New explicit solutions for the Vakhnenko and a generalized form of the nonlinear heat conduction equations via Exp-function method

International Journal of Nonlinear Sciences and Numerical Simulation

Volumn 10, Issue 10, 2009, Pages 1307-1318

  Kabir, M M ^a,b   Khajeh, A ^a  

^a ISLAMIC AZAD UNIVERSITY   (Iran)

^b UNIVERSITY OF MOHAGHEGH ARDABILI   (Iran)

Author keywords

Exp function method; Nonlinear heat conduction; Travelling wave solutions; Vakhnenko equation

Indexed keywords

EID: 74549120915     PISSN: 15651339     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (53)

1. Wadati M. The exact solution of the modified Korteweg-de Vries equation. J Phys Soc Japan 1972;32:1681-7.
2. Wadati M. Introduction to solitons. Pramana: J Phys 2001;57(5-6):841-7.
3. Ablowitz MJ, Segur H. Solitons and inverse scattering transform. Philadelphia: SIAM; 1981.
4. Vakhnenko VO, Parkes EJ, Morrison AJ. A Backlund transformation and the inverse scattering transform method for the generalized Vakhnenko equation. Chaos, Solitons & Fractals 2003;17(4):683-92.
5. Vakhnenko VO, Parkes EJ. The calculation of multi-soliton solutions of the Vakhnenko equation. Chaos, Soliton & Fractals 2002;13:1819-26.
6. Malfliet W. Solitary wave solutions of nonlinear wave equations. Am J Phys 1992;60:650-4.
7. Khater AH, Malfliet W, Callebaut DK, Kamel ES. The tanh method, a simple transformation and exact analytical solutions for nonlinear reaction-diffusion equations. Chaos, Solitons & Fractals 2002;14(2):513-22.
8. Wazwaz AM. Two reliable methods for solving variants of the KdV equation with compact and noncompact structures. Chaos, Solitons & Fractals 2006;28(2):454-62.
9. Bai CL, Zhao H. Generalized extended tanh-function method and its application. Chaos, Solitons & Fractals 2006;27:1026-35.
10. Abdou MA, Soliman AA. Modified extended tanh-function method and its application on nonlinear physical equations. Phys Lett A 2006;353(6):487-92.
11. Ibrahim RS, El-Kalaawy OH. Extended tanh-function method and reduction of nonlinear Schrodinger-type equations to a quadrature. Chaos, Solitons & Fractals 2007;31(4):1001-8.
12. El-Wakil SA, Abdou MA. New exact travelling wave solutions using modified extended tanh-function method. Chaos, Solitons & Fractals 2007;31(4):840-52.
13. Elwakil SA, El-Labany SK, Zahran MA, Sabry R. Modified extended tanh-function method and its applications to nonlinear equations. Appl Math Comput 2005;161(2):403-12.
14. Yan Z. Constructing exact solutions for two-dimensional nonlinear dispersion Boussinesq equation II: solitary pattern solutions. Chaos, Solitons & Fractals 2003;18(4):869-80.
15. Yusufoglu E, Bekir A. The tanh and the sine-cosine methods for exact solutions of the MBBM and the Vakhnenko equations. Chaos, Solitons & Fractals 38 (2008) 1126-1133.
16. Wazwaz AM, Helal MA. Nonlinear variants of the BBM equation with compact and noncompact physical structures. Chaos, Solitons & Fractals 2005;26(3):767-76.
17. Wazwaz AM. A sine-cosine method for handling nonlinear wave equations. Math Comput Model 2004;40:499-508.
18. Wazwaz AM, The tanh method and the sine-cosine method for solving the KP-MEW equation, Int. J. Comput. Math. 82 (2) (2005) 235-246.
19. He JH. New interpretation of homotopy perturbation method. Int J Mod Phys B 2006;20 (18) :2561-8.
20. He JH. Application of homotopy perturbation method to nonlinear wave equations. Chaos, Solitons & Fractals 2005;26(3):695-700.
21. He JH. Limit cycle and bifurcation of nonlinear problems. Chaos, Solitons & Fractals 2005;26(3):827-33.
22. He JH. Homotopy perturbation method for bifurcation of nonlinear problems. Int J Nonlinear Sci Numer Simul 2005;6(2):207-8.
23. Siddiqui AM, Mahmood R, Ghori QK. Thin film flow of a third grade fluid on a moving belt by He's homotopy perturbation method. Int J Nonlinear Sci Numer Simul 2006;7(1):7-14.
24. Siddiqui AM, Ahmed M, Ghori QK. Couette and Poiseuille flows for non-Newtonian fluids. Int J Nonlinear Sci Numer Simul 2006;7(1):15-26.
25. Liao SJ. A second-order approximate analytical solution of a simple pendulum by the process analysis method. ASME J. Appl. Mech. 59 (1992) 970-975.
