An accurate Chebyshev pseudospectral scheme for multi-dimensional parabolic problems with time delays

Ali H Bhrawy,Fouad Mallawi,Mohamed A Abdelkawy

doi:10.1186/s13661-015-0364-y

Abstract

In this paper, the Chebyshev Gauss-Lobatto pseudospectral scheme is investigated in spatial directions for solving one-dimensional, coupled, and two-dimensional parabolic partial differential equations with time delays. For the one-dimensional problem, the spatial integration is discretized by the Chebyshev pseudospectral scheme with Gauss-Lobatto quadrature nodes to provide a delay system of ordinary differential equations. The time integration of the reduced system in temporal direction is implemented by the continuous Runge-Kutta scheme. In addition, the present algorithm is extended to solve the coupled time delay parabolic equations. We also develop an efficient numerical algorithm based on the Chebyshev pseudospectral algorithm to obtain the two spatial variables in solving the two-dimensional time delay parabolic equations. This algorithm possesses spectral accuracy in the spatial directions. The obtained numerical results show the effectiveness and highly accuracy of the present algorithms for solving one-dimensional and two-dimensional partial differential equations.

Highlights

In recent years there has been a high level of interest in employing spectral methods for numerically solving many types of integral and differential equations, due to their high accuracy and ease of applying them for finite and infinite domains [ – ]
The spatial direction is collocated at (N – ) collocation points of the Chebyshev Gauss-Lobatto quadrature nodes. This scheme has the advantage of reducing the one-dimensional parabolic partial differential equations (PDEs) into a system of (N – ) ordinary differential equations (ODEs) with time delays in the time direction, that can be solved by continuous Runge-Kutta (RK) method
We extend the application of this algorithm to solve the coupled time delay parabolic equations at (N – ) collocation nodes, which provides a system of ( N – ) ODEs with time delays

Summary

Introduction

In recent years there has been a high level of interest in employing spectral methods for numerically solving many types of integral and differential equations, due to their high accuracy and ease of applying them for finite and infinite domains [ – ]. The spatial direction is collocated at (N – ) collocation points of the Chebyshev Gauss-Lobatto quadrature nodes This scheme has the advantage of reducing the one-dimensional parabolic PDEs into a system of (N – ) ODEs with time delays in the time direction, that can be solved by continuous Runge-Kutta (RK) method. This algorithm is developed to solve the two-dimensional time delay parabolic equations, in which the two spatial variables are collocated at (N – ) × (M – ) Chebyshev Gauss-Lobatto quadrature nodes This provides a system of (N – ) × (M – ) ODEs with time delays. Applying the Chebyshev pseudospectral approximation [ , ] for the coupled parabolic PDEs with discrete time delay ( ), at the nodes of Chebyshev-Gauss-Lobatto quadrature.

The matrix formulation of the previous systems is

Assume that the absolute error is given by