Chebyshev Wavelet Method for Numerical Solutions of Coupled Burgers Equation

Abstract

This paper deals with the numerical solutions of one dimensional time dependent coupled Burgers' equation with suitable initial and boundary conditions by using Chebyshev wavelets in collaboration with a collocation method. The proposed method converts coupled Burgers' equations into system of algebraic equations by aid of the Chebyshev wavelets and their integrals which can be solved easily with a solver. Benchmarking of the proposed method with exact solution and other known methods already exist in the literature is made by three test problems. The feasibility of the proposed method is demonstrated by test problems and indicates that the proposed method gives accurate results in short cpu times. Computer simulations show that the proposed method is computationally cheap, fast and quite good even in the case of less number of collocation points.

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).…”

“…The error norms L 1 and L 2 for Example 3, corresponding to different choices of spatial grid size at t = 1, by setting Dt = 0.01 and l = 0.1,0.4, are listed in Table 7. In Tables 8 and 9, a comparison of computational outcomes with CVM (Oruç et al, 2019) is presented for different values of M at t = 3 and 5 using fixed Dt and l . It is revealed that our commuted results exhibit a better agreement with the analytical exact solution as compared to CVM.…”

“…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. Oruç et al (2019) used the Chebyshev wavelets method (CVM) to solve 1D timedependent coupled Burgers' equations. Angstmann et al (2020) proposed an explicit numerical approach for solving non-linear Burgers' equations.…”

An efficient numerical scheme based on Lucas polynomials for the study of multidimensional Burgers-type equations

“…Oruc et al [25,26] applied a unified finite difference Chebyshev wavelet approach for time fractional Burger equations. The same authors studied the Chebyshev wavelet method for approximation of coupled Burgers equations [27]. Ali et al [4] applied a meshfree collocation method based on the Crank-Nicolson method for time discretization and radial basis function for space discretization to solve two-dimensional coupled Burger equations, whereas the meshless method of radial basis functions (RBFs) and local RBFs were described in [28,29] for approximate solutions of Burger-type equations.…”

New Analytical Solutions for Time-Fractional Kolmogorov-Petrovsky-Piskunov Equation with Variety of Initial Boundary Conditions

“…A.K. Gupta and S.S. Ray studied the solution of fractional fifth-order Sawada-Kotera equation using second kind Chebyshev wavelet method [14] and many others [16][17][18][19].…”