Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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APPLICATION OF EXP-FUNCTION METHOD FOR A CLASS OF NONLINEAR PDE'S ARISING IN MATHEMATICAL PHYSICS

  • Parand, Kourosh (Department of Computer Sciences, Shahid Beheshti University) ;
  • Amani Rad, Jamal (Department of Computer Sciences, Shahid Beheshti University) ;
  • Rezaei, Alireza (Department of Computer Sciences, Shahid Beheshti University)
  • Received : 2010.04.29
  • Accepted : 2010.08.09
  • Published : 2011.05.30
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Abstract

In this paper we apply the Exp-function method to obtain traveling wave solutions of three nonlinear partial differential equations, namely, generalized sinh-Gordon equation, generalized form of the famous sinh-Gordon equation, and double combined sinh-cosh-Gordon equation. These equations play a very important role in mathematical physics and engineering sciences. The Exp-Function method changes the problem from solving nonlinear partial differential equations to solving a ordinary differential equation. Mainly we try to present an application of Exp-function method taking to consideration rectifying a commonly occurring errors during some of recent works.

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References

  1. M. A. Abdou, A. A. Soliman, and S. T. El-Basyony, New application of exp-function method for improved boussinesq equation, Physics Letters, Section As : General, Atomic and Solid State Physics 369 (2007), 469-475.
  2. M.J. Ablowitz, B.M.Herbst, and C. Schober, On the numerical solution of the sinh-gordon equation, Journal of Computational Physics 126 (1996), 299-314. https://doi.org/10.1006/jcph.1996.0139
  3. MJ. Ablowitz and PA. Clarkson, Solitons, nonlinear evolution equations and inverse scatting, Cambridge University Press, 1991.
  4. L.M.B. Assas, New exact solutions for the kawahara equation using exp-function method, Journal of Computational and Applied Mathematics 233 (2009), 97-102. https://doi.org/10.1016/j.cam.2009.07.016
  5. M.E. Berberler and A. Yildirim, He's homotopy perturbation method for solving the shock wave equation, Applicable Analysis 88 (2009), 997-1004. https://doi.org/10.1080/00036810903114767
  6. J. Biazar and Z. Ayati, Extension of the exp-function method for systems of two- dimensional burger's equations, Computers and Mathematics with Applications 58 (2009), 2103-2106. https://doi.org/10.1016/j.camwa.2009.03.003
  7. J. Biazar, F. Badpeimaa, and F. Azimi, Application of the homotopy perturbation method to zakharov-kuznetsov equations, Computers and Mathematics with Applications 58 (2009), 2391-2394. https://doi.org/10.1016/j.camwa.2009.03.102
  8. L. Bougoffa and A. Khanfer, Particular solutions to equations of sine-gordon type, Journal of Applied Mathematics and Computing 32 (2010), 303-309. https://doi.org/10.1007/s12190-009-0252-7
  9. C.Koroglu and T.zis, A novel traveling wave solution for ostrovsky equation using expfunction method, Computers and Mathematics with Applications 58 (2009), 2142-2146. https://doi.org/10.1016/j.camwa.2009.03.028
  10. M. Dehghan and A. Shokri, Numerical solution of the nonlinear klein-gordon equation using radial basis functions, Journal of Computational and Applied Mathematics 230 (2009), 400-410. https://doi.org/10.1016/j.cam.2008.12.011
  11. G. Domairry, A. G. Davodi, and A. G. Davodi, Solutions for the double sine-gordon equation by exp-function, tanh, and extended tanh methods, Numerical Methods for Partial Differential Equations 26 (2010), 384-398.
  12. A. Ebaid, Exact solitary wave solutions for some nonlinear evolution equations via exp- function method, Physics Letters, Section As : General, Atomic and Solid State Physics 365 (2007), 213-219.
  13. A.E.H. Ebaid, Generalization of he's exp-function method and new exact solutions for burger's equation, Zeitschrift fur Naturforschung - Section A Journal of Physical Sciences 64 (2009), 604-608.
