Nonlinear Space Time Fractional Differential Equations Research Articles

The kink solitary wave and numerical solutions for conformable non-linear space–time fractional differential equations

This paper is about the conformable non-linear space–time fractional Telegraph and the non-linear space–time fractional Kolmogorov–Petrovskii–Piskunov (KPP) equations. To construct some real and analytical wave solutions of both equations the simplest equation method is applied. As a consequence, a series of analytical wave solutions are obtained as well as verify these solutions by using Mathematica software. These solutions denote that this method is useful and straightforward. We show some graphs in 2-D and 3-D of our solution to illustrate the behaviour of waves. Finally, we introduce the numerical solutions for the conformable non-linear space–time fractional Telegraph and the non-linear space–time fractional KPP equations using the finite difference method and make comparisons with analytical solutions.

Some optical soliton solutions of space-time conformable fractional Schrödinger-type models

In this article, we introduce a family of nonlinear (1+1) dimensions Schrödinger-type models with space-time fractional evolution in the sense of a conformable fractional derivative. We apply the modified Kudryashov method in context of fractional complex transformation and seek a variety of optical soliton solutions for these equations. The modified Kudryashov method is efficient and consistent for solving nonlinear space-time fractional differential equations.

Exact solitary wave solutions for a system of some nonlinear space–time fractional differential equations

We have enumerated new and exact general wave solutions, along with multiple exact soliton solutions of space–time nonlinear fractional differential equations (FDE), namely Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZKBBM), foam drainage and symmetric regularised long-wave (SRLW) equations, by employing a relatively new technique called ( $$G^\prime /G, 1/G)$$ -expansion method. Also, based on fractional complex transformation and the properties of the modified Riemann–Liouville fractional-order operator, the fractional partial differential equations transform into a form of ordinary differential equation (ODE). This method is a recollection of the commutation of the well-appointed ( $$G^\prime /G$$ )-expansion method introduced by Wang et al, Phys. Lett. A 372, 417 (2008) In this paper, it is mentioned that the two-variable ( $$G^\prime /G, 1/G$$ )-expansion method is more legitimate, modest, sturdy and effective in the sense of theoretical and pragmatical point of view. Lastly, the peculiarities of these analytic solutions are illustrated graphically by utilising the computer symbolic programming Wolfram Mathematica.

The generalized Kudryashov method for nonlinear space–time fractional partial differential equations of Burgers type

The nonlinear space–time fractional differential equations (FDE) of Burgers’ type play an important role for describing many phenomena in applied sciences. This paper develops a generalized version of the modified Kudryashov method to obtain the exact solutions for FDE of Burgers’ type. The FDE are firstly reduced to a set of ordinary differential equations by means of a fractional complex transformation and, afterward, they are solved by applying the generalized Kudryashov method (GKM). Several kinds of traveling wave solutions such as kink and singular kink waves are illustrated with various figures and discussed. The results demonstrate the efficiency of the GKM in tackling space–time partial FDE.

Some exact solutions of the nonlinear space–time fractional differential equations

ABSTRACTIn this article, we construct exact soliton solutions of the nonlinear space–time fractional Klein–Gordon equation, Hirota–Satsuma coupled KdV (HSC KdV) system, coupled Whitham–Broer–Kaup (WBK) system, and KdV–Zakharov–Kuznetsov (KdV-ZK) equation using exponential rational function method. Our solutions include soliton-like, singular soliton, and kink solutions. Two-dimensional and three-dimensional graphical representation of all solutions is also presented to understand the accuracy of the technique. Moreover, a comparison of the recent solutions is made with some already existing solutions. It has been shown that the suggested method will be helpful in finding the solutions of different kind of nonlinear fractional differential equations emerging in mathematical physics. Throughout the paper, all the calculations are made with the aid of the Maple software.

