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

Journal of Computational and Applied Mathematics

Volumn 222, Issue 2, 2008, Pages 333-350

  Khater, A H ^a   Temsah, R S ^a   Hassan, M M ^b  

^a Beni Suef University   (Egypt)

^b Minia University ^*  (Egypt)

Author keywords

1D Burgers' equation; 2D Burgers' equation; Chebyshev spectral collocation method; Coupled Burgers' equations; KdV Burgers' equation; Numerical solutions; System of 2D Burgers' equations

Indexed keywords

CHEBYSHEV APPROXIMATION; COMPUTATIONAL FLUID DYNAMICS; DIFFERENCE EQUATIONS; DIFFERENTIAL EQUATIONS; DIFFERENTIATION (CALCULUS); IMAGE SEGMENTATION; NONLINEAR EQUATIONS; NUMERICAL ANALYSIS; NUMERICAL METHODS; ORDINARY DIFFERENTIAL EQUATIONS; RUNGE KUTTA METHODS;

1D BURGERS' EQUATION; 2D BURGERS' EQUATION; CHEBYSHEV SPECTRAL COLLOCATION METHOD; COUPLED BURGERS' EQUATIONS; KDV-BURGERS' EQUATION; NUMERICAL SOLUTIONS; SYSTEM OF 2D BURGERS' EQUATIONS;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 53449090789     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2007.11.007     Document Type: Article

References (49)

1. Abdou M.A., and Soliman A.A. Variational iteration method for solving Burgers and coupled Burgers equations. J. Comput. Appl. Math. 181 2 (2005) 245-251
2. Adomian G. The diffusion-Brusselator equation. Comput. Math. Appl. 29 (1995) 1-3
3. Bahadir A.R. A fully implicit finite-difference scheme for two-dimensional Burgers' equations. Appl. Math. Comput. 137 (2003) 131-137
4. Burger J.M. A mathematical model illustrating the theory of turbulence. Adv. Appl. Mech. I (1948) 171-199
5. Clenshaw C.W., and Curtis A.R. A method for numerical integration on an automatic computer. Numer. Math. 2 (1960) 197-205
6. Cole J.D. On a quasilinear parabolic equations occurring in aerodynamics. Quart. Appl. Math. 9 (1951) 225-236
7. Debtnath L. Nonlinear Partial Differential Equations for Scientist and Engineers (1997), Birkhauser, Boston
8. Elbarbary E.M.E., and El-Kady M. Chebyshev finite difference approximation for the boundary value problems. Appl. Math. Comput. 139 (2003) 513-523
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. 158 (2004) 101-109
10. Fletcher C.A.J. Generating exact solutions of the two-dimensional Burgers' equation. Int. J. Numer. Methods Fluids 3 (1983) 213-216
11. Goyon O. Multilevel schemes for solving unsteady equations. Int. J. Numer. Methods Fluids 22 (1996) 937-959
12. Helal M.A., and Mehanna M.S. A comparison between two different methods for solving KdV-Burgers equation. Chaos Solitons Fractals 28 (2006) 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. 25 (1988) 33-48
14. Jain P.C., and Holla D.N. Numerical solution of coupled Burgers Δ equations. Int. J. Numer. Methods Eng. 12 (1978) 213-222
15. Kaya D. An application of the decomposition method for the KdVB equation. Appl. Math. Comput. 152 (2004) 279-288
16. Kaya D. An explicit solution of coupled viscous burgers equation by the decomposition method. Int. J. Math. Math. Sci. 27 (2001) 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. 65 (2000) 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. 140 (2002) 435-467
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. 189 (2006) 387-411
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. 51 (2004) 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 62 (2000) 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. 58 (1997) 51-69
23. Khater A.H., and Hassan A.A. Travelling and periodic wave solutions of some nonlinear wave equations. Z. Naturforsch. A 59 (2004) 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. 74 (2005) 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 70 (2005) 221-226
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. 21 (1998) 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. 73 (2004) 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 14 (2002) 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. 140 (2002) 469-477
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 64 (2004) 247-258
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. 72 (2003) 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 312 (2002) 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 312 (2002) 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. 95 (1998) 29-43
35. Khater A.H., Shamardan A.B., and Ibrahim R.S. Chebyshev spectral collocation methods for nonlinear isothermal magnetostatic atmospheres. J. Comput. Appl. Math. 115 (2000) 309-329
36. Khater A.H., and Temsah R.S. Numerical solutions of some nonlinear evolution equations by Chebyshev spectral collocation methods. Inter. J. Comput. Math. 84 (2007) 305-316
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. 165 (2005) 237-249
39. Radwan S.F. Comparison of higher-order accurate schemes for solving the two-dimensional unsteady Burgers equation. J. Comput. Appl. Math. 174 (2004) 383-397
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 361 (2006) 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 10 (2005) 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. 69 (1995) 299-311
45. Wazwaz A.M. A comparison between Adomian's decomposition methods and Taylor series method in the series solutions. Appl. Math. Comput. 97 (1998) 37-44
46. Wazwaz A.M. A new algorithm for calculating Adomian polynomials for nonlinear operators. Appl. Math. Comput. 111 (2000) 53-69
47. Wazwaz A.M. The modified decomposition method and Pade approximants for solving the Thomas-Fermi equation. Appl. Math. Comput. 105 (1999) 11-19
48. Wazwaz A.M. A reliable modification of Adomian decomposition method. Appl. Math. Comput. 102 (1999) 77-86
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. 9 (1992) 157-181