article

A Chebyshev spectral collocation method for solving Burgers'-type equations

Authors:

M. M. HassanAuthors Info & Claims

Journal of Computational and Applied Mathematics, Volume 222, Issue 2

Pages 333 - 350

https://doi.org/10.1016/j.cam.2007.11.007

Published: 20 December 2008 Publication History

Abstract

In this paper, we elaborated a spectral collocation method based on differentiated Chebyshev polynomials to obtain numerical solutions for some different kinds of nonlinear partial differential equations. The problem is reduced to a system of ordinary differential equations that are solved by Runge-Kutta method of order four. Numerical results for the nonlinear evolution equations such as 1D Burgers', KdV-Burgers', coupled Burgers', 2D Burgers' and system of 2D Burgers' equations are obtained. The numerical results are found to be in good agreement with the exact solutions. Numerical computations for a wide range of values of Reynolds' number, show that the present method offers better accuracy in comparison with other previous methods. Moreover the method can be applied to a wide class of nonlinear partial differential equations.

References

[1]

Abdou, M.A. and Soliman, A.A., Variational iteration method for solving Burgers and coupled Burgers equations. J. Comput. Appl. Math. v181 i2. 245-251.

Digital Library

[2]

Adomian, G., The diffusion-Brusselator equation. Comput. Math. Appl. v29. 1-3.

[3]

Bahadir, A.R., A fully implicit finite-difference scheme for two-dimensional Burgers' equations. Appl. Math. Comput. v137. 131-137.

Digital Library

[4]

Burger, J.M., A mathematical model illustrating the theory of turbulence. Adv. Appl. Mech. I. 171-199.

[5]

Clenshaw, C.W. and Curtis, A.R., A method for numerical integration on an automatic computer. Numer. Math. v2. 197-205.

Digital Library

[6]

Cole, J.D., On a quasilinear parabolic equations occurring in aerodynamics. Quart. Appl. Math. v9. 225-236.

[7]

Debtnath, L., Nonlinear Partial Differential Equations for Scientist and Engineers. 1997. Birkhauser, Boston.

Digital Library

[8]

Elbarbary, E.M.E. and El-Kady, M., Chebyshev finite difference approximation for the boundary value problems. Appl. Math. Comput. v139. 513-523.

Digital Library

[9]

El-Sayed, S.M. and Kaya, D., On the numerical solution of the system of two-dimensional Burgers' equations by the decomposition method. Appl. Math. Comput. v158. 101-109.

[10]

Fletcher, C.A.J., Generating exact solutions of the two-dimensional Burgers' equation. Int. J. Numer. Methods Fluids. v3. 213-216.

[11]

Goyon, O., Multilevel schemes for solving unsteady equations. Int. J. Numer. Methods Fluids. v22. 937-959.

[12]

Helal, M.A. and Mehanna, M.S., A comparison between two different methods for solving KdV-Burgers equation. Chaos Solitons Fractals. v28. 320-326.

[13]

Ibrahim, M.A.K. and Temsah, R.S., Spectral methods for some singularly perturbed problems with initial and boundary layers. Int. J. Comput. Math. v25. 33-48.

[14]

Jain, P.C. and Holla, D.N., Numerical solution of coupled Burgers Δ equations. Int. J. Numer. Methods Eng. v12. 213-222.

[15]

Kaya, D., An application of the decomposition method for the KdVB equation. Appl. Math. Comput. v152. 279-288.

[16]

Kaya, D., An explicit solution of coupled viscous burgers equation by the decomposition method. Int. J. Math. Math. Sci. v27. 675-680.

[17]

Khater, A.H., Callebaut, D.K. and El-Kalaawy, O.H., Backlund transformations and exact solutions for a nonlinear elliptic equation modelling isothermal magnetostatic atmosphere. IMA J. Appl. Math. v65. 97-108.

[18]

Khater, A.H., Callebaut, D.K., Malfliet, W. and Kamel, A.S., Backlund transformations and exact solutions for some nonlinear evolution equations in solar magnetostatic models. J. Comput. Appl. Math. v140. 435-467.

Digital Library

[19]

Khater, A.H., Callebaut, D.K. and Sayed, S.M., Exact solutions for some nonlinear evolution equations which describe pseudo-spherical surfaces. J. Comput. Appl. Math. v189. 387-411.

Digital Library

[20]

Khater, A.H., Callebaut, D.K. and Sayed, S.M., Conservation laws for some nonlinear evolution equations which describe pseudo-spherical surfaces. J. Geom. Phys. v51. 332-352.

[21]

Khater, A.H., Callebaut, D.K. and Seadawy, A.R., General soliton solutions of an n-dimensional complex Ginzburg-Landau equation. Phys. Scripta A. v62. 353-357.

[22]

Khater, A.H., Ibrahim, R.S., Shamardan, A.B. and Callebaut, D.K., Backlund transformations and Painleve analysis: Exact solutions for a Grad-Shafranov-type magnetohydrodynamic equilibrium. IMA J. Appl. Math. v58. 51-69.

[23]

Khater, A.H. and Hassan, A.A., Travelling and periodic wave solutions of some nonlinear wave equations. Z. Naturforsch. A. v59. 389-396.

[24]

Khater, A.H., Hassan, A.A. and Temsah, R.S., Exact solutions with Jacobi elliptic functions of two nonlinear models for ion-acoustic plasma waves. J. Phys. Soc. Jpn. v74. 1431-1435.

[25]

Khater, A.H., Hassan, A.A. and Temsah, R.S., Cnoidal wave solutions for a class of fifth-order KdV equations. Math. Comput. Simulation. v70. 221-226.

Digital Library

[26]

Khater, A.H., Helal, M.A. and El-Kalaawy, O.H., Backlund transformations: Exact solutions for the KdV and the Calogero-Degasperis-Fokas mKdV equations. Math. Methods Appl. Sci. v21. 719-731.

