On the Approximate Solution of Partial Integro-Differential Equations Using the Pseudospectral Method Based on Chebyshev Cardinal Functions

Fairouz Tchier,Ioannis Dassios,Ferdous Tawfiq,Lakhdar Ragoub

doi:10.3390/math9030286

Abstract

In this paper, we apply the pseudospectral method based on the Chebyshev cardinal function to solve the parabolic partial integro-differential equations (PIDEs). Since these equations play a key role in mathematics, physics, and engineering, finding an appropriate solution is important. We use an efficient method to solve PIDEs, especially for the integral part. Unlike when using Chebyshev functions, when using Chebyshev cardinal functions it is no longer necessary to integrate to find expansion coefficients of a given function. This reduces the computation. The convergence analysis is investigated and some numerical examples guarantee our theoretical results. We compare the presented method with others. The results confirm the efficiency and accuracy of the method.

Highlights

In this paper, we apply the pseudospectral method based on Chebyshev cardinal functions to solve one-dimensional partial integro-differential equations (PIDEs)
The pseudospectral methods are a special type of numerical method that used scientific computing and applied mathematics to solve partial differential equations
While introducing the Chebyshev cardinal functions, the pseudospectral method applies to obtain the approximate solution of PIDEs (1)

Summary

Introduction

We apply the pseudospectral method based on Chebyshev cardinal functions to solve one-dimensional partial integro-differential equations (PIDEs). The Galerkin method is a class of spectral techniques that convert a continuous operator problem to a discrete problem In other words, this scheme applies the method of variation of parameters to function space by transforming the equation to a weak formulation. The pseudospectral methods are a special type of numerical method that used scientific computing and applied mathematics to solve partial differential equations. We introduce a simple numerical method with high accuracy To this end, while introducing the Chebyshev cardinal functions, the pseudospectral method applies to obtain the approximate solution of PIDEs (1).

Chebyshev Cardinal Functions

Pseudospectral Method

Convergence Analysis

Proposed Method

Conclusions