Application of the Generalized Differential Quadrature Method in Solving Burgers' Equations

Abstract

The aim of this paper is to obtain numerical solutions of the one-dimensional, two-dimensional and coupled Burgers' equations through the generalized differential quadrature method (GDQM). The polynomial-based differential quadrature (PDQ) method is employed and the obtained system of ordinary differential equations is solved via the total variation diminishing Runge—Kutta (TVD-RK) method. The numerical solutions are satisfactorily coincident with the exact solutions. The method can compete against the methods applied in the literature.

Septic B-spline collocation method for numerical solution of the coupled Burgers’ equations

“…Spline functions theory is very active field of approximate theory in partial differential equations. Many researchers have proposed numerical solution for the nonlinear equations including Burger equations, such as, Galerkin B-Spline-collocation method (Bryan et al, 2017), exponential cubic B-spline differential quadrature method (Korkmaz and Akmaz, 2015), trigonometric cubic B-spline differential quadrature method (Korkmaz and Akmaz, 2018), cubic Bspline collocation method (Sharifi and Rashidinia, 2016), B-spline collocation and self-adapting differential evolution (jDE) algorithm (Luo et al, 2018), fourth-order cubic B-spline collocation method (Rohila and Mittal, 2018), cubic B-spline collocation scheme (Mittal and Arora, 2011), non-polynomial spline method (Ali et al, 2015), collocation method with cubic trigonometric B-spline (Raslan et al, 2016), collocation method with quintic B-spline method (Raslan et al, 2017), generalized differential quadrature method (Mokhtari et al, 2011), exponential cubic B-spline finite element method (Ersoy and Dag, 2015), B-spline Differential Quadrature Method (Bashan et al, 2015), and the Galerkin quadratic Bspline finite element method (Kutluay and Ucar, 2013). The septic B-spline approach has been used to establish approximate solutions for several partial differential equations (Ramadan et al, 2005;El-Danaf, 2008;Soliman and Hussien, 2005;Quarteroni et al, 2007;Karakoc and Zeybek, 2016;Geyikli and Karakoc, 2011).…”

Numerical simulation of the coupled viscous Burgers equation using the Hermite formula and cubic B-spline basis functions

“…Further numerical solution of the CVBE using cubic B-spline functions is obtained by Mittal and Arora [12]. Mokhtari et al [13] presented generalized differential quadrature method for the Burgers' equation. Srivastava et al [14][15][16] proposed various finite difference methods for the two dimensional CVBE.…”

A quartic trigonometric tension b-spline algorithm for nonlinear partial differential equation system

“…The difficulty in the numerical method of CBE arises due to the nonlinear terms and viscosity parameters. Many numerical techniques have constructed to get the solution of the CBE fourth order accurate compact ADI scheme (Radwan, 1999), spectral collocation method using chebyshev polynomials (Khater et al , 2008), the Fourier pseudospectral method (Rashid and Ismail, 2009), the differential transform method via Taylor series formula (Liu and Hou, 2011), generalized differential quadrature method (Mokhtari et al , 2011), a robust technique for solving optimal control of CBE (Sadek and Kucuk, 2011), a differential quadrature method (Mittal and Jiwari, 2012), B-spline finite element method (Kutluay and Ucar, 2013; Mittal and Arora, 2011; Mittal and Tripathi, 2014; Raslan et al , 2016; Shallal et al , 2019; Onarcan and Hepson, 2018), a mesh free interpolation method (Islam et al , 2009), a fully implicit finite-difference method (Srivastava et al , 2013), a composite numerical scheme based on finite difference (Kumar and Pandit, 2014), logarithmic finite-difference method (Srivastava et al , 2014). Also, finite difference and differential quadrature methods (Bashan et al , 2015; Bashan, 2020; Ersoy et al , 2018; Karakoc et al , 2014; Ucar et al , 2019) are applied to Burgers and modified Burgers equations.…”