Journal of Mathematical Research and Applications       
Journal of Mathematical Research and Applications
Frequency: Quarterly
Website: www.academicpub.org/jmra/
The Modified Simple Equation Method Applied to Nonlinear Two Models of Diffusion-Reaction Equations
Full Paper(PDF, 297KB)
Abstract:
In this paper, we employ the modified simple equation method (MSEM) to find the exact solutions of some nonlinear partial differential equations (PDEs), namely two different models of nonlinear diffusion- reaction equations arising in many physical problems and including one dimensional turbulence, sound and shock waves in viscous medium etc. The applicability of this method for constructing these exact solutions is demonstrated. This paper is qualified because finding the exact solutions for these two nonlinear equations is difficult. The MSEM is useful to find the exact solutions of nonlinear evolution reaction diffusion equations in mathematical physics and engineering problems. These solutions are kink and anti-kink shaped soliton solutions.
Keywords:Nonlinear Diffusion-Reaction Equations; Modified Simple Equation Method; Traveling Wave Solutions; Exact Solutions; Solitary Wave Solution
Author: Elsayed M. E. Zayed1
1.Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt
References:
  1. M. J. Ablowitz, H. Segur, “Solitons and Inverse Scattering Transform,” SIAM, Philadel- phia, 1981.
  2. W. Malfliet, “Solitary wave solutions of nonlinear wave equation,” Am. J. Phys., vol. 60, pp. 650-654, 1992.
  3. W. Malfliet, W. Hereman, “The tanh method: Exact solutions of nonlinear evolution and wave equations,” Phys. Scr., vol. 54, pp. 563-568, 1996.
  4. A. M. Wazwaz, “The tanh method for travelling wave solutions of nonlinear equations,” Appl. Math. Comput., vol. 154, pp. 714-723, 2004.
  5. S. A. EL-Wakil, M.A. Abdou, “New exact travelling wave solutions using modified extented tanh-function method,” Chaos, Solitons and Fractals, vol. 31, pp. 840-852, 2007.
  6. E. Fan, “Extended tanh-function method and its applications to nonlinear equations," Phys. Lett. A, vol. 277, pp. 212-218, 2000.
  7. A. M. Wazwaz, “The extended tanh method for abundant solitary wave solutions of nonlinear wave equations,” Appl. Math. Comput., vol. 187, pp. 1131-1142, 2007.
  8. A. M. Wazwaz, “Exact solutions to the double sinh-Gordon equation by the tanh-method and a variable separated ODE method,” Comput. Math. Appl., vol. 50, pp. 1685-1696, 2005.
  9. A. M. Wazwaz, “A sine-cosine method for handling nonlinear wave equations,” Math. Comput. Modelling, vol. 40, pp. 499-508, 2004.
  10. C. Yan, “A simple transformation for nonlinear waves,” Phys. Lett. A, vol. 224, pp. 77-84, 1996.
  11. E. Fan, H.Zhang, “A note on the homogeneous balance method,” Phys. Lett. A, vol. 246, pp. 403-406, 1998.
  12. M. L. Wang, “Exact solutions for a compound KdV-Burgers equation,” Phys. Lett. A, vol. 213, pp. 279-287, 1996.
  13. C. Q. Dai, J. F. Zhang, “Jacobian elliptic function method for nonlinear differential difference equations,” Chaos, Solitons and Fractals, vol. 27, pp. 1042-1049, 2006.
  14. E. Fan, J .Zhang, “Applications of the Jacobi elliptic function method to special-type nonlinear equations,” Phys. Lett. A, vol. 305, pp. 383-392, 2002.
  15. S. Liu, Z. Fu, S. Liu, and Q. Zhao, “Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations,” Phys. Lett. A, vol. 289, pp. 69-74, 2001.
  16. X. Q. Zhao, H. Y. Zhi, and H. Q. Zhang, “Improved Jacobi elliptic function method with symbolic computation to construct new double -periodic solutions for the generalized Ito system,” Chaos, Solitons and Fractals, vol. 28, pp. 112-126, 2006.
  17. M. A. Abdou, “The extended F-expansion method and its application for a class of nonlinear evolution equations,” Chaos, Solitons and Fractals, vol. 31, pp. 95-104, 2007.
  18. Y. J. Ren, H. Q. Zhang, “A generalized F-expansion method to find abundant families of Jacobi elliptic function solutions of the (2+1)-dimensional Nizhnik-Novikov-Veselov equation,” Chaos, Solitons and Fractals, vol. 27, pp. 959-979, 2006.
  19. J. L. Zhang, M. L. Wang, Y. M. Wang, and Z. D. Fang, “The improved F-expansion method and its applications,” Phys. Lett. A, vol. 350, pp. 103-109, 2006.
