A Differential Quadrature Based Approach for Volterra Partial
Integro-Differential Equation with a Weakly Singular Kernel

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.

Recently, Korkmaz et al. [Korkmaz and Akmaz (2018)] introduced the CTBS-DQ method for the solution of non-conservative linear transport problems based on second order advection-diffusion equation. Singh et al. [Singh and Kumar (2018)] extended the method for the solution of one and two dimensional coupled viscous Burgers' equations. Tasmir et al. [Tamsir, Dhiman and Srivastava (2018)] used the CTBS-DQ method for second order nonlinear Fisher's reaction-diffusion equations.
The aim of this work is to initiate the CTBS-DQ approach for the solution of PDIEs (1)-(3). Besides other challenges in solving PDEs, PIDEs of the form (1) have additional two major issues, the singular kernel and the memory term. These issues also require numerical treatment, which further affect stability and accuracy of a numerical method.
Rest of the paper is outlined as follows: Section 2 describes development of the proposed technique by coupling CTBS functions with differential quadrature method. Section 3 implements the CTBS-DQ method using test problems. This section also provides detail error analysis, conditioning, eigenvalues, comparison with some existing methods, and computational efficiency in order to establish the current approach. Section 4 concludes the findings and outcomes of the paper.
The DQ method approximates kth order derivative of the function u(x,r) from its values at where a k ð Þ ij , k=1, 2, …, are kth order weighting coefficients which are determined by test functions. Different basis functions were considered as test functions for determination of the weighting coefficients such as cubic trigonometric B-spline functions [Korkmaz and Akmaz (2018); Singh and Kumar (2018)], Lagrange polynomials [Quan and Chang (1989)], quintic B-spline functions [Mittal and Dahiya (2016)], sinc functions [Korkmaz and Dag (2011b)], cubic B-spline functions [Arora and Singh (2013); Bashan (2019b)], radial basis functions [Lin, Zhao, Watson et al. (2020); Shu, Ding and Yeo (2003)], Fourier expansion [Shu and Chew (1997)], and polynomial basis [Korkmaz and Dag (2011a)]. Accuracy of DQ solution depends on the accuracy of weighting coefficients and as well as on the selection of nodal points x i . Moreover, the DQ solution declines with increasing the number of nodes which is a limitation of this method. Various researchers have provided different techniques such as using explicit formulae for computation of weighting coefficients of higher order derivatives and non-uniform nodes to circumvent this problem, which leads to improvement in accuracy of the DQ solution [Bert and Malik (1996)]. Shu [Shu (2000)] established two approaches (i) a recurrence formula (ii) matrix multiplication, for finding the weighting coefficients of derivatives of higher order. Recently, Lin et al. [Lin, Zhao, Watson et al. (2020)] presented an improved radial basis functions based DQ method using ghost points for the solution of 2D and 3D elliptic boundary value problems. CTBS function denoted by B i (x) are given by Abd Hamid et al. [Abd Hamid, Majid and Ismail (2010)] and Abbas et al. [Abbas, Majid, Ismail et al. (2014a)]: The following modified CTBS functions are used as test functions [Arora and Joshi (2018)]: T 1 ðxÞ¼ðB 1 þ2B 0 ÞðxÞ; T 2 ðxÞ¼ðB 2 ÀB 0 ÞðxÞ; T l ðxÞ¼B l ðxÞ; for l¼3; 4; …; NÀ2; T NÀ1 ðxÞ¼ðB NÀ1 ÀB Nþ1 ÞðxÞ; which form a basis in region [a, b].

Problem 2
We consider Eqs.

Problem 3
In this test problem we take Eqs.