Melike Kaplan

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.

Regarding on the exact solutions for the nonlinear fractional differential equations

Abstract In this work, we have considered the modified simple equation (MSE) method for obtaining exact solutions of nonlinear fractional-order differential equations. The space-time fractional equal width (EW) and the modified equal width (mEW) equation are considered for illustrating the effectiveness of the algorithm. It has been observed that all exact solutions obtained in this paper verify the nonlinear ordinary differential equations which was obtained from nonlinear fractional-order differential equations under the terms of wave transformation relationship. The obtained results are shown graphically.

Exact Travelling Wave Solutions of the Nonlinear Evolution Equations by Auxiliary Equation Method

Abstract The auxiliary equation method presents wide applicability to handling nonlinear wave equations. In this article, we establish new exact travelling wave solutions of the nonlinear Zoomeron equation, coupled Higgs equation, and equal width wave equation. The travelling wave solutions are expressed by the hyperbolic functions, trigonometric functions, and rational functions. It is shown that the proposed method provides a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering. Throughout the article, all calculations are made with the aid of the Maple packet program.

Auxiliary equation method for time-fractional differential equations with conformable derivative

Abstract In this paper, the auxiliary equation method is applied to obtain analytical solutions of (2xa0+xa01)-dimensional time-fractional Zoomeron equation and the time-fractional third order modified KdV equation in the sense of the conformable fractional derivative. Given equations are converted to the nonlinear ordinary differential equations of integer order; and then, the resulting equations are solved using a novel analytical method called the auxiliary equation method. As a result, some exact solutions for them are successfully established. The exact solutions obtained by the proposed method indicate that the approach is easy to implement and effective.

A novel modified simple equation method and its application to some nonlinear evolution equation systems

In this paper, the modified simple equation (MSE) method is used to construct exact solutions of the nonlinear Drinfeld-Sokolