Analytic Solutions of the Space-Time Fractional Combined KdV-mKdV Equation

Abstract

The fractional mapping method is proposed to solve fractional differential equations. To illustrate the effectiveness of the method, we discuss the space-time fractional combined KdV-mKdV equation. Many types of exact analytical solutions are obtained. The solutions include generalized trigonometric and hyperbolic functions solutions. These solutions are useful to understand the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time.

A Table Lookup Method for Exact Analytical Solutions of Nonlinear Fractional Partial Differential Equations

“…In this section, we will apply the table lookup method to the following space-time fractional combined KdV-mKdV equation [37]:…”

“…It can be observed from (30)-(41) that we have successfully obtained twelve exact analytical solutions of the space-time fractional combined KdV-mKdV equation. In comparison, [37] using the fractional mapping method only obtained five analytical solutions; thus, the proposed lookup table method is more concise and more effective.…”

The Extended Fractional (DξαG/G)-Expansion Method and Its Applications to a Space-Time Fractional Fokas Equation

“…Thus, it is an important and significant task to find more exact solutions of different forms for the FPDEs. In recent decades, many mathematicians and physicists have made significant achievements and also presented some effective methods, for example, the fractional subequation method [1][2][3], the Jacobi elliptic equation method [4], the fractional mapping method [5], the ( / )-expansion method [6], the extended fractional Riccati expansion method [7], the first integral method [8], and the fractional complex transform [9]. Due to these methods, various exact solutions or numerical solutions of FPDEs have been established successfully.…”

A new method for exact solutions of variant types of time‐fractional Korteweg‐de Vries equations in shallow water waves

“…The KdV‐mKdV equation mainly describes the propagation of bounded particle of the atmosphere dust‐acoustic solitary waves, internal solitary waves in shallow seas and ion acoustic waves in plasmas with negative ions . Here, the governing equation is the time‐fractional combined KdV‐mKdV equation , which is given as where a , b , γ are the constants and α is the fractional order whose range is 0 < α ≤1.…”

“…The approximate solution of KdVB equation has been studied by Kaya [20] and Wang et al [21]. The Exact travelling solutions of KdVB equation have been established by Feng [22], Jeffrey and Xu [23], Bikbaev [24], Bona and Schonbek [25], Feng and Knobel [26].The KdV-mKdV equation mainly describes the propagation of bounded particle of the atmosphere dust-acoustic solitary waves, internal solitary waves in shallow seas and ion acoustic waves in plasmas with negative ions [27][28][29]. Here, the governing equation is the time-fractional combined KdV-mKdV equation [29], which is given as Dt u C auu x C bu 2 u x C u xxx D 0( 1.2) where a, b, are the constants and˛is the fractional order whose range is 0 <˛Ä 1.…”