Yusuf Gurefe

Classification of Exact Solutions for Some Nonlinear Partial Differential Equations with Generalized Evolution

We obtain the classification of exact solutions, including soliton, rational, and elliptic solutions, to the one-dimensional general improved Camassa Holm KP equation and KdV equation by the complete discrimination system for polynomial method. In discussion, we propose a more general trial equation method for nonlinear partial differential equations with generalized evolution.

The Extended Trial Equation Method for Some Time Fractional Differential Equations

Nonlinear fractional partial differential equations have been solved with the help of the extended trial equation method. Based on the fractional derivative in the sense of modified Riemann-Liouville derivative and traveling wave transformation, the fractional partial differential equation can be turned into the nonlinear nonfractional ordinary differential equation. For illustrating the reliability of this approach, we apply it to the generalized third order fractional KdV equation and the fractional equation according to the complete discrimination system for polynomial method. As a result, some new exact solutions to these nonlinear problems are successfully constructed such as elliptic integral function solutions, Jacobi elliptic function solutions, and soliton solutions.

New Exact Solutions of the Time-Fractional Nonlinear Dispersive KdV Equation

In this study, new version of the extended trial equation method is applied the nonlinear fractional partial differential equations. The fractional partial differential equations can be turned into the nonlinear non-fractional ordinary differential equations by the fractional derivative and traveling wave transformation. So, we find some traveling wave solutions to the time-fractional nonlinear dispersive KdV equation by the using of the complete discrimination system for polynomial method. As a result, these exact solutions to this nonlinear problem are constructed such as single king solution and hyperbolic function solutions.

A multiple extended trial equation method for the fractional Sharma-Tasso-Olver equation

In this paper, a multiple extended trial equation method is proposed to seek exact solutions of nonlinear time-fractional equation. The validity and advantages of the proposed method are illustrated by its application to the Sharma-Tasso-Olver equation. As a result, various complexiton solutions consisting of hyperbolic functions, rational functions and their mixture with parameters are obtained. When some parameters are taken as special values, the known double solitary-like wave solutions are derived from the double hyperbolic function solution.

A new approach to Kudryashov's method for solving some nonlinear physical models

The study of nonlinear evolution equations has become very important in the recent years. There are a lot of nonlinear evolution equations that are solved using different mathematical methods. For these physical problems, soliton solutions, compactons, cnoidal waves, singular solitons and the other solutions have been found. These types of solutions appear in various areas of applied sciences and engineering. In this paper, we consider the

An application of the new function method to the generalized double sinh-Gordon equation

This paper applies a new approach including the trial equation u′=f(cosh (nu)) based on the hyperbolic function cosh in order to find new traveling wave solutions to the generalized sinh-Gordon equation. By the using of this method, we obtain a new elliptic integral function solution. Also, this solution can be converted into Jacobi elliptic function solution by a simple transformation.

A Generalization on Weighted Means and Convex Functions with respect to the Non-Newtonian Calculus

This paper is devoted to investigating some characteristic features of weighted means and convex functions in terms of the non-Newtonian calculus which is a self-contained system independent of any other system of calculus. It is shown that there are infinitely many such useful types of weighted means and convex functions depending on the choice of generating functions. Moreover, some relations between classical weighted mean and its non-Newtonian version are compared and discussed in a table. Also, some geometric interpretations of convex functions are presented with respect to the non-Newtonian slope. Finally, using multiplicative continuous convex functions we give an application.

New 2-point implicit block multistep method for multiplicative initial value problems

In this study, new method is constructed to find numerical solutions of multiplicative initial value problems. This method called as multiplicative 2-point implicit block method is applied to some problems, and the obtained results are analyzed and compared with the results in literature. Also it is shown that this method is g-stable.

Traveling wave solutions of the (N + 1)-dimensional sine-cosine-Gordon equation

In this study, an isteresting method that assumes φ′ = f (sin(φ)) is used for solving the (N + 1)-dimensional sine-cosine-Gordon equation. A new elliptic integral function solution is obtained by this technique, and then this solution is converted into the Jacobi elliptic function solution. This methodology can be applied to solve other types of sine-cosine-Gordon equations.

Traveling wave solutions by extended trial equation method

This paper obtains soliton solutions, rational function, elliptic integral function F solution and Jacobi elliptic function solution by using the extended trial equation method based on the complete discrimination system for polynomial method. It is seen that this method is an effective mathematical tool for nonlinear differential equations.