The Exp-function method for solving nonlinear space–time fractional differential equations in mathematical physics

Using the Exp-function method, we derive exact solutions of the nonlinear space–time fractional Telegraph equation and space–time fractional KPP equation. As a result, we obtain many exact analytical solutions including hyperbolic function. The fractional derivative is described in Jumarie’s modified Riemann–Liouville sense. This method is very effective and convenient for solving nonlinear fractional differential equations.

Exact solutions of some fractional differential equations by various expansion methods

In this paper, we construct the exact solutions of some nonlinear spacetime fractional differential equations involving modified Riemann-Liouville derivative in mathematical physics and applied mathematics; namely the fractional modified Benjamin-Bona- Mahony (mBBM) and Kawahara equations by using G'/G and (G'/G, 1/G)-expansion methods.

The G ′ / G , 1 / G $\boldsymbol {\left (G^{\prime }/G,1/G\right )}$ -expansion method for solving nonlinear space–time fractional differential equations

The G ′ / G , 1 / G $\boldsymbol {\left (G^{\prime }/G,1/G\right )}$ -expansion method for solving nonlinear space–time fractional differential equations

A new fractional Biswas–Milovic model with its periodic soliton solutions

The present study introduces a general form of fractional Biswas–Milovic (FBM) equation. After this, some exact solutions for FBM equation are obtained by Exp-function and sine–cosine methods. This work illustrates the validity and great potential of the Exp-function and sine–cosine approaches for the nonlinear space–time fractional differential equations with complex valued solutions. One can see that the methods are relatively easy and efficient to use.

Solving Space-Time Fractional Differential Equations by Using Modified Simple Equation Method

In this article, we establish new and more general traveling wave solutions of space-time fractional Klein–Gordon equation with quadratic nonlinearity and the space-time fractional breaking soliton equations using the modified simple equation method. The proposed method is so powerful and effective to solve nonlinear space-time fractional differential equations by with modified Riemann–Liouville derivative.

Analytical solution of nonlinear space–time fractional differential equations using the improved fractional Riccati expansion method

In this paper, the improved fractional Riccati expansion method is proposed to solve fractional differential equations. The method is applied to solve space–time fractional modified Korteweg–de Vries equation, space–time fractional modified regularized long-wave equation, time fractional biological population model, and space–time fractional Klein–Gordon equation. The obtained solutions include generalized trigonometric and hyperbolic functions solutions. Among these solutions, some are found for the first time.

Bäcklund transformation of fractional Riccati equation and its applications to the space–time FDEs

In this paper, the Bäcklund transformation of fractional Riccati equation is presented to establish traveling wave solutions for two nonlinear space–time fractional differential equations in the sense of modified Riemann–Liouville derivatives, namely, the space–time fractional generalized reaction duffing equation and the space–time fractional diffusion reaction equation with cubic nonlinearity. The proposed method is effective and convenient for solving nonlinear evolution equations with fractional order. Copyright © 2014 John Wiley & Sons, Ltd.

Generalized exp-function method for non-linear space-time fractional differential equations

A generalized exp-function method is proposed to solve non-linear space-time fractional differential equations. The basic idea of the method is to convert a fractional partial differential equation into an ordinary equation with integer order derivatives by fractional complex transform. To illustrate the effectiveness of the method, space-time fractional asymmetrical Nizhnik-Novikor-Veselov equation is considered. The fractional derivatives in the present paper are in Jumarie?s modified Riemann-Liouville sense.

Solution of Nonlinear Space-Time Fractional Differential Equations Using the Fractional Riccati Expansion Method

The fractional Riccati expansion method is proposed to solve fractional differential equations. To illustrate the effectiveness of the method, space-time fractional Korteweg-de Vries equation, regularized long-wave equation, Boussinesq equation, and Klein-Gordon equation are considered. As a result, abundant types of exact analytical solutions are obtained. These solutions include generalized trigonometric and hyperbolic functions solutions which may be useful for further understanding of the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time. The periodic and kink solutions are founded as special case.