Chebyshev Wavelet Method for Numerical Solutions of Coupled Burgers' Equation

Year 2019, Volume: 48 Issue: 1, 1 - 16, 01.02.2019

Ö. Oruç F. Bulut A. Esen

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.

Keywords

Chebyshev wavelet method, Chebyshev collocation, coupled Burgers' equation, nonlinear phenomena, numerical solution

References

• R. Abazari and A. Borhanifar, Numerical study of the solution of the Burgers and coupled Burgers equations by a differential transformation method, Comput. Math. Appl. 59, 2711-2722, 2010.
• M.A. Abdou and A.A. Soliman, Variational iteration method for solving Burger’s and coupled Burger’s equations, J. Comput. Appl. Math. 181, 245-251, 2005.
• H. Adibi and P. Assari Chebyshev wavelet method for numerical solution of Fredholm integral equations of the first kind. Math. Prob. Eng. 2010, 1-17, 2010, doi:10.1155/2010/138408.
• E. Babolian and F. Fattahzadeh Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration, Appl. Math. Comput. 188, 417- 426, 2007.
• C. Basdevant, M. Deville, P. Haldenwang and J. M. Lacroix Spectral and finite difference solutions of the Burgers’ equation, Comput. Fluids 14, 23-41, 1986.
• F. Bulut, Ö. Oruç and A. Esen, Numerical Solutions of Fractional System of Partial Differential Equations By Haar Wavelets, Comput. Model. Eng. Sci. 108 (4), 263-284, 2015.
• I. Celik, Haar wavelet approximation for magnetohydrodynamic flow equations, Appl. Math. Model. 37, 3894-3902, 2013.
• I. Celik, Chebyshev Wavelet collocation method for solving generalized Burgers-Huxley equation, Math. Methods. Appl. Sci. 39 (3), 366-377, 2015.
• C. Chen and C.H. Hsiao, Haar wavelet method for solving lumped and distributed parameter systems, IEE Proc., Control Theory Appl. 144, 87-94, 1997.
• I. Daubechies, Ten Lectures on Wavelet, SIAM, Philadelphia, 1992.
• M. Dehghan, A. Hamidi and M. Shakourifar, The solution of coupled Burgers’ equations using Adomian-Pade technique, Appl. Math. Comput. 189, 1034-1047, 2007.
• S.E. Esipov, Coupled Burgers’ equations: a model of polydispersive sedimentation, Phys. Rev. Lett. 52, 3711-3718, 1995.
• A.K. Gupta and S. Saha Ray, Travelling wave solution of fractional KdV-Burger- Kuramoto equation describing nonlinear physical phenomena, AIP Adv. 4, 2014, doi: 10.1063/1.4895910.097120-1-11.
• M.H. Heydari, M.R. Hooshmandasl and F.M. Maalek Ghaini, A new approach of the Chebyshev wavelets method for partial differential equations with boundary conditions of the telegraph type, Appl. Math. Model. 38, 1597-1606, 2014.
• J. D. Hunter, Matplotlib: A 2D graphics environment, Comput. Sci. Eng. 9 (3), 90-95, 2007.
• I.E. Inan, D. Kaya and Y. Ugurlu, Auto-Bäcklund transformation and similarity reductions for coupled Burger’s equation, Appl. Math. Comput. 216, 2507-2511, 2010.
• S. Islam, S. Haq and M. Uddin, A mesh free interpolation method for the numerical solution of the coupled nonlinear partial differential equations, Eng. Anal. Bound. Elem. 33, 399-409, 2009.
• R. Jiwari, A Haar wavelet quasilinearization approach for numerical simulation of Burgers’ equation, Comput. Phys. Commun. 183, 2413-2423, 2012.
• D. Kaya, An explicit solution of coupled viscous Burgers’ equation by the decomposition method. Int. J. Math. Math. Sci. 27 (11), 675-680, 2001.
• A. Kelleci and A. Yıldırım, An efficient numerical method for solving coupled Burgers’ equation by combining homotopy perturbation and pade techniques, Numer. Methods Partial Differ. Equ. 27 (4), 982-995, 2011.
• A.H. Khater, R.S. Temsah and M.M. Hassan, A Chebyshev spectral collocation method for solving Burgers’-type equations, J. Comput. Appl. Math. 222 (2), 333-350, 2008.
• M. Kumar and S. Pandit, A composite numerical scheme for the numerical simulation of coupled Burgers’ equation, Comput. Phys. Commun. 185 (3), 809-817, 2014.
