Numerical Solution of the Coupled Viscous Burgers’ Equation Using Differential Quadrature Method Based on Fourier Expansion Basis

Abstract

The differential quadrature method based on Fourier expansion basis is applied in this work to solve coupled viscous Burgers' equation with appropriate initial and boundary conditions. In the first step for the given problem we have discretized the interval and replaced the differential equation by the Differential quadrature method based on Fourier expansion basis to obtain a system of ordinary differential equation (ODE) then we implement the numerical scheme by computer programing and perform numerical solution. Finally the validation of the present scheme is demonstrated by numerical example and compared with some existing numerical methods in literature. The method is analyzed for stability and convergence. It is found that the proposed numerical scheme produces a good result as compared to other researcher's result and even generates a value at the nodes or mesh points that the results have not seen yet. 822Applied Mathematics proximate solution for problems which are difficult to solve analytically. The de-

New cubic B-spline approximation technique for numerical solutions of coupled viscous Burgers equations

“…where Y m and y m denote the approximate and exact solutions to the unknown functions at m th knot, respectively. The computational outcomes are obtained and compared with those methods already available in literature like the Chebyshev spectral collocation method (CSCM) (Khater et al, 2008), Fourier pseudo-spectral method (FPM) (Rashid and Ismail, 2009), cubic B-spline collocation method (CBSCM) (Mittal and Arora, 2011), differential quadrature method (DQM) (Mittal and Jiwari, 2012), CBS-based differential quadrature method (CBS-DQM) (Mittal and Tripathi, 2014), quintic B-spline collocation method (QBSCM) (Raslan et al, 2017), Fourier expansion basis differential quadrature method (FEB-DQM) (Jima et al, 2018), Chebyshev wavelets method (CVM) (Oruç et al, 2019) and septic B-spline collocation method (SpBSM) (Shallal et al, 2019).…”

“…Example 1: Consider the following system of non-linear CVBEs (Rashid and Ismail, 2009;Mittal and Arora, 2011;Mittal and Jiwari, 2012;Mittal and Tripathi, 2014;Raslan et al, 2017;Jima et al, 2018;Shallal et al, 2019):…”

“…Table 1 reports the maximum absolute error in the approximate solutions of u and v at time levels t = 0.1, 0.5, 1.0, 2.0 and 3.0 using M = 50. The comparison of computational outcomes with FPM (Rashid and Ismail, 2009), CBSCM (Mittal and Arora, 2011), DQM (Mittal and Jiwari, 2012), CBS-DQM (Mittal and Tripathi, 2014), QBSCM (Raslan et al, 2017), FEB-DQM (Jima et al, 2018) and SpBSM (Shallal et al, 2019) shows the superior accuracy of the presented method because of its new approximation for the second derivative. In Table 2, the error norm L 1 , at t = 1, is listed corresponding to various choices of time step.…”

“…Al-Jawary et al (2018) studied the exact and approximate solutions of one-, two-and three-dimensional (3D) non-linear Burgers' equations. Jima et al (2018) used a Fourier expansion basis to develop a differential quadrature method for the numerical approximation of CVBEs. Kutluay et al (2019) presented a Galerkin finite element method based on the Hopf-Cole transformation for solving a system of two-dimensional (2D) CVBEs.…”

Synchronization of the nonlinear advection‐diffusion‐reaction processes

“…These models are characterized by the reaction and diffusion or by the interaction between advection and diffusion. In recent years, many researchers have paid particular attention to solving these problems for both conceptual understanding of physical flows and testing various numerical methods 1‐8 . It is still crucial to further investigate such problems by reducing the computational difficulties in capturing numerical solutions and keeping the real features of nature at a low viscosity value at various free parameters.…”