A Legendre–Petrov–Galerkin method for solving Volterra integro-differential equations with proportional delays. Issue 5 (4th May 2019) - Cai, Haotao; Chen, Yanping; Huang, Yunqing
ABSTRACT: In this paper, a fully discrete Legendre–Petrov–Galerkin method is presented for solving functional Volterra integro-differential equations with vanishing delays. This method produces a fully discrete linear system. We prove that this system has a unique solution for sufficiently large n, where n +1 denotes the order of the system. Moreover, we prove that the approximate solution and its corresponding derivative function arrive at an optimal convergence orderO ( n − m − 1 ) andO ( n − m ) inL 2 norm, where m +1 is the regularity of the original solution. In addition, we prove that for sufficiently large n, the spectral condition number of the corresponding linear system is uniformly bounded. Numerical examples are presented to demonstrate the effectiveness of the proposed method.