• S. Kutluay and Y. Ucar, Numerical solutions of the coupled Burgers’ equation by the Galerkin quadratic B-spline finite element method, Math. Methods Appl. Sci. 36 (17) 2403-2415, 2013.
• U. Lepik, Numerical solution of differential equations using Haar wavelets, Math. Comput. Simul. 68, 127-143, 2005.
• U. Lepik, Application of the Haar wavelet transform to solving integral and differential Equations, Proc. Estonian Acad. Sci. Phys. Math. 56 (1), 28-46, 2007.
• U. Lepik, Numerical solution of evolution equations by the Haar wavelet method, Appl. Math. Comput. 185, 695-704, 2007.
• U. Lepik, Solving PDEs with the aid of two-dimensional Haar wavelets, Comput. Math. Appl. 61, 1873-1879, 2011.
• R.C. Mittal and G. Arora, Numerical solution of the coupled viscous Burgers’ equation, Commun. Nonlinear Sci. Numer. Simul. 16, 1304-1313, 2011.
• R.C. Mittal, H. Kaur, and V. Mishra, Haar wavelet-based numerical investigation of coupled viscous Burgers’ equation, Int. J. Comput. Math. 92 (8), 1643-1659, 2015, doi: 10.1080/00207160.2014.957688
• J. Nee and J. Duan, Limit set of trajectories of the coupled viscous Burgers’ equations, Appl. Math. Lett. 11 (1), 57-61, 1998.
• Ö. Oruç, F. Bulut and A. Esen, A haar wavelet-finite difference hybrid method for the numerical solution of the modified Burgers’ equation, J. Math. Chem. 53 (7), 1592-1607, 2015.
• Ö. Oruç, F. Bulut and A. Esen, Numerical Solutions of Regularized Long Wave Equation By Haar Wavelet Method, Mediterr. J. Math. 13 (5), 3235-3253, 2016.
• A. Rashid, M. Abbas, A.I.Md. Ismail and A. Abd Majid, Numerical solution of the coupled viscous Burgers equations by Chebyshev-Legendre Pseudo-Spectral method, Appl. Math. Comput. 245, 372-381, 2014.
• A. Rashid and A.I.B.M. Ismail, A Fourier pseudospectral method for solving coupled viscous Burgers equations, Comput. Methods Appl. Math. 9 (4), 412-420, 2009.
• M. Razzaghi and S. Yousefi, Legendre wavelets direct method for variational problems, Math. Comput. Simul. 53, 185-192, 2000.
• M. Razzaghi and S. Yousefi, Legendre wavelets operational matrix of integration, Int. J. Syst. Sci. 32 (4), 49-502, 2001.
• Z. Rong-Pei, Y. Xi-Jun and Z. Guo-Zhong, Local discontinuous Galerki nmethod for solving Burgers and coupled Burgers equations, Chin. Phys. B. 20 (11), 110205-1-6, 2011.
• P.K. Sahu and S. Saha Ray, Legendre wavelets operational method for the numerical solutions of nonlinear Volterra integro-differential equations system, Appl. Math. Comput. 256, 715-723, 2015.
• P.K. Sahu and S. Saha Ray, Two dimensional Legendre wavelet method for the numerical solutions of fuzzy integro-differential equations, J. Intell. Fuzzy Syst. 28, 1271- 1279, 2015.
• Z. Shi, Y. Cao, and Q.J. Chen, Solving 2D and 3D Poisson equations and biharmonic equations by the Haar wavelet method, Appl. Math. Model. 36, 5143-5161, 2012.
• A.A. Soliman, The modified extended tanh-function method for solving Burgers-type equations, Physica A 361, 394-404, 2006.
• V.K. Srivastava, M.K. Awasthi and M. Tamsir, A fully implicit Finite-difference solution to one dimensional Coupled Nonlinear Burgers’ equations, Int. J. Math. Sci. 7 (4) 2013.
• O.V. Vasilyev and S. Paolucci, A Dynamically Adaptive Multilevel Wavelet Collocation Method for Solving Partial Differential Equations in a Finite Domain, J. Comput. Phys. 125, 498-512, 1996.
• Y.Wang and Q. Fan, The second kind Chebyshev wavelet method for solving fractional differential equations, Appl. Math. Comput. 218, 8592-8601, 2012.
• C. Yang and J. Hou, Chebyshev wavelets method for solving Bratu’ s problem, Bound. Value. Probl. 142, 1-9, 2013.
• F. Zhou and X. Xu, Numerical solution of the convection diffusion equations by the second kind Chebyshev wavelets, Appl. Math. Comput. 247, 353-367, 2014.
• L. Zhu and Q. Fan, Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet, Commun. Nonlinear Sci Numer. Simul. 17, 2333-2341, 2012.