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

## TL;DR

This paper presents a sparse spectral method leveraging bivariate orthogonal polynomials on a triangle for efficiently solving Volterra integral equations with proven exponential convergence.

## Contribution

The paper introduces a novel sparse spectral approach using orthogonal polynomials on a triangle domain for Volterra equations, demonstrating high efficiency and convergence.

## Key findings

- Achieves exponential convergence in solving Volterra equations
- Demonstrates effectiveness on first and second kind equations
- Proves convergence leveraging Toeplitz operator connections

## 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.