Skip to content
Research Article Open access CC BY 3.0

Fractional Sub-equation Method and Analytical Solutions to the Hirota-satsuma Coupled KdV Equation and Coupled mKdV Equation

H. Yépez-Martínez, J. M. Reyes, I. O. Sosa

Journal of Advances in Mathematics and Computer Science · pp. 572–589 · Published 21 Nov 2013

10.9734/BJMCS/2014/7059

Abstract

The fractional sub-equation method is proposed to construct analytical solutions of nonlinear fractional partial differential equations (FPDEs), involving Jumarie’s modified Riemann-Liouville derivative. The fractional sub-equation method is applied to the space-time fractional generalized Hirota-Satsuma coupled KdV equation and coupled mKdV equation. The analytical solutions show that the fractional sub-equation method is very effective for the fractional coupled KdV and mKdV equations. The solutions are compared with that of the extended tanh-function method. New exact solutions are found for the coupled mKdV equation.

Fractional sub-equation method Analytical solutions Nonlinear KdV and mKdV fractional equations.

References (45)

  1. 1 Fractional Differential Equations [DOI]
  2. 2 An Introduction to the Fractional Calculus and Fractional Differential Equations
  3. 3 Homotopy perturbation technique [DOI]
  4. 4 Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies)
  5. 5 Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results [DOI]
  6. 6 Advances in Fractional Calculus : Theoretical Developments and Applications in Physics and Engineering [DOI]
  7. 7 Fractional Calculus: Models and Numerical Methods [DOI]
  8. 8 Solitary wave solutions for variant Boussinesq equations [DOI]
  9. 9 Generalized fractional calculus and applications
  10. 10 Soliton solutions of a coupled Korteweg-de Vries equation [DOI]
  11. 11 A new operational matrix for solving fractional-order differential equations [DOI]
  12. 12 Finite Element Method for the Space and Time Fractional Fokker–Planck Equation [DOI]
  13. 13 Physics of Fractal Operators
  14. 14 Advanced Local Fractional Calculus and Its Applications
  15. 15 Fractional sub-equation method and its applications to nonlinear fractional PDEs [DOI]
  16. 16 A generalized differential transform method for linear partial differential equations of fractional order [DOI]
  17. 17 Modified extended tanh-function method for solving nonlinear partial differential equations [DOI]
  18. 18 Fractional variational iteration method and its application [DOI]
  19. 19 The variational iteration method: An efficient scheme for handling fractional partial differential equations in fluid mechanics [DOI]
  20. 20 Numerical approaches to fractional calculus and fractional ordinary differential equation [DOI]
  21. 21 Soliton solutions for a generalized Hirota–Satsuma coupled KdV equation and a coupled MKdV equation [DOI]
  22. 22 Local Fractional Functional Analysis and Its Applications
  23. 23 Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation [DOI]
  24. 24 The Adomian decomposition method for solving partial differential equations of fractal order in finite domains [DOI]
  25. 25 Analytical solution of the linear fractional differential equation by Adomian decomposition method [DOI]
  26. 26 Fractional partial differential equations and modified Riemann-Liouville derivative new methods for solution [DOI]
  27. 27 Existence and uniqueness for -type fractional neutral differential equations [DOI]
  28. 28 Qualitative analysis and traveling wave solutions for the perturbed nonlinear Schrödinger's equation with Kerr law nonlinearity [DOI]
  29. 29 New exact solutions to the perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity [DOI]
  30. 30 Adomian’s decomposition method for solving an intermediate fractional advection–dispersion equation [DOI]
  31. 31 A GENERALIZED EXP-FUNCTION METHOD FOR FRACTIONAL RICCATI DIFFERENTIAL EQUATIONS
  32. 32 A generalized Hirota–Satsuma coupled Korteweg–de Vries equation and Miura transformations [DOI]
  33. 33 A finite difference scheme for fractional sub-diffusion equations on an unbounded domain using artificial boundary conditions [DOI]
  34. 34 New exact solutions to the perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity via modified trigonometric function series method [DOI]
  35. 35 Fractional Sub-Equation Method and its Applications to the Space–Time Fractional Differential Equations in Mathematical Physics [DOI]
  36. 36 The modified G ′ G -expansion method and traveling wave solutions of nonlinear the perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity [DOI]
  37. 37 The modified extended tanh-function method for solving coupled MKdV and coupled Hirota–Satsuma coupled KdV equations [DOI]
  38. 38 A Note on Exact Traveling Wave Solutions of the Perturbed Nonlinear Schrödinger's Equation with Kerr Law Nonlinearity [DOI]
  39. 39 The Fractional Sub-Equation Method and Exact Analytical Solutions for Some Nonlinear Fractional PDEs [DOI]
  40. 40 The Extended Fractional Subequation Method for NonlinearFractional Differential Equations [DOI]
  41. 41 Approximate Analytic Solutions of Time-Fractional Hirota-Satsuma Coupled KdV Equation and Coupled MKdV Equation [DOI]
  42. 42 Homotopy perturbation method for a type of nonlinear coupled equations with parameters derivative [DOI]
  43. 43 Solving Famous Nonlinear Coupled Equations with Parameters Derivative by Homotopy Analysis Method [DOI]
  44. 44 Numerical Solution of Hirota-Satsuma Coupled MKdV Equation with Quantic B-Spline Collocation Method [DOI]
  45. 45 Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models [DOI]

Cited by 18

Study on Fractional Differential Equations with Modified Riemann–Liouville Derivative via Kudryashov Method

E. Aksoy, A. Bekir, A. Cevikel · International journal of nonlinear sciences and numerical simulation · 2019

Closed form solutions to the coupled space-time fractional evolution equations in mathematical physics through analytical method

M. N. Islam, M. Akbar · JOURNAL OF MECHANICS OF CONTINUA AND MATHEMATICAL SCIENCES · 2018

Series Solution for the Time-Fractional Coupled mKdV Equation Using the Homotopy Analysis Method

J. Gómez-Aguilar, H. Yépez-Martínez, R. Escobar-Jiménez · 2016

Fractional Subequation Method

· Computational Fractional Dynamical Systems · 2022

Closed form soliton solutions to the space-time fractional foam drainage equation and coupled mKdV evolution equations

Onur Alp İlhan, Fatma Berna Benli, M. Nurul Islam · International Journal of Nonlinear Sciences and Numerical Simulation · 2021

Showing 10 of 18 known citations — external sources report more than can currently be individually listed.

Article metrics

Real usage data collected on this platform.

49

Page views

0

PDF downloads

18

Outbound clicks

18

Citations

Views over time

Views by country

Approximate, from request IP at view time — not citizenship or institution. Countries with fewer than 5 views are grouped as "Other".

Traffic sources

Referring site, by host.

Views and downloads exclude known bots/crawlers. Citations combines this platform's own DOI-resolved index with each external source's own reported total — see Cited by above for individually listed citing works. Last refreshed 0 seconds ago.