A novel numerical method and its convergence for nonlinear delay Volterra integro‐differential equations

Abstract

In this paper, we suggest a convergent numerical method for solving nonlinear delay Volterra integro‐differential equations. First, we convert the problem into a continuous‐time optimization problem and then use a shifted pseudospectral method to discrete the problem. Having solved the last problem, we can achieve the pointwise and continuous approximate solutions for the main delay Volterra integro‐differential equations. Here, we analyze the convergence of the method and solve some numerical examples to show the efficiency of the method.

Numerical Solution of Reaction–Diffusion Equations with Convergence Analysis

“…We fully show that the approximate solutions are convergent to the exact solution when the number of collocation points tends to infinity. Note that spectral collocation methods have high accuracy and exponential convergence and, up to now many researchers utilized them to solve different continuous-time problems involving the ordinary and partial differential equations [12,13,18].…”

On the convergence of Jacobi‐Gauss collocation method for linear fractional delay differential equations

“…They reduce their equation to a system of algebraic equations. Mahmoudi et al 12 used spectral collocation methods to numerically solve nonlinear delay Volterra integro‐differential equations. They converted their problem into the continuous‐time optimization problem and then approximated the optimal solution by using a interpolating polynomial.…”