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

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 tanhfunction method. New exact solutions are found for the coupled mKdV equation.

Q-homotopy analysis method for time-fractional Newell–Whitehead equation and time-fractional generalized Hirota–Satsuma coupled KdV system

“…Some exact solutions of system (2) were constructed by Saberi and Hejazi using the invariant subspace method with Caputo sense [38]. Martínez, Reyes and Sosa [39] obtained the analytical solutions by applying the sub-equation method for the time-space fractional generalized HS-cKdVS.…”

Two efficient computational technique for fractional nonlinear Hirota–Satsuma coupled KdV equations

“…Likewise, HS-cKdV equations have been studied previously by various techniques. He and Wu (2006) used variational iteration method, Kaya (2004) used an Adomain decomposition method, Ganji and Rafei (2006) used homotopy perturbation method, Abbasbandy (2007) used homotopy analysis method, Yu et al (2005) used Jacobi elliptic function method, Yang and Zhang (2005) used Projective Riccati equation method, Zayad et al (2004) used algebraic method, Abzari and Abzari (2012) used reduced differential transform method, Lu (2012) used first integral method for time fractional HS-cKdV equation, Yepez-Martinez et al (2014) used sub equation method, Lie and Li (2013) used generalized two-dimensional differential transform method and Merdan (2015) used variational iteration method to get the numerical solution to the HS-cKdV equation with modified Riemann-Liouville derivative. However, the fractional model of HS-cKdV equations has not been studied by q-homotopy analysis Sumudu transform method (q-HASTM) and RPSM.…”