Close Form Solutions of the Fractional Generalized Reaction Duffing Model and the Density Dependent Fractional Diffusion Reaction Equation

Abstract

Abstract:The two variable ( ⁄ , 1 ⁄ )-expansion method is significant for finding the exact traveling wave solution to nonlinear evolution equations (NLEEs) in mathematical physics, applied mathematics and engineering. In this article, we exert the two variable ( ⁄ , 1 ⁄ )-expansion method for investigating the fractional generalized reaction Duffing model and density dependent fractional diffusion reaction equation and obtain exact solutions containing parameters. When the parameters are taken particular values, traveling wave solutions are transferred into the solitary wave solutions. The two variable ( ′ ⁄ , 1 ⁄ )-expansion method is the generalization of the original ( ⁄ )-expansion method established by Wang et al [21].

Diverse Precise Traveling Wave Solutions Possessing Beta Derivative of the Fractional Differential Equations Arising in Mathematical Physics

“…We have also depicted some of the obtained solutions graphically (3D surface graphs and 2D line plots) and concluded that the results we obtained are accurate, efficient, and versatile in mathematical physics. It is worth to noticing that compared to previous works [25,26,44,45]; the results obtained in this paper are presented for the first time. Lastly, it can be concluded that our offered methods are more effective, reliable, and powerful, which give bounteous consistent solutions to NLPFDEs arise in different fields of nonlinear sciences.…”

“…Density dependent fractional diffusion reaction equation which is widely used in mathematical biology in the form [44,45]…”

“…The core aim of this work is to establish the exact traveling wave solutions of the fractional generalized reaction doffing model arising in mathematical biology [44,45] and the density dependent fractional diffusion-reaction equation with the beta-derivative based on three different methods, the Generalized tanh (GT) method [46], Generalized Bernoulli (GB) sub-ODE method [47], and Riccati-Bernoulli (RB) sub-ODE method [48]. These methods are the most direct and effective algebraic methods used for obtaining the exact traveling wave solutions of nonlinear partial differential equations.…”

“…In [49], Jafari et al applied the fractional subequation method to construct exact solutions of the fractional generalized reaction Duffing model and in [50], Eslami et al applied the first integral method to obtain the exact solutions of fractional generalized reaction Duffing model and the exact solutions of fractional diffusionreaction equation. Uddin et al [44] obtained the close form solutions of the fractional generalized reaction Duffing model and the density dependent fractional diffusion reaction equation by using the ðG ′ /G, 1/GÞ-expansion method. In [51] Xia et al applied hyperbolic function to obtain new explicit and exact travelling wave solutions for a class of nonlinear evolution equations.…”

New Explicit Solutions to the Fractional-Order Burgers’ Equation

“…e major challenges, however, are that there is no unified numerical or analytical approach that can investigate all sorts of nonlinear fractional-order differential equations. us, several numerical and theoretical methods for finding solutions for NLFDEs have been established, for example, the differential transformation method [6,7], the variational iteration method [8][9][10], the fractional subequation method [11], the Kudryashov [12] method, the homotopy perturbation method [13,14], the homotopy analysis method [15], the exp-function method [16,17], the (G ′ /G)-expansion method and its various modification [18][19][20][21][22], the Chelyshkov polynomial method [23,24], the multiple exp-function method [25], the finite difference method [26], the finite element method [27], the first integral method [28,29], the modified simple equation method [30], the reproducing kernel method [31], the two variables ((G ′ /G), (1/G))-expansion method [32,33], and the Picard technique [34].…”