  14. E.Yusufoglu and A. Bekir, The variational iteration method for solitary patterns solutions of gbbm equation, Physics Letters A 367 (2007), 461-464. https://doi.org/10.1016/j.physleta.2007.03.045
  15. Z. Fu, S. Liu, and S. Liu, Exact solutions to double and triple sinh-gordon equations, Z. Naturforsch 59 (2004), 933-937.
  16. Gegenhasi, Xing-Biao Hu, and Hong-Yan Wang, A (2+1)-dimensional sinh-gordon equation and its pfaffan generalization, Physics Letters A 360 (2007), 439-447. https://doi.org/10.1016/j.physleta.2006.07.031
  17. V. I. Gromak, Solutions of the third painleve equation, Differential Equations 9 (1973), 1599-1600.
  18. I. Hashim, M. S. M. Noorani, and M. R. S. Hadidi, Solving the generalized burger's- huxley equation using the adomian decomposition method, Mathematical and Computer Modelling 43 (2006), 1404-1411. https://doi.org/10.1016/j.mcm.2005.08.017
  19. J. H. He and M. A. Abdou, New periodic solutions for nonlinear evolution equations using exp-function method, Chaos, Solitons and Fractals 34 (2007), 1421-1429. https://doi.org/10.1016/j.chaos.2006.05.072
  20. J. H. He and X. H.Wu, Exp-function method for nonlinear wave equations, Chaos, Solitons and Fractals 30 (2006), 700-708. https://doi.org/10.1016/j.chaos.2006.03.020
  21. J. H. He and X. H.Wu, Exp-function method and its application to nonlinear equations, Chaos, Solitons and Fractals 38 (2008), 903-910. https://doi.org/10.1016/j.chaos.2007.01.024
  22. J. H. He and L. Zhang, Generalized solitary solution and compacton-like solution of the jaulent-miodek equations using the exp-function method, Physics Letters, Section As : General, Atomic and Solid State Physics 372 (2008), 1044-1047.
  23. W. Hu, Z. Deng, S. Han, and W. Fan, The complex multi-symplectic scheme for the generalized sinh-gordon equation, Science in China, Series G: Physics, Mechanics and Astronomy 52 (2009), 1618-1623. https://doi.org/10.1007/s11433-009-0190-2
  24. E. Infeld and G. Rowlands, Nonlinear waves, Solitons and Chaos,Cambridge, England, 2000.
  25. N. A. Kudryashov, Seven common errors in ?nding exact solutions of nonlinear differential equations, Commun Nonlinear Sci Numer Simulat 14 (2009), 3507-3529. https://doi.org/10.1016/j.cnsns.2009.01.023
  26. T. Ozis and C. Koroglu, A novel approach for solving the fisher equation using exp-function method, Physics Letters, Section As : General, Atomic and Solid State Physics 372 (2008), 3836-3840.
  27. K. Parand, M. Dehghan, A.R. Rezaei, and S.M. Ghaderi, An approximation algorithm for the solution of the nonlinear lane-emden type equations arising in astrophysics using hermite functions collocation method, Computer Physics Communications (2010).
  28. K. Parand and M. Razzaghi, Rational chebyshev tau method for solving volterra's population model, Appl. Math. Comput 149 (2004), 893-900. https://doi.org/10.1016/j.amc.2003.09.006
  29. K. Parand and A. Taghavi, Rational scaled generalized laguerre function collocation method for solving the blasius equation, Journal of Computational and Applied Mathematics 233 (2009), 980-989. https://doi.org/10.1016/j.cam.2009.08.106
  30. J.K. Perring and T.H. Skyrme, A model unified field equation, Nuel. Phys. 31 (1962), 550-555.
  31. A. Polyanin and V.F. Zaitsev, Handbook of nonlinear partial differential equations, CRC, Boca Raton, FL, 2004.
  32. M.M. Rashidi, D.D. Ganji, and S. Dinarvand, Explicit analytical solutions of the generalized burger and burger-fisher equations by homotopy perturbation method, Numerical Methods for Partial Differential Equations 25 (2009), 409-417. https://doi.org/10.1002/num.20350
  33. W. Rui, B. He, and Y.Long, Double periodic wave solutions and breather-soliton solutions of the (n+1)-dimensional sinh-gordon equation, Journal of Physics: Conference Series 96 (2008), 012048.
