卷积型Volterra积分微分方程的一种快速算法
A Fast Algorithm for Convolutional Volterra Integral Differential Equations
摘要: 卷积型Volterra积分微分方程是一类重要问题，广泛应用于生物学、经济学，本文研究了一种快速算法求解卷积型Volterra积分微分方程。该方法采用多步配置方法对卷积型Volterra积分微分方程进行离散，结合卷积核的特性，得到离散方程的线性系统由Toeplitz矩阵、对角矩阵和稀疏矩阵组合而成。考虑Toeplitz矩阵与向量的快速算法，设计系数矩阵与向量的快速计算格式。本文利用GMRES算法与快速计算格式结合，获得一种快速求解线性系统的改进算法，并通过实验验证改进算法的高效性。
Abstract: Convolutional Volterra integral differential equations are an important class of problems, widely used in biology, economics, among other fields. This study presents a fast algorithm for solving convolutional Volterra integral differential equations. The method involves discretizing the equations using a multi-step collocation approach, combined with the characteristics of the convolution kernels, resulting in a linear system of equations composed of Toeplitz matrices, diagonal matrices, and sparse matrices. Considering fast algorithms for Toeplitz matrices and vectors, a fast computation format for the coefficient matrix and vector is designed. By combining the GMRES algorithm with the fast computation format, an improved algorithm for efficiently solving linear systems is obtained. Experimental results verify the effectiveness of the improved algorithm.
文章引用：李海洋, 胡怀青, 刘婧雅. 卷积型Volterra积分微分方程的一种快速算法[J]. 运筹与模糊学, 2023, 13(5): 5571-5580. https://doi.org/10.12677/ORF.2023.135556

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