We present a sparse spectral method for nonlinear integro-differential Volterra equations based on the Volterra operator’s banded sparsity structure when acting on specific Jacobi polynomial bases. The method is not restricted to convolution-type kernels of the form K(x, y) = K(x − y) but instead works for general kernels at competitive speeds and with exponential convergence. We provide various numerical experiments based on an open-source implementation for problems with and without known analytic solutions and comparisons with other methods.

中文翻译：

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具有通用核的非线性积分-微分Volterra方程的快速稀疏谱方法

当对特定的Jacobi多项式基进行操作时，我们基于Volterra算子的带状稀疏结构，提出了一种非线性积分-微分Volterra方程的稀疏谱方法。该方法不限于形式为K（x，y）= K（x - y）的卷积型内核，而是适用于具有竞争力的速度和指数收敛的一般内核。我们提供基于开放源代码实现的各种数值实验，以解决有无已知解析解以及与其他方法进行比较的问题。