  34. F. Shakeri and M. Dehghan, Numerical solution of the kleingordon equation via he's variational iteration method, Nonlinear Dynamics 51 (2008), 89-97.
  35. B.C. Shin, M.T. Darvishi, and A. Barati, Some exact and new solutions of the nizhnik- novikov-vesselov equation using the exp-function method, Computers and Mathematics with Applications 58 (2009), 2147-2151. https://doi.org/10.1016/j.camwa.2009.03.006
  36. Sirendaoreji and S. Jiongu, A direct method for solving sine-gordon type equations, Physics Letters A 298 (2002), 133-139. https://doi.org/10.1016/S0375-9601(02)00513-3
  37. A.A. Soliman and M.A. Abdou, Numerical solutions of nonlinear evolution equations using variational iteration method, Journal of Computational and Applied Mathematics 207 (2007), 111-120. https://doi.org/10.1016/j.cam.2006.07.016
  38. S. Tang and W. Huang, Bifurcations of travelling wave solutions for the generalized double sinh-gordon equation, Applied Mathematics and Computation 189 (2007), 1774-1781. https://doi.org/10.1016/j.amc.2006.12.082
  39. F. Tascan and A. Bekir, Analytic solutions of the (2 + 1)-dimensional nonlinear evolu- tion equations using the sine-cosine method, Applied Mathematics and Computation 215 (2009), 3134-3139. https://doi.org/10.1016/j.amc.2009.09.027
  40. M. Tatari, M. Dehghan, and M. Razzaghi, Application of the adomian decomposition method for the fokker-planck equation, Mathematical and Computer Modelling 45 (2007), 639-650. https://doi.org/10.1016/j.mcm.2006.07.010
  41. M. Tatari and M. M. Dehghan, A method for solving partial differential equations via radial basis functions: Application to the heat equation, Engineering Analysis with Boundary Elements 34 (2010), 206-212. https://doi.org/10.1016/j.enganabound.2009.09.003
  42. A. M. Wazwaz, New solitary wave solutions to the modified kawahara equation, Physics Letters, Section A: General, Atomic and Solid State Physics 360 (2007), 588-592.
  43. A.M. Wazwaz, Exact solutions to the double sinh-gordon equation by the tanh method and a variable separated ode method, Computers and Mathematics with Applications 50 (2005), 1685-1696. https://doi.org/10.1016/j.camwa.2005.05.010
  44. A.M. Wazwaz, The tanh method: exact solutions of the sine-gordon and the sinh-gordon equations, Applied Mathematics and Computation 167 (2005), 1196-1210. https://doi.org/10.1016/j.amc.2004.08.005
  45. A.M. Wazwaz, Exact solutions for the generalized sine-gordon and the generalized sinh-gordon equations, Chaos, Solitons and Fractals 28 (2006), 127-135. https://doi.org/10.1016/j.chaos.2005.05.017
  46. A.M. Wazwaz, The variable separated ode and the tanh methods for solving the combined and the double combined sinh-cosh-gordon equations, Applied Mathematics and Computation 177 (2006), 745-754. https://doi.org/10.1016/j.amc.2005.09.101
  47. G.B. Whitham, Linear and nonlinear waves, Wiley-Interscience, New York, 1999.
  48. ZK. Yan, A sinh-gordon equation expansion method to construct doubly periodic solutions for nonlinear differential equations, Chaos, Solitons and Fractals 16 (2003), 291-297. https://doi.org/10.1016/S0960-0779(02)00321-1
  49. H. Zhang, New exact solutions for the sinh-gordon equation, Chaos, Solitons and Fractals 28 (2006), 489-496. https://doi.org/10.1016/j.chaos.2005.07.005
  50. S. Zhang, Application of exp-function method to a kdv equation with variable coeffcients, Physics Letters, Section A: General, Atomic and Solid State Physics 365 (2007), 448-453.
  51. S. Zhang, Exp-function method for solving maccari's system, Physics Letters, Section A: General, Atomic and Solid State Physics 371 (2007), 65-71.
  52. S. Zhang, Application of exp-function method to high-dimensional nonlinear evolution equation, Chaos, Solitons and Fractals 38 (2008), 270-276. https://doi.org/10.1016/j.chaos.2006.11.014