Ozkan Guner

Bright and dark soliton solutions for some nonlinear fractional differential equations

In this work, we propose a new approach, namely ansatz method, for solving fractional differential equations based on a fractional complex transform and apply it to the nonlinear partial space–time fractional modified Benjamin–Bona–Mahoney(m BBM) equation, the time fractional m Kd V equation and the nonlinear fractional Zoomeron equation which gives rise to some new exact solutions. The physical parameters in the soliton solutions: amplitude, inverse width, free parameters and velocity are obtained as functions of the dependent model coefficients. This method is suitable and more powerful for solving other kinds of nonlinear fractional PDEs arising in mathematical physics. Since the fractional derivatives are described in the modified Riemann–Liouville sense.

A novel method for nonlinear fractional differential equations using symbolic computation

In this paper, a new approach, namely an ansatz method is applied to find exact solutions for nonlinear fractional differential equations in the sense of modified Riemann–Liouville derivative. Based on a nonlinear fractional complex transformation, a certain fractional partial differential equation can be turned into another ordinary differential equation of integer order. For illustrating the validity of this method, we apply it to solve the fractional-order biological population model and the space–time fractional modified equal width equation, and as a result, some dark soliton solutions for them are established.

Different methods for ( 3 + 1 ) -dimensional space-time fractional modified KdV-Zakharov-Kuznetsov equation

In this paper, the fractional derivatives in the sense of modified Riemann-Liouville derivative and the exp-function method, the ( G G ) -expansion method and the generalized Kudryashov method?are used to construct exact solutions for ( 3 + 1 ) -dimensional space-time fractional modified KdV-Zakharov-Kuznetsov equation. This fractional equation can be turned into another nonlinear ordinary differential equation by fractional complex transformation and then these three methods are applied to solve it. As a result, some new exact solutions are obtained. The three methods demonstrate power, reliability and efficiency.

Dark Soliton Solutions of Space-Time Fractional Sharma–Tasso–Olver and Potential Kadomtsev–Petviashvili Equations

Dark soliton solutions for space-time fractional Sharma–Tasso–Olver and space-time fractional potential Kadomtsev–Petviashvili equations are determined by using the properties of modified Riemann–Liouville derivative and fractional complex transform. After reducing both equations to nonlinear ODEs with constant coefficients, the ansatz is substituted into the resultant nonlinear ODEs. The coefficients of the solutions in the ansatz are calculated by algebraic computer computations. Two different solutions are obtained for the Sharma–Tasso–Olver equation as only one solution for the potential Kadomtsev–Petviashvili equation. The solution profiles are demonstrated in 3D plots in finite domains of time and space.

Traveling wave solutions for time-dependent coefficient nonlinear evolution equations

In this article, we establish exact solutions for variable-coefficient modified KdV equation, variable-coefficient KdV equation, and variable-coefficient diffusion–reaction equations. The modified sine-cosine method is used to construct exact periodic solutions. These solutions may be important for the explanation of some practical physical problems. The obtained results show that the modified sine-cosine method provides a powerful mathematical tool for solving nonlinear equations with variable coefficients.

Soliton and periodic solutions for time-dependent coefficient non-linear equation

In this article, we establish exact solutions for the generalized (3+1)-dimensional variable coefficient Kadomtsev–Petviashvili (GVCKP) equation. Using solitary wave ansatz in terms of functions and the modified sine–cosine method, we find exact analytical bright soliton solutions and exact periodic solutions for the considered model. The physical parameters in the soliton solutions are obtained as function of the dependent model coefficients. The effectiveness and reliability of the method are shown by its application to the GVCKP equation.

Exact solutions to the time-fractional differential equations via local fractional derivatives

Abstract This article utilizes the local fractional derivative and the exp-function method to construct the exact solutions of nonlinear time-fractional differential equations (FDEs). For illustrating the validity of the method, it is applied to the time-fractional Camassa–Holm equation and the time-fractional-generalized fifth-order KdV equation. Moreover, the exact solutions are obtained for the equations which are formed by different parameter values related to the time-fractional-generalized fifth-order KdV equation. This method is an reliable and efficient mathematical tool for solving FDEs and it can be applied to other non-linear FDEs.

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

Abstract 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.

The exp-function method for some time-fractional differential equations

In this article, the fractional derivatives in the sense of modified Riemann-Liouville derivative and the Exp-function method are employed for constructing the exact solutions of nonlinear time fractional partial differential equations in mathematical physics. As a result, some new exact solutions for them are successfully established. It is indicated that the solutions obtained by the Exp-function method are reliable, straightforward and effective method for strongly nonlinear fractional partial equations with modified Riemann-Liouville derivative by Jumarie U+02BC s. This approach can also be applied to other nonlinear time and space fractional differential equations.

Exp-Function Method and Fractional Complex Transform for Space-Time Fractional KP-BBM Equation

In the present article, Hes fractional derivative, the ansatz method, the (G/G)-expansion method, and the exp-function method are used to construct the exact solutions of nonlinear space-time fractional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony (KP-BBM). As a result, different types of exact solutions are obtained. Also we have examined the relation between the solutions obtained from the different methods. These methods are an efficient mathematical tool for solving fractional differential equations (FDEs) and it can be applied to other nonlinear FDEs.