Vol.124, No.3, 2020, pp.915-935, doi:10.32604/cmes.2020.011218
OPEN ACCESS
ARTICLE
A Differential Quadrature Based Approach for Volterra Partial Integro-Differential Equation with a Weakly Singular Kernel
• Siraj-ul-Islam1, Arshed Ali2,*, Aqib Zafar1, Iltaf Hussain1
1 Department of Basic Sciences and Islamiat, University of Engineering and Technology Peshawar, Khyber Pakhtunkhwa, Pakistan.
2 Department of Mathematics, Islamia College Peshawar, Khyber Pakhtunkhwa, Pakistan.
* Corresponding Author: Arshed Ali. Email: arshad.ali@icp.edu.pk.
Received 27 April 2020; Accepted 16 June 2020; Issue published 21 August 2020
Abstract
Differential quadrature method is employed by numerous researchers due to its numerical accuracy and computational efficiency, and is mentioned as potential alternative of conventional numerical methods. In this paper, a differential quadrature based numerical scheme is developed for solving volterra partial integro-differential equation of second order having a weakly singular kernel. The scheme uses cubic trigonometric B-spline functions to determine the weighting coefficients in the differential quadrature approximation of the second order spatial derivative. The advantage of this approximation is that it reduces the problem to a first order time dependent integro-differential equation (IDE). The proposed scheme is obtained in the form of an algebraic system by reducing the time dependent IDE through unconditionally stable Euler backward method as time integrator. The scheme is validated using a homogeneous and two nonhomogeneous test problems. Conditioning of the system matrix and numerical convergence of the method are analyzed for spatial and temporal domain discretization parameters. Comparison of results of the present approach with Sinc collocation method and quasi-wavelet method are also made.
Keywords
Partial integro-differential equation, differential quadrature, cubic trigonometric B-spline functions, weakly singular kernel.