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Multiple Soliton Solutions for Nonlinear Differential Equations with a New Version of Extended F-Expansion Method

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Abstract

In this article, a new version of extended F-expansion method is suggested. A multiple Jacobi elliptic functions are presented in the solution function. We achieve analytical solutions of the (1 + 1)-dimensional dispersive long wave (DLW) equation and (2 + 1)-dimensional Kadomtsev–Petviashvili (KP) equation using the new version of extended F-expansion method. Then, the single and the combined non-degenerative Jacobi elliptic function solutions and their degenerate solutions to the above-mentioned class of nonlinear evolution equations are obtained.

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References

  1. Hu XB, Ma WX (2002) Application of Hirota’s bilinear formalism to the Toeplitz lattice-some special soliton-like solutions. Phys Lett A 293:161–165

    Article  MathSciNet  MATH  ADS  Google Scholar 

  2. Zhang Y (2016) Lie symmetry analysis and exact solutions of the Sharma–Tasso–Olever equation. IAENG Int J Appl Math 46(2):1–5

    MathSciNet  Google Scholar 

  3. Liu CS (2006) Trial equation method for nonlinear evolution equations with rank inhomogeneous: mathematical discussions and applications. Commun Theor Phys 45(2):219–223

    Article  ADS  Google Scholar 

  4. Gurefe Y, Sonmezoglu A, Misirli E (2011) Application of trial equation method to the nonlinear partial differential equations arising in mathematical physics. Pramana J Phys 77(6):1023–1029

    Article  ADS  Google Scholar 

  5. Pandir Y, Gurefe Y, Kadak U, Misirli E (2012) Classifications of exact solutions for some nonlinear partial differential equations with generalized evolution. Abstr Appl Anal 2012:1–16

    Article  MathSciNet  MATH  Google Scholar 

  6. Gurefe Y, Misirli E, Sonmezoglu A, Ekici M (2013) Extended trial equation method to generalized nonlinear partial differential equations. Appl Math Comput 219(10):5253–5260

    MathSciNet  MATH  Google Scholar 

  7. Chen HT, Hong-Qing Z (2004) New double periodic and multiple soliton solutions of the generalized (2 + 1)-dimensional Boussinesq equation. Chaos Solitons Fractals 20:765–769

    Article  MathSciNet  MATH  ADS  Google Scholar 

  8. Chen Y, Yan Z (2006) The Weierstrass elliptic function expansion method and its applications in nonlinear wave equations. Chaos Solitons Fractals 29:948–964

    Article  MathSciNet  MATH  ADS  Google Scholar 

  9. Abdou MA (2008) Further improved F-expansion and new exact solutions for nonlinear evolution equations. Nonlinear Dyn 52:227–288

    Article  MathSciNet  Google Scholar 

  10. Cai G, Wang Q, Huang J (2006) A modified F-expansion method for solving breaking soliton equation. Int J Nonlinear Sci 2:122–128

    Google Scholar 

  11. Zhang S, Xia T (2007) A generalized F-expansion method with symbolic computation exactly solving Broer–Kaup equations. Appl Math Comput 189:949–955

    MathSciNet  MATH  Google Scholar 

  12. Zhang S, Xia T (2008) An improved generalized F-expansion method and its application to the (2 + 1)-dimensional KdV equations. Commun Nonlinear Sci Num Simul 13:1294–1301

    Article  MathSciNet  MATH  Google Scholar 

  13. Zeng X, Wang DS (2009) A generalized extended rational expansion method and its application to (1 + 1)-dimensional dispersive long wave equation. Appl Math Comput 212:296–304

    MathSciNet  MATH  Google Scholar 

  14. Chen C, Lou SY (2003) Soliton excitations and periodic waves without dispersion relation in shallow water system. Chaos Solitons Fractals 16:27–35

    Article  MathSciNet  MATH  ADS  Google Scholar 

  15. Khan K, Akbar MA (2014) Solitary wave solutions of some coupled nonlinear evolution equations. J Sci Res 6(2):273–284

    Article  Google Scholar 

  16. Das P, Natesan S (2014) Optimal error estimate using mesh equidistribution technique for singularly perturbed system of reaction-diffusion boundary value problems. Appl Math Comput 249:265–277

    MathSciNet  MATH  Google Scholar 

  17. Das P, Natesan S (2015) Adaptive mesh generation for singularly perturbed fourth order ordinary differential equations. Int J Comput Math 92(3):562–578

    Article  MathSciNet  MATH  Google Scholar 

  18. Das P, Natesan S (2013) Numerical solution of a system of singularly perturbed convection diffusion boundary value problems using mesh equidistribution technique. Aust J Math Anal Appl 10(1):1–17

    MathSciNet  MATH  Google Scholar 

  19. Das P, Natesan S (2013) Richardson extrapolation method for singularly perturbed convection-diffusion problems on adaptively generated mesh. CMES Comput Model Eng Sci 90(6):463–485

