Numerical solution of Volterra partial integro-differential equations based on sinc-collocation method

Atefeh Fahim,Mohammad Ali Fariborzi Araghi,Mehdi Jalalvand,Jalil Rashidinia

doi:10.1186/s13662-017-1416-7

Abstract

We provide the numerical solution of a Volterra integro-differential equation of parabolic type with memory term subject to initial boundary value conditions. Finite difference method in combination with product trapezoidal integration rule is used to discretize the equation in time and sinc-collocation method is employed in space. A weakly singular kernel has been viewed as an important case in this study. The convergence analysis has been discussed in detail, which shows that the approach exponentially converges to the solution. Furthermore, numerical examples and illustrations are presented to prove the validity of the suggested method.

Highlights

IntroductionConstruction of precise numerical methods for integro-differential equations is still a challenge owing to the weak singularity of the kernel k that contains sharp states of transitions in the solution
We consider a Volterra integro-differential equation with memory term of the form t ut(x, t) = k (t – s)uxx(x, s) ds + f (x, t), x ∈, t ∈ J, ( )subjected to initial and boundary conditions u(a, t) = u(b, t) =, t ∈ J, ( )u(x, ) = u (x), x ∈, where = [a, b] ⊆ R and J [, T]
We extend the sinc-collocation method for solving partial integro-differential equations of type ( )

Summary

Introduction

Construction of precise numerical methods for integro-differential equations is still a challenge owing to the weak singularity of the kernel k that contains sharp states of transitions in the solution. This lack of smoothness of the solution near t = results in a decay in the order of the practical performance of familiar timestepping methods for equation ( ). The sinc approximation has been studied by many authors to solve various equations such as integral equations [ ], ordinary differential equations [ ], partial differential equations [ – ], integro-differential equations [ ], and so on, due to high accuracy, exponential rate of convergence, and near optimality of this method [ ]. In Section , we develop the sinc collocation method to solve Volterra partial integro-differential equations. The sinc-collocation algorithm is described for solving equation ( )

Discretization in time

Numerical results

Conclusions