Adem C. Cevikel

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.

A procedure to construct exact solutions of nonlinear fractional differential equations.

We use the fractional transformation to convert the nonlinear partial fractional differential equations with the nonlinear ordinary differential equations. The Exp-function method is extended to solve fractional partial differential equations in the sense of the modified Riemann-Liouville derivative. We apply the Exp-function method to the time fractional Sharma-Tasso-Olver equation, the space fractional Burgers equation, and the time fractional fmKdV equation. As a result, we obtain some new exact solutions.

Solutions for fuzzy matrix games

In this paper, we deal with two-person zero-sum games with fuzzy payoffs and fuzzy goals. We have presented two models for studying two-person zero-sum matrix games with fuzzy payoffs and fuzzy goals. We assume that each player has a fuzzy goal for each of the payoffs. We obtained that the fuzzy relation approach and the max-min solution are equivalent.

Efficient solutions of systems of fractional PDEs by the differential transform method

In this paper we obtain approximate analytical solutions of systems of nonlinear fractional partial differential equations (FPDEs) by using the two-dimensional differential transform method (DTM). DTM is a numerical solution technique that is based on the Taylor series expansion which constructs an analytical solution in the form of a polynomial. The traditional higher order Taylor series method requires symbolic computation. However, DTM obtains a polynomial series solution by means of an iterative procedure. The fractional derivatives are described in the Caputo fractional derivative sense. The solutions are obtained in the form of rapidly convergent infinite series with easily computable terms. DTM is compared with some other numerical methods. Computational results reveal that DTM is a highly effective scheme for obtaining approximate analytical solutions of systems of linear and nonlinear FPDEs and offers significant advantages over other numerical methods in terms of its straightforward applicability, computational efficiency, and accuracy.

Dark soliton and periodic wave solutions of nonlinear evolution equations

This paper studies the Kudryashov-Sinelshchikov and Jimbo-Miwa equations. Subsequently, we formally derive the dark (topological) soliton solutions for these equations. By using the sine-cosine method, some additional periodic solutions are derived. The physical parameters in the soliton solutions of the ansatz method, amplitude, inverse width and velocity, are obtained as functions of the dependent model coefficients.PACS Codes:02.30.Jr, 02.70.Wz, 05.45.Yv, 94.05.Fg.

Bright and Dark Soliton Solutions of the (2 + 1)-Dimensional Evolution Equations

AbstractIn this paper, we obtained the 1-soliton solutions of the (2+1)-dimensional Boussinesq equation and the Camassa–Holm–KP equation. By using a solitary wave ansatz in the form of sechp function, we obtain exact bright soliton solutions and another wave ansatz in the form of tanhp function we obtain exact dark soliton solutions for these equations. The physical parameters in the soliton solutions are obtained nonlinear equations with constant coefficients.

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.

Various methods for solving time fractional KdV-Zakharov-Kuznetsov equation

This paper presents the exact analytical solution of the (3+1)-dimensional time fractional KdV-Zakharov-Kuznetsov (KdV-ZK) equation with the help of the Kudryashov method, the exp-function method and the functional variable method. The fractional derivatives are described in Jumarie’s sense.

Exact solutions of the (3+1)-dimensional space-time fractional Jimbo-Miwa equation

Exact solutions of the (3+1)-dimensional space-time fractional Jimbo-Miwa equation are studied by the generalized Kudryashov method, the exp-function method and the (G′/G)-expansion method. The solutions obtained include the form of hyperbolic functions, trigonometric and rational functions. These methods are effective, simple, and many types of solutions can be obtained at the same time.

New Solitons and Periodic Solutions for (2+1)-dimensional Davey-Stewartson Equations

In this paper, we find exact solutions for the (2+1)-dimensional Davey-Stewartson (DS) equations. The sine-cosine, tanh-coth, and exp-function methods are used to construct periodic and soliton solutions of these equations. Many new families of exact solutions of the DS equations are successfully obtained. These solutions may be of significant importance for the explanation of some practical physical problems.