Özkan Güner

Fractional Complex Transform and exp-Function Methods for Fractional Differential Equations

The exp-function method is presented for finding the exact solutions of nonlinear fractional equations. New solutions are constructed in fractional complex transform to convert fractional differential equations into ordinary differential equations. The fractional derivatives are described in Jumaries modified Riemann-Liouville sense. We apply the exp-function method to both the nonlinear time and space fractional differential equations. As a result, some new exact solutions for them are successfully established.

The First Integral Method for Exact Solutions of Nonlinear Fractional Differential Equations

In this paper, we establish exact solutions for some nonlinear fractional differential equations (FDEs). The first integral method with help of the fractional complex transform (FCT) is used to obtain exact solutions for the time fractional modified Korteweg–de Vries (fmKdV) equation and the space–time fractional modified Benjamin–Bona–Mahony (fmBBM) equation. This method is efficient and powerful in solving kind of other nonlinear FDEs.

Exact Solutions of the Space Time Fractional Symmetric Regularized Long Wave Equation Using Different Methods

We apply the functional variable method, exp-function method, and -expansion method to establish the exact solutions of the nonlinear fractional partial differential equation (NLFPDE) in the sense of the modified Riemann-Liouville derivative. As a result, some new exact solutions for them are obtained. The results show that these methods are very effective and powerful mathematical tools for solving nonlinear fractional equations arising in mathematical physics. As a result, these methods can also be applied to other nonlinear fractional differential equations.

Singular and non-topological soliton solutions for nonlinear fractional differential equations

In this article, the fractional derivatives are described in the modified Riemann–Liouville sense. We propose a new approach, namely an ansatz method, for solving fractional differential equations (FDEs) based on a fractional complex transform and apply it to solve nonlinear space–time fractional equations. As a result, the non-topological as well as the singular soliton solutions are obtained. This method can be suitable and more powerful for solving other kinds of nonlinear fractional FDEs arising in mathematical physics.

Analytical Approach for the Space-time Nonlinear Partial Differential Fractional Equation

Abstract In this paper, the fractional derivatives in the sense of modified Riemann–Liouville derivative and the functional variable method, exp-function method and -expansion method are used to construct exact solutions for some nonlinear partial fractional differential equations via the nonlinear space-time fractional (2 + 1)-dimensional breaking soliton equations. These methods are also applied to derive a variety of travelling wave solutions with distinct physical structures for this nonlinear fractional equation. As a result, some new exact solutions for them are obtained. The three methods demonstrate power, reliability and efficiency.

Functional variable method for the nonlinear fractional differential equations

In this type functional variable method has been used to private type of nonlinear fractional differential equations. The main property of the method demonstrate in its flexibility and ability to solve nonlinear equations accurately, efficiency and conveniently. The fractional derivatives are described in the modified Riemann-Liouville sense. Three examples, are presented to show the application of the present technique. As a result, periodic and hyperbolic solutions are obtained.

Bright and dark soliton solutions for variable-coefficient diffusion–reaction and modified Korteweg–de Vries equations

In this paper, by using a solitary wave ansatz in the form of sechp and tanhp functions, we obtain the exact bright and dark soliton solutions for the considered model, respectively. The topological (dark) and non-topological (bright) soliton solutions to the variable-coefficient diffusion–reaction equation and the variable-coefficient modified Korteweg–de Vries equation with power law nonlinearity are obtained by using the solitary wave ansatz method. The parametric conditions for the formation of soliton pulses are determined. Note that it is always useful and desirable to construct exact analytical solutions, especially soliton-type envelope ones, for understanding most nonlinear physical phenomena.

Exact solutions of some fractional differential equations arising in mathematical biology

In the last decades Exp-function method has been used for solving fractional differential equations. In this paper, we obtain exact solutions of fractional generalized reaction Duffing model and nonlinear fractional diffusion–reaction equation. The fractional derivatives are described in the modified Riemann–Liouville sense. The fractional complex transform has been suggested to convert fractional-order differential equations with modified Riemann–Liouville derivatives into integer-order differential equations, and the reduced equations can be solved by symbolic computation.

A note on exp-function method combined with complex transform method applied to fractional differential equations

Abstract In this article, the fractional derivatives in the sense of modified Riemann–Liouville and the exp-function method are used to construct exact solutions for some nonlinear partial fractional differential equations via the nonlinear fractional Liouville equation and nonlinear fractional Zoomeron equation. These nonlinear fractional equations can be turned into another nonlinear ordinary differential equation by complex transform method. This method is efficient and powerful in solving wide classes of nonlinear fractional order equations. The exp-function method appears to be easier and more convenient by means of a symbolic computation system.

A generalized fractional sub-equation method for nonlinear fractional differential equations

This letter studies some nonlinear fractional differential equations. The sub-equation method is used for finding exact solutions of these equations. Meanwhile, the traveling wave transformation method has been used to convert fractional order partial differential equation to fractional order ordinary differential equation. Calculations in this method are simple and effective mathematical tool for solving fractional differential equations in science and engineering. The power of this manageable method is presented by applying it to several examples. This approach can also be applied to other nonlinear fractional differential equations.