. 65N35
  2. 45D05

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Information & Authors


Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 1993 - 2018
ISSN (online): 1095-7170


Submitted: 10 June 2019
Accepted: 20 April 2020
Published online: 29 June 2020


  1. Volterra integral equations
  2. spectral methods
  3. sparse operators
  4. orthogonal polynomials

MSC codes

  1. 65N35
  2. 45D05



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