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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com


A Method for Solving Nonlinear Differential Equations: An Application to λφ4 Model

Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 163--171 | DOI:10.5890/DNC.2015.06.004

Danilo V. Ruy

Instituto de Física Teórica-UNESP, Rua Dr Bento Teobaldo Ferraz 271, Bloco II São Paulo, 01140-070, Brazil

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Abstract

Recently, it has been great interest in the development of methods for solving nonlinear differential equations directly. Here, it is shown an algorithm based on Pad′e approximants for solving nonlinear partial differential equations without requiring a one-dimensional reduction. This method is applied to the λφ4 model in 4 dimensions and new solutions are obtained.

Acknowledgments

I am thankful to H. Aratyn, J. F. Gomes and A. H. Zimerman for discussions. The author also thanks FAPESP (2010/18110-9) for financial support.

References

  1. [1]  Parkes, E.J. and Duffy, B.R. (1996), An automated tanh-function method for finding solitary wave solutions to nonlinear evolution equations, Computer Physics Communications, 98, 288–300.
  2. [2]  Malfliet,W. and Hereman,W. (1996), The Tanh method: I. Exact solutions of nonlinear evolution and wave equations, Physica Scripta, 54, 563–8.
  3. [3]  Kudryashov, N.A. (1990), Exact solutions of the generalized Kuramoto-Sivashinsky equation, Physics Letters A, 147, 281-91.
  4. [4]  Kudryashov, N.A.(2005), Simplest equation method to look for exact solutions of nonlinear differential equations, Chaos, Solitons & Fractals , 24, 1217–1231.
  5. [5]  Parkes, E.J., Duffy, B.R. and Abbott, P C. (2002), The Jacobi elliptic-function method for finding periodic-wave solutions to nonlinear evolution equations, Physics Letters A, 295, 280-6.
  6. [6]  Vitanov, N.K. (2010), Application of simplest equations of Bernouli and Riccati kind for obtaining exact travelling wave solutions for a class of PDEs with polynomial nonlinearity, Communications in Nonlinear Science and Numerical Simulation, 15, 2050-60.
  7. [7]  Vitanov, N.K. (2011), Modified method of simplest equation: powerful tool for obtaining exact and approximate traveling-wave solutions of nonlinear PDEs, Communications in Nonlinear Science and Numerical Simulation, 16, 1176-85.
  8. [8]  Vitanov, N.K. (2011), On modified method of simplest equation for obtaining exact and approximate of nonlinear PDEs: the role of the simplest equation, Communications in Nonlinear Science and Numerical Simulation, 16, 4215- 31.
  9. [9]  Biswas, A. (2009), Solitary wave solution for the generalized Kawahara equation, Applied Mathematics Letters , 22, 208-10.
  10. [10]  Biswas, A., Petkovich, M.D., and Milovich, D. (2011), Topological and non-topological exact soliton of the power law KdV equation, Communications in Nonlinear Science and Numerical Simulation, 15(11), 3263–9.
  11. [11]  He, J.H. and Wu, X.H. (2006), Exp-function method for nonlinear wave equations, Chaos, Solitons & Fractals, 30, 700-8.
  12. [12]  Wu, X.H. and He, J.H. (2007), Solitary solutions, periodic solutions and compaction - like solutions using the Exp - function method, Computers & Mathematics with Applications , 54.
  13. [13]  Ma, W-X., Huang, T. and Zhang, Y.(2010), A multiple Exp - function method for nonlinear differential equations and its application, Physica Scripta , 82, 005003.
  14. [14]  Ma, W-X., and Fan, E. (2011), Linear Superposition principle applying to Hirota bilinear equations, Computers & Mathematics with Applications, 61, 950-9.
  15. [15]  Wang, M.L., Li, X. and Zhang, J. (2008), The G’/G - expansion method and evolution equation in mathematical physics, Physics Letters A, 372, 417-21.
  16. [16]  Ebadi, G. and Biswas, A. (2011), The G0/G method and topological solution of the K(m,n) equation, Communications in Nonlinear Science and Numerical Simulation, 16, 2377-82.
  17. [17]  Ma, W-X. and Lee, J.-H. (2009), A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo - Miwa equation, Chaos, Solitons & Fractals, 42, 1356-63.
  18. [18]  Aratyn, H., Gomes, J.F., Ruy, D.V., and Zimerman, A.H. (2013), Rational solutions from Padé approximants for the generalized Hunter-Saxton equation, Journal of Physics: Conference Series, 474, 012006.
  19. [19]  Chaffy-Camus, C. (1988), Convergence uniforme d’une nouvelle classe d’approximants de Padé à plusieurs variables, C. R. Acad. Sci. Paris, Ser I, 306, 387-92.
  20. [20]  Chisholm, J.S.R. (1973), Rational approximants defined from double power series, Mathematics of Computation, 27, 841-8.
  21. [21]  Cuyt, A. (1983),Multivariate Padé approximants, Journal of Mathematical Analysis and Applications , 96, 283-293.
  22. [22]  Guillaume, P. (1997), Nested Multivariate Padé Approximants, Journal of Computational and Applied Mathematics, 82, 149-58.
  23. [23]  Guillaume, P., Huard, A., and Robin, V. (1998), Generalized Multivariate Padé Approximants, Journal of Approximation Theory, 95 (2), 203-14.
  24. [24]  Guillaume, P. and Huard, A. (2000), Multivariate Padé approximation, Journal of Computational and Applied Mathematics, 121, 197-219.
  25. [25]  Levin, D. (1976), General order Padé-type rational rational approximants defined from double power series, Journal of the Institute of Mathematics and its Applications, 18, 395-07.
  26. [26]  Dashen, R., Hasslacher, B. and Neveu, A. (1974), Nonperturbative methods and extended-hadron models in field theory. II. Two-dimensional models and extended hadrons, Physical Review D, 10, 4130-8.
  27. [27]  Carrillo, J.A.E., Maia Jr., A., and Mostepanenko, V.M. (2000), Jacobi Elliptic Solution of λφ4 Theory in a Finite Domain, International Journal of Modern Physics A, 15, 2645.