Open Access

ARTICLE

A Comparative Numerical Study of Parabolic Partial Integro-Differential Equation Arising from Convection-Diffusion

Kamil Khan¹, Arshed Ali^1,*, Fazal-i-Haq², Iltaf Hussain³, Nudrat Amir⁴
1 Department of Mathematics, Islamia College, Peshawar, 25000, Pakistan
2 Department of Mathematics, Statistics and Computer Science, The University of Agriculture, Peshawar, 25000, Pakistan
3 Department of Basic Sciences & Islamiat, University of Engineering & Technology, Peshawar, 25000, Pakistan
4 Department of Basic Sciences & Humanities, CECOS University of Information Technology and Emerging Sciences, Peshawar, 25000, Pakistan
* Corresponding Author: Arshed Ali. Email:

Computer Modeling in Engineering & Sciences 2021, 126(2), 673-692. https://doi.org/10.32604/cmes.2021.012730

Received 10 July 2020; Accepted 10 November 2020; Issue published 21 January 2021

Abstract

This article studies the development of two numerical techniques for solving convection-diffusion type partial integro-differential equation (PIDE) with a weakly singular kernel. Cubic trigonometric B-spline (CTBS) functions are used for interpolation in both methods. The first method is CTBS based collocation method which reduces the PIDE to an algebraic tridiagonal system of linear equations. The other method is CTBS based differential quadrature method which converts the PIDE to a system of ODEs by computing spatial derivatives as weighted sum of function values. An efficient tridiagonal solver is used for the solution of the linear system obtained in the first method as well as for determination of weighting coefficients in the second method. An explicit scheme is employed as time integrator to solve the system of ODEs obtained in the second method. The methods are tested with three nonhomogeneous problems for their validation. Stability, computational efficiency and numerical convergence of the methods are analyzed. Comparison of errors in approximations produced by the present methods versus different values of discretization parameters and convection-diffusion coefficients are made. Convection and diffusion dominant cases are discussed in terms of Peclet number. The results are also compared with cubic B-spline collocation method.

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Keywords

Partial integro-differential equation; convection-diffusion; collocation method; differential quadrature; cubic trigonometric B-spline functions; weakly singular kernel

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