The Extended Trial Equation Method for Some Time Fractional Differential Equations

Abstract

Nonlinear fractional partial differential equations have been solved with the help of the extended trial equation method. Based on the fractional derivative in the sense of modified Riemann-Liouville derivative and traveling wave transformation, the fractional partial differential equation can be turned into the nonlinear nonfractional ordinary differential equation. For illustrating the reliability of this approach, we apply it to the generalized third order fractional KdV equation and the fractionalKn,nequation according to the complete discrimination system for polynomial method. As a result, some new exact solutions to these nonlinear problems are successfully constructed such as elliptic integral function solutions, Jacobi elliptic function solutions, and soliton solutions.

New Exact Solutions of the New Hamiltonian Amplitude-Equation and Fokas Lenells Equation

“…In this study, GKM [10][11][12] and ETEM [14][15][16][17][18][19][20][21] will be investigated to find exact solutions of the new Hamiltonian amplitude equation and Fokas-Lenells equation.…”

Counterexamples on Jumarie’s three basic fractional calculus formulae for non-differentiable continuous functions

“…The formula (3) has been applied to solve the exact solutions to some nonlinear fractional order differential equations(see, for example, Refs. [6][7][8][9]).…”

High‐order compact finite difference and laplace transform method for the solution of time‐fractional heat equations with dirchlet and neumann boundary conditions

“…A considerable effort has been contributed by Bhrawy et al [12][13][14][15][16] focusing on the hybridization of the shifted Legendre tau method with spectral methods for fractional differential equations. More recent work toward the analytic solution of fractional partial differential equations has been conducted by various researchers (see [17][18][19]) and numerical solutions are sought by [20][21][22][23][24], these methods include the use of homotopy analysis method, a novel method involving expressing the solution as a finite sum of arbitrary degree, two-dimensional Legendre functions, an iterative Laplace transform method and the so-called trial equation method [22]. Jacobs and Harley [25,26] previously coupled the Laplace transform with finite-difference discretization and Chebyshev-collocation.…”