Numerical Solution of the Burgers' Equation Based on Sinc Method

Subject Areas : Applied Mathematics

Ghasem Kazemi Gelian ¹ , Mohammad Ali Fariborzi Araghi ²

1 - Department of Mathematics, Shirvan Branch, Islamic Azad University, Shirvan, Iran
2 - Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran.

Received: 2018-12-05 Accepted : 2019-03-04 Published : 2019-05-01

Keywords: ‎Single‎ ‎Exponential Transformation, Burgers' Equation‎, ‎Sinc method‎,

Abstract :

Burgers' equation arises in various areas of applied mathematics,‎‎such as modeling of dynamics‎, ‎heat conduction‎, ‎and acoustic‎‎waves‎ Also‎, ‎this equation has a large variety of applications in‎‎the modeling of water in unsaturated soil‎, ‎dynamics of soil‎‎water‎, ‎models of traffic‎, ‎turbulence and fluid flow‎, ‎mixing and‎‎turbulent diffusion. Many researchers tried to find analytic and numerical solutions of‎‎ this equation by different methods.Sinc method is a powerful numerical tool for finding fast and‎‎accurate solution in various areas of problems.‎In this paper‎, ‎numerical solution of Burgers' equation‎‎is considered by applying Sinc method‎. ‎For this purpose‎, ‎we apply‎‎Sinc method in cooperative with a classic finite difference‎‎formula to‎ ‎Burgers'equation‎. ‎‎The purpose of this paper is to extend the application of the‎‎sinc method for solving Burgers'equation by considering stability‎‎analysis of the method.‎ Numerical examples are provided to verify the validity of proposed method

References:

[1] S. Abbasbandy, M. T. Darvishi, A numerical solution of Burgers'equation
by modied Adomian method, Appl. Math. Comput, 163 (2005)1265-1272.

[2] M. A. Abdou, A. A. Soliman, Variational iteration method for solving
Burgers and coupled Burgers' equations, J. Comput. Appl. Math. 181
(2005) 245-251.

[3] R. Bahadir, M. Saglam, A mixed nite dierence and boundary element

approach to one-dimensional Burgers' equation, Appl. Math. Comput, 160
(2005)663-673.

[4] J. M. Burgers, A mathematical model illustrating the theory of turbulence,

Adv. Appl. Mech, 1 (1948) 171-199.

[5] E. Hopf, The partial dierential equation Ut + UUx = cUxx, Commun.
Pure Appl. Math, 3 (1950) 201-230.

[6] D. Khojasteh Salkuyeh, F. Saadati Sharafeh, On the numerical solution
of the Burgers' equation, International Journal of Computer Mathematics
86 (2009)1334-1344.

[7] S. Kutluaya, A. Esena, I. Dagb, Numerical solutions of the Burgers'
equation by the least-squares quadratic B-spline nite element method,
J. Comput.Appl. Math, 167 (2004) 21-33.

[8] J. Lund, K. Bowers, Sinc Methods for Quadrature and Dierential
Equations, SIAM, Philadelphia, 1992.[9]Z. S. Lu, An explicit Backlund transformation of Burgers' equation with

[9] Z. S. Lu, An explicit Backlund transformation of Burgers' equation with
applications, Commun. Theor. Phys, 44 (2005) 987-989.

[10]V. A. Matveev, M. A. Salle, Darboux Transformation and Solitons,
Springer-Verlag, Berlin, Heidelberg, 1991.

[11] Ozis, E. N. Aksan, A nite element approach for solution of
Burgers'equation, Appl. Math. Comput, 139 (2003) 417-428.

[12]A. A. Soliman, The modied extended tanh-function method for solving
Burgers-type equations, Physica A, 361 (2006) 394-404.

[13] Siraj-ul-Islam, Sirajul Haq, Arshed Ali, Meshfree method for the numerical
solution of the RLW equation, J. Comput. Appl. Math, 223 (2009) 9971012.

[14]F. Stenger, Numerical Methods Based on Sinc and Analytic Functions,
Springer, 1993.

[15] J. W. Thomas, Numerical Partial Dierential Equations: Finite Dierence
Methods, Springer-Verlag, New York, 1995.

[16] A. M. Wazwaz, The tanh and the sine-cosine methods for a reliable
treatment of the modied equal width equation and its variants, Commun,
Nonlinear Sci. Numer. Simul, 11 (2006) 148-160.