On the solutions of the nonlinear fractional differential equations via the modified trial equation method

Abstract

Communicated by I. StratisIn this study, the nonlinear fractional partial differential equations have been defined by the modified Riemann-Liouville fractional derivative. By using this fractional derivative and traveling wave transformation, the nonlinear fractional partial differential equations have been converted into nonlinear ordinary differential equations. The modified trial equation method is implemented to obtain exact solutions of the nonlinear fractional Klein-Gordon equation and fractional clannish random walker's parabolic equation. As a result, some exact solutions including single kink solution and periodic and rational function solutions of these equations have been successfully obtained.

Solitons and other solutions of (3 + 1)-dimensional space–time fractional modified KdV–Zakharov–Kuznetsov equation

“…Seeking different types of solutions to FDEs for such phenomena has become the subject of interest for researchers. Thus, a lot of powerful mathematical methods have been applied to obtain exact analytic solutions of FDEs, namely, the extended tanh-function method [6,7], the exp-function method [8,9], the sub-equation method [10,11], the improved tan(φ /2)-expansion method [12,13], the (G /G)-expansion method [14,15], the modified trial equation method [16,17], the new extended direct algebraic method [18,19], the extended sinh-Gordon equation expansion method [20,21], the unified method [22] and so on.…”

On the iterative learning control of fractional impulsive evolution equations in Banach spaces

“…Fractional differential equations have been recognized as one of the best tools to be applied in interdisciplinary field such as viscoelastic materials and electromagnetic problems. For more details, one can refer to [18][19][20][21][22][23][24][25][26][27][28][29] and the references given therein. Recently, there have been some advances in ILC theory of fractional differential systems [30][31][32][33][34][35].…”

Nonautonomous first integrals and general solutions of the KdV‐Burgers and mKdV‐Burgers equations with the source

“…18 Now, this method is well known and has been improved by other scientists. 19,20 In this paper, we apply a modification of the simplest equation method. Let us formulate the algorithm used later in this work for finding the first integrals and general solutions of nonlinear differential equations (more details this algorithm can be found in recent paper 21 ).…”