[27]

Khater, A.H., Malfliet, W., Callebaut, D.K. and Abdul-Aziz, S.F., On the similarity solutions of the inhomogeneous nonlinear diffusion equation via symmetry method. J. Phys. Soc. Jpn. v73. 42-48.

[28]

Khater, A.H., Malfliet, W., Callebaut, D.K. and Kamel, E.S., The tanh method, a simple transformation and exact analytical solutions for nonlinear reaction-diffusion equations. Chaos Solitons Fractals. v14. 513-522.

[29]

Khater, A.H., Malfliet, W., Callebaut, D.K. and Kamel, E.S., Travelling wave solutions of some classes of nonlinear evolution equations in (1+1) and (2+1) dimensions. J. Comput. Appl. Math. v140. 469-477.

Digital Library

[30]

Khater, A.H., Malfliet, W. and Kamel, E.S., Travelling wave solutions of some classes of nonlinear evolution equations in (1+1) and higher dimensions. Math. Comput. Simulation. v64. 247-258.

Digital Library

[31]

Khater, A.H., Malfliet, W., Moussa, M.H.M. and Abdul-Aziz, S.F., On the similarity solutions of the Calogero-Degasperis-Fokas modified KdV equation via symmetry method. J. Phys. Soc. Jpn. v72. 2523-2529.

[32]

Khater, A.H., Moussa, M.H.M. and Abdul-Aziz, S.F., Potential symmetries and invariant solutions for the inhomogeneous nonlinear diffusion equation. Physica A. v312. 99-108.

[33]

Khater, A.H., Moussa, M.H.M. and Abdul-Aziz, S.F., Potential symmetries and invariant solutions for the generalized one-dimensional Fokker-Planck equation. Physica A. v312. 395-408.

[34]

Khater, A.H., Shamardan, A.B., Farag, M.H. and Abel-Hamid, A.H., Analytical and numerical solutions of a quasilinear parabolic optimal control problem. J. Comput. Appl. Math. v95. 29-43.

Digital Library

[35]

Khater, A.H., Shamardan, A.B. and Ibrahim, R.S., Chebyshev spectral collocation methods for nonlinear isothermal magnetostatic atmospheres. J. Comput. Appl. Math. v115. 309-329.

Digital Library

[36]

Khater, A.H. and Temsah, R.S., Numerical solutions of some nonlinear evolution equations by Chebyshev spectral collocation methods. Inter. J. Comput. Math. v84. 305-316.

Digital Library

[37]

Logan, J.D., An Introduction to Nonlinear Partial Differential Equations. 1994. Wiley-Interscience, New York.

[38]

Özis, T., Esen, A. and Kutluay, S., Numerical solution of Burgers' equation by quadratic B-spline finite elements. Appl. Math. Comput. v165. 237-249.

[39]

Radwan, S.F., Comparison of higher-order accurate schemes for solving the two-dimensional unsteady Burgers equation. J. Comput. Appl. Math. v174. 383-397.

Digital Library

[40]

Rivlin, T.J., Chebyshev Polynomials. 1990. John Wiley & Sons, Inc.

[41]

Soliman, A.A., The modified extended tanh-function method for solving Burgers-type equations. Physica A. v361. 394-404.

[42]

Wazwaz, A.M., Exact solutions with solitons and periodic structures for the Zakharov-Kuznetsov (ZK) equation and its modified form. Commun. Nonlinear Sci. Numer. Simulation. v10. 597-606.

[43]

Wazwaz, A.M., Partial differential equations: methods and applications. 2002. Balkema Publishers, The Netherlands.

[44]

Wazwaz, A.M., The decomposition for approximate solution of the Goursat problem. Appl. Math. Comput. v69. 299-311.

Digital Library

[45]

Wazwaz, A.M., A comparison between Adomian's decomposition methods and Taylor series method in the series solutions. Appl. Math. Comput. v97. 37-44.

Digital Library

[46]

Wazwaz, A.M., A new algorithm for calculating Adomian polynomials for nonlinear operators. Appl. Math. Comput. v111. 53-69.

Digital Library

[47]

Wazwaz, A.M., The modified decomposition method and Pade approximants for solving the Thomas-Fermi equation. Appl. Math. Comput. v105. 11-19.

[48]

Wazwaz, A.M., A reliable modification of Adomian decomposition method. Appl. Math. Comput. v102. 77-86.

Digital Library

[49]

Wubs, F.W. and de Goede, E.D., An explicit-implicit method for a class of time-dependent partial differential equations. Appl. Numer. Math. v9. 157-181.

Digital Library

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Index Terms

A Chebyshev spectral collocation method for solving Burgers'-type equations
1. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
      1. Partial differential equations

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cover image Journal of Computational and Applied Mathematics

Journal of Computational and Applied Mathematics Volume 222, Issue 2

December, 2008

507 pages

ISSN:0377-0427

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Copyright © Elsevier B.V. © 2007.

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Elsevier Science Publishers B. V.

Netherlands

Publication History

Published: 20 December 2008

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https://dl.acm.org/doi/10.1007/s00366-024-01963-7
Mohammadi MShirzadi A(2023)A Meshless Runge–Kutta Method for Some Nonlinear PDEs Arising in PhysicsComputational Mathematics and Modeling10.1007/s10598-023-09579-033:3(375-387)Online publication date: 11-Apr-2023
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Mohanty RSharma S(2021)A new high-accuracy method based on off-step cubic polynomial approximations for the solution of coupled Burgers’ equations and Burgers–Huxley equationEngineering with Computers10.1007/s00366-020-00982-437:4(3049-3066)Online publication date: 1-Oct-2021
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