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.

中文翻译：

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基于三角形正交多项式的Volterra积分方程的稀疏谱方法

我们介绍并分析了使用三角形域上的二元正交多项式求解 Volterra 积分方程的稀疏谱方法。Volterra 算子在加权 Jacobi 基础上的稀疏性用于实现高效率和指数收敛。讨论之后是使用已知解析解的第一类和第二类示例 Volterra 积分方程的方法演示以及面向应用的数值实验。我们证明了第一类和第二类问题的收敛性，前者建立在与 Toeplitz 算子的联系上。