Some Nonlinear Fractional PDEs Involvingβ-Derivative by Using Rationalexp−Ωη

Abstract

In this article, some new nonlinear fractional partial differential equations (PDEs) (the space-time fractional order Boussinesq equation; the space-time (2 + 1)-dimensional breaking soliton equations; and the space-time fractional order SRLW equation) have been considered, in which the treatment of these equations in the diverse applications are described. Also, the fractional derivatives in the sense of β-derivative are defined. Some fractional PDEs will convert to consider ordinary differential equations (ODEs) with the help of transformation β-derivative. These equations are analyzed utilizing an integration scheme, namely, the rational exp−Ωη-expansion method. Different kinds of traveling wave solutions such as solitary, topological, dark soliton, periodic, kink, and rational are obtained as a by product of this scheme. Finally, the existence of the solutions for the constraint conditions is also shown. The outcome indicates that some fractional PDEs are used as a growing finding in the engineering sciences, mathematical physics, and so on.

A New Version of the Generalized F-Expansion Method for the Fractional Biswas-Arshed Equation and Boussinesq Equation with the Beta-Derivative

“…Thus, none of the solution functions obtained by the various methods in Ref. [37][38][39][40][41] articles contain the solutions found by the method used in this article. Owing to the F ′ /F and F/F ′ terms contained in the finite series in the applied method, various rational solution combinations of the doubleperiod Jacobi elliptic functions, which have not yet been found in the literature, have been reached.…”

“…where ϕ = ϕðx, tÞ is a complex function [37][38][39][40]; k 1 and k 2 are the parameters of the group velocity dispersion and the spatiotemporal dispersion, respectively; l 1 and l 2 are the parameters of the third-order dispersion and the spatiotemporal third-order dispersion, respectively; ε is the parameter of the self-steepening effect; and μ and θ present the parameters of the nonlinear dispersions. Also, we will research the exact solutions of the Boussinesq equation with the betaderivative [41] A…”

Jacobi Elliptic Function Solutions of Space-Time Fractional Symmetric Regularized Long Wave Equation

“…An important one of these equations is the fractional SRLW equation. So far, the solutions of the space-time fractional SRLW equation has been investigated by utilizing the sub-equation method [10], functional variable method [11], exp-function method [11], (G /G)-expansion method [11], tanh-coth method [2], tan-cot method [2], sech-csch method [2] and sec-csc method [2], a novel (G /G)-expansion method [12], Riccati equation method [13], rational (G /G)-expansion method [14], improved F -expansion method [15], the extended Jacobi elliptic function expansion method [16], the auxiliary equation method [17], new extended direct algebraic method [18], improved Bernoulli sub-equation function method [19], modified extended tanh method [20], rational exp(−Ω(η))-expansion method [21], (G /G, 1/G)-expansion method [22], extended auxiliary equation mapping method [23], (D α G/G)-expansion method [24], modified Kudryashov method [25], and the fractional (D α ξ G/G)-expansion method [26]. Among these methods, rational (G /G)-expansion, new extended direct algebraic, improved Bernoulli sub-equation function, and modified extended tanh methods include the conformable derivatives.…”