26. Liao SJ. An explicit, totally analytic approximation of Blasius' viscous flow problems. Int. J. Non-Linear Mech. 34 (1999) 759-778.
27. He JH. Comparison of homotopy perturbation method and homotopy analysis method. Applied Mathematics and Computation 156 (2004) 527-539.
28. Tari H, Ganji DD, Rostamian M. Approximate solutions of K (2, 2), KdV and modified KdV equations by variational iteration method, homotopy perturbation method and homotopy analysis method. Int J Nonlin Sci Num 2007;8(2):203-10.
29. Abbasbandy S. Solitary wave solutions to the modified form of Camassa-Holm equation by means of the homotopy analysis method. Chaos, Solitons & Fractals 2009;39(1):428-35.
30. Fan E, Zhang H. A note on the homogeneous balance method. Phys Lett A 1998;246:403-6.
31. Wang ML. Exact solutions for a compound KdV-Burgers equation. Phys Lett A 1996;213:279.
32. Hirota R. Exact solution of Korteweg-de Vries equation for multiple collisions of solitons. Phys Rev Lett 1971;27:1192-4.
33. Tanoglu G. Solitary wave solution of nonlinear multi-dimensional wave equation by bilinear transformation method. Commun Nonlinear Sci Numer Simul 2007;12:1195-201.
34. Dai CQ, Zhang JF. Jacobian elliptic function method for nonlinear differential-difference equations. Chaos, Solitons & Fractals 2006;27:1042-9.
35. Fan E, Zhang J. Applications of the Jacobi elliptic function method to special-type nonlinear equations. Phys Lett A 2002;305:383-92.
36. Zhao XQ, Zhi HY, Zhang HQ. Improved Jacobi-function method with symbolic computation to construct new doubleperiodic solutions for the generalized Ito system. Chaos, Solitons & Fractals 2006;28:112-26.
37. He JH, Wu XH. Exp-function method for nonlinear wave equations. Chaos, Solitons & Fractals 2006;30(3):700-8.
38. He JH, Xu F. Nonlinearity as a sensitive informative marker in the ENSO model, Computers and Mathematics with Applications 2009, doi:10.1016/j.camwa. 2009.03.067
39. He JH, Zhang LN. Generalized solitary solution and compacton-like solution of the Jaulent-Miodek equations using the Expfunction method. Phys Lett A 2008;372(7):1044-7.
40. Bekir A, Boz A. Exact solutions for a class of nonlinear partial differential equations using Exp-function method. Int J Non Sci Numer Simul 2007;8(4):505-12.
41. Zhang S. Exp-function method for solving Maccari's system. Physics Letters A 371 (2007) 65-71.
42. Zhang S. Application of Exp-function method to high dimensional nonlinear evolution equation. Chaos, Solitons & Fractals 2008;38(1):270-6.
43. Borhanifar A, Kabir M. M, Vahdat L. Maryam, New periodic and soliton wave solutions for the generalized Zakharovsystem and (2 + 1)-dimensional Nizhnik-Novikov-Veselov system. Chaos, Solitons & Fractals 2009, doi:10.1016/j.chaos.2009.03.064
44. Borhanifar A, Kabir M. M, New periodic and soliton solutions by application of Expfunction method for nonlinear evolution equations, J. Computational and Applied Mathematics 2008, doi:10.1016/j.cam. 2008.10.052
45. Ostrovsky LA. Nonlinear internal waves in a rotating ocean. Oceanology 1978;18:119-25.
46. Parkes E. J, Explicit solutions of the reduced Ostrovsky equation. Chaos, Solitons and Fractals 31 (2007) 602-610.
47. Vakhnenko VO, Parkes EJ. The two loop soliton of the Vakhnenko equation. Nonlinearity 1998;11:1457-64.
48. Yusufoglu E, Bekir A. A travelling wave solution to the Ostrovsky equation. Applied Mathematics and Computation 186 (2007) 256-260.
49. Wazwaz AM. The tanh method for generalized forms of nonlinear heat conduction and Burgers-Fisher equations. Applied Mathematics and Computation 169 (2005) 321-338.
50. Fan E. Traveling wave solutions for nonlinear equations using symbolic computation, Comput. Math. Appl. 43 (2002) 671-680.
51. Wu XH, He JH. Solitary solutions, periodic solutions and compacton-like solutions using the Exp-function method. Computers and Mathematics with Applications 54 (2007) 966-986.
52. Zhou XW, Wen YX, He JH. Exp-function method to solve the nonlinear dispersive K (m, n) equations. Int J Non Sci Numer Simul 9 (2008) 301-306.
53. He JH. An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering. Int J Modern Physics B 22 (2008) 3487-3578.