Comparative Numerical Solutions of Stiff Ordinary Differential Equations Using Magnus Series Expansion Method

Abstract

In this paper, we establish exact-special solutions of the generalized nonlinear dispersion GNLS(m,n,k,l) equation. We usethe ansatz method for acquiring the compactons, solitary patterns, solitons and other types of solutions

Keywords

• Generalized nonlinear dispersion, Ansatz method, solitary patterns

Compact and noncompact structures of the nonlinearly dispersive GNLS(m,n,k,l) equation

Abstract

References

1. Hereman W., Banerjee P.P., Korpel A., Assanto, G., van Immerzeele A., Meerpoel,A., Exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method. Journal Physics A: Mathematical and General 19(1986)607-628.
2. Wang Deng-S., Complete integrability and the Miura transformation of a coupled KdV equation, Applied Mathematics Letters 23(2010)665-669.
3. Geng X., H, G., Darboux transformation and explicit solutions for the Satsuma–Hirota coupled equation, Applied Mathematics and Computation 216(2010)2628-2634.
4. Cesar A., G´omez S., Alvaro H. S., The Cole–Hopf transformation and improved tanh–coth method applied to new integrable system (KdV6), Applied Mathematics and Computation 204(2008)957-962.
5. Lei Y., Fajiang Z., Yinghai W., The homogeneous balance method, Lax pair, Hirota transformation and a general fifth-order KdV equation, Chaos, Solitons and Fractals 13(2002)337-340.
6. Wang M. L., Wang Y.M., A new B¨acklund transformation and multi-soliton solutions to the KdV equation with general variable coefficients. Physics Letters A 287(2001)211-216.
7. Tas¸can F., Bekir A., Analytic solutions of the (2 + 1)-dimensional nonlinear evolution equations using the sine–cosine method, Applied Mathematics and Computation 215(2009)3134-3139.
8. Wang M.L., Zhou Y.B., Li Z.B., Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Physics Letters A 216(1996)67-75.

9. Fan E.G., Auto-B¨acklund transformation and similarity reductions for general variable coefficient KdV equations, Physics Letters A 294(2002)26-30.
10. Parkes E.J., Duffy B.R., An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations. Computers Physics Communications 98(1996)288-300.
11. Wu, X. H. and He, J. H. Solitary solutions, periodic solutions and compacton-like solutions using the Exp-function method. Computer Mathematics with Application 54 (2007) 966–986.)
12. Zhang S., Ba JM., Sun YN., Dong L., A Generalized ( G/G-expansion method for the nonlinear Schr¨odinger equation with ′ variable coefficients, Z. Naturforsch 64(2009)691-696.
13. U. Goktas, E. Hereman Symbolic computation of conserved densities for systems of nonlinear evolution equations J. Symb. Comput. , 24(5)(1997)591–622.
14. De Angelis C., Self-trapped propagation in the nonlinear cubic-quintic equation: a variational schrodinger approach, iEEE Journal of Quantum Electronics 30(1994)818-821.
15. Yan Z,Envelope compactons and solitary patterns, Physics Letters A 355(2006)212-215.
16. P. G. Agrawal, Nonlinear Fiber Optics, (Academic Press, New York, 1989).
17. Zheng W, An envelope solitary-wave solution for a generalized nonlinear Schr¨odinger equation, Journal of Physics A 27(1994)931-934.
18. Hirota R,Exact envelope soliton solutions of the waveequation, Journal of Mathematical Physics 14(1973)805-810.
19. Yan Z, Envelope compact and solitary pattern structures for the GNLS(m,n,p,q) equations, Physics Letters A 357(2006)196–203.
20. Lai S, Wu Y, Zhou Y., Some physical structures for the (2+1) dimensional Boussinesq water equation with positive and negative exponents, Computers & Mathematics with Applications 56(2008)339–345.
21. Lai S, Wu Y., The compact and noncompact structures for two types of generalized Camassa-Holm-KP equations, Mathematical and Computer Modelling 47(2008)1089–1098.
22. Lai S, Wu Y., Wiwatanapataphee B., On exact travelling wave solutions for two types of nonlinear K(n,n) equations and a generalized KP equation, Journal of Computational and Applied Mathematics 212(2008)291–299.
23. Lai SJ., Different physical structures of solutions for a generalized Boussinesq wave equation, Computational & Applied Mathematics 231(2009)311–318.
24. Clarkson PA, Mansifield E. Symmetry reductions and exact solutions of a class of nonlinear heat equations, Physica D 70(1993)250-288.
25. Kudryashov N.A, E. D. Zargaryan, Solitary waves in active dissipative media, Journal of Physics A 29(1996)8067-8077.
26. He JH. Asymptotic methods for solitary solutions and compactons, Abstract and Applied Analysis (2012)916793.
27. Inc M, On new exact special solutions of GNLS(m,n,p,q) equations, Modern Physics Letter B 24(2010)1769-1783.
28. Inc M, New compact and noncompact solutions of the K(k,n) equations,Chaos, Solitons & Fractals29(2006)895–903.
29. Inc M, New exact solitary pattern solutions of the nonlinearly dispersive R(m,n) equations,Chaos, Solitons & Fractals 29(2006)499–505.
30. Abdou M.A., New exact travelling wave solutions for the generalized nonlinear Schroedinger equation with a source, Chaos, Solitons and Fractals 38(2008)949–955.
31. Sun C, Gao H, Exact solitary and periodic wave solutions for a generalized nonlinear Schr¨odinger equation, Chaos, Solitons and Fractals 39(2009)2399–2410.
32. Yomba E, The sub-ODE method for finding exact travelling wave solutions of generalized nonlinear Camassa–Holm, and generalized nonlinear Schr¨odinger equations, Physics Letters A 372(2008)215–222.