    MathSciNet  MATH  Google Scholar 

  20. Das P, Natesan S (2013) A uniformly convergent hybrid scheme for singularly perturbed system of reaction-diffusion Robin type boundary value problems. J. Appl Math Comput 41(1–2):447–471

    Article  MathSciNet  MATH  Google Scholar 

  21. Das P, Natesan S (2012) Higher order parameter uniform convergent schemes for Robin type reaction–diffusion problems using adaptively generated grid. Int J Comput Methods 9(4):1250052

    Article  MathSciNet  MATH  Google Scholar 

  22. Klein C, Roidot K (2011) Fourth order time-stepping for Kadomtsev–Petviashvili and Davey–Stewartson equations. SIAM J Sci Comput 33(6):3333–3356

    Article  MathSciNet  MATH  Google Scholar 

  23. Roozi A, Mahmeiani AG (2011) The (G′/G)-expansion method for (2 + 1)-dimensional Kadomtsev–Petviashvili equation. World Appl Sci J 13(10):2231–2234

    Google Scholar 

  24. Ablowitz MJ, Demirci A, Ma YP (2016) Dispersive shock waves in the Kadomtsev-Petviashvili and two dimensional Benjamin–Ono equations. Physica D 333:84–98

    Article  MathSciNet  MATH  ADS  Google Scholar 

  25. Kadomtsev BB, Petviashvili VI (1970) On the stability of solitary waves in weakly dispersing media. Sov Phys Dokl 15(6):539–541

    MATH  ADS  Google Scholar 

  26. Ablowitz MJ, Segur H (1979) On the evolution of packets of water waves. J Fluid Mech 92:691–715

    Article  MathSciNet  MATH  ADS  Google Scholar 

  27. Das P, Vigo-Aguiar J (2019) Parameter uniform optimal order numerical approximation of a class of singularly perturbed system of reaction diffusion problems involving a small perturbation parameter. J Comput Appl Math 354:533–544

    Article  MathSciNet  MATH  Google Scholar 

  28. Das P (2019) An a posteriori based convergence analysis for a nonlinear singularly perturbed system of delay differential equations on an adaptive mesh. Numer Algorithm 81:465–487

    Article  MathSciNet  MATH  Google Scholar 

  29. Das P (2018) A higher order difference method for singularly perturbed parabolic partial differential equations. J Differ Equ Appl 24(3):452–477

    Article  MathSciNet  MATH  Google Scholar 

  30. Chandru M, Das P, Ramos H (2018) Numerical treatment of two-parameter singularly perturbed parabolic convection diffusion problems with non-smooth data. Math Methods Appl Sci 41(14):5359–5387

    Article  MathSciNet  MATH  Google Scholar 

  31. Chandru M, Prabha T, Das P, Shanthi V (2019) A numerical method for solving boundary and interior layers dominated parabolic problems with discontinuous convection coefficient and source terms. Differ Equ Dyn Syst 27(1–3):91–112

    Article  MathSciNet  MATH  Google Scholar 

  32. Das P, Mehrmann V (2016) Numerical solution of singularly perturbed parabolic convection–diffusion reaction problems with two small parameters. BIT Numer Math 56:51–76

    Article  MATH  Google Scholar 

  33. Das P (2015) Comparison of a priori and a posteriori meshes for singularly perturbed nonlinear parameterized problems. J Comput Appl Math 290:16–25

    Article  MathSciNet  MATH  Google Scholar 

  34. Das P, Mehrmann V (2015) Upwind based parameter uniform convergence analysis for two parametric parabolic convection diffusion problems by moving mesh methods. Proc Appl Math Mech 15:591–592

    Article  Google Scholar 

  35. Das P, Rana S, Ramos H (2019) Homotopy perturbation method for solving Caputo type fractional order Volterra–Fredholm integro-differential equations. Comput Math Methods 1(e1047):1–9. https://doi.org/10.1002/cmm4.1047

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Arts and Sciences, Bozok University, 66100, Yozgat, Turkey

    Yusuf Pandir

  2. The Graduate School of Natural and Applied Sciences, Bozok University, 66100, Yozgat, Turkey

    Nail Turhan

Authors
  1. Yusuf Pandir
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  2. Nail Turhan
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Correspondence to Yusuf Pandir.

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Significance Statement

In this study, we attempted to construct a more general expression of the F-expansion methods such as improved F-expansion method, generalized F-expansion method, extended F-expansion method that found earlier analytical solutions. Through our proposed method, we have obtained more general solutions of the results from the other methods.

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Pandir, Y., Turhan, N. Multiple Soliton Solutions for Nonlinear Differential Equations with a New Version of Extended F-Expansion Method. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 91, 495–501 (2021). https://doi.org/10.1007/s40010-020-00687-9

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  • DOI: https://doi.org/10.1007/s40010-020-00687-9

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