  20. J. H. He, X. H. Wu, “Exp-function method for nonlinear wave equations,” Chaos, Solitons and Fractals, vol.30, pp. 700-708, 2006.
  21. H. Aminikhad, H. Moosaei, and M. Hajipour, “Exact solutions for nonlinear partial differential equations via Exp-function method,” Numer, Methods Partial Differential. Equations, vol. 26, pp. 1427-1433, 2009.
  22. Z. Y. Zhang, “New exact traveling wave solutions for the nonlinear Klein-Gordon equation,” Turk. J. Phys., vol. 32, pp. 235-240, 2008.
  23. M. L. Wang, J. L. Zhang, and X. Z. Li, “The (G’/G) - expansion method and traveling wave solutions of nonlinear evolutions equations in mathematical physics,” Phys. Lett. A, vol. 372, pp. 417-423, 2008.
  24. S. Zhang, J. L. Tong, and W.Wang, “A generalized (G’/G) - expansion method for the mKdV equation with variable coefficients,” Phys. Lett A, vol. 372, pp. 2254-2257, 2008.
  25. E. M. E. Zayed, K. A. Gepreel, “The (G’/G) - expansion method for finding traveling wave solutions of nonlinear partial differentialequations in mathematical physics,” J. Math. Phys, vol. 50, 013502-013513, 2009.
  26. E. M. E. Zayed, “The (G’/G) - expansion method and its applications to some nonlinear evolution equations in mathematical physics,” J. Appl. Math. Computing, vol. 30, pp. 89-103, 2009.
  27. A. J. M.Jawad, M. D. Petkovic, and A. Biswas, “Modified simple equation method for nonlinear evolution equations,” Appl. Math. Comput., vol. 217, pp. 869-877, 2010.
  28. E. M. E. Zayed, “A note on the modified simple equation method applied to Sharam- Tasso-Olver equation,” Appl. Math. Comput, vol. 218, pp. 3962-3964, 2011.
  29. E. M. E. Zayed, S. A. Hoda Ibrahim, “Exact solutions of nonlinear evolution equation in mathematical physics using the modified simple equation method,” Chin. Phys. Lett., vol. 29, 060201-4, 2012.
  30. E. M. E. Zayed, A. H. Arnous, “Exact solutions of the nonlinear ZK-MEW and the potential YTSF equations using the modified simple equation method,” AIP Conf. Proc., vol. 1479, pp. 2044-2048, 2012.
  31. E. M. E. Zayed, S. A. Hoda Ibrahim, “Modified simple equation method and its applications for some nonlinear evolution equations in mathematical physics,” Int. J. Comput. Appl., vol. 67, pp. 39-44, 2013.
  32. E. M. E. Zayed, S. A. Hoda Ibrahim, “Exact solutions of Kolmogorov-Petrovskii- Piskunov equation using the modified simple equation method,” Acta Math. Appl. Sinica, English Series, vol. 30, no.3, 2014.
  33. W. X. Ma, B. Fuchssteiner, “Explicit and exact solutions of KPP equation,” Int. J. Nonl. Mech, vol. 31, pp. 329-338, 1996.
  34. W. X. Ma, “Generalized bilinear differential equations,” Stud. Nonl. Sci., pp. 140-144, 2011.
  35. W. X. Ma, “Bilinear equations and resonant solutions characterized by bell polynomial,” reports on Math. Phys, vol. 72, pp. 41-56, 2013.
  36. W. X. Ma, J. H. Lee, “A transformed rational function method and exact solutions to the (3+1)-dimensional Jimbo Miwa equation,” Chaos, Solitons and Fractals, vol. 42, pp. 1356-1363, 2009.
  37. W. X. Ma, Z. Zhu, “Solving the (3+1)-dimensional generalized KP and BKP by the multiple exp-function method,” Appl. Math. Comput, vol. 218, pp. 11871-11879, 2012.
  38. D. Ludwig, D. D Jones, and C. S. Holling, “Qualitative Analysis of Insect Outbreak Systems: The Spruce Budworm and Forest,” J. Anim. Eco., vol. 47, pp. 315-332, 1978.
  39. L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers, Birkhauser, Boston, 1997.
  40. X. Y. Wang, Z. S. Zhu, and Y. K. Lu, “Solitary wave solutions of the generalized Burgers- Huxley equation,” J. Phys. A: Math. Gen., vol. 23, pp. 271-274, 1990.
  41. R. Kumar, R S Kaushal, and A. Prasad, “Solitary wave solutions of selective nonlinear diffusion-reaction equations using homogeneous balance method,” Pramana J. Phys., vol. 75, pp. 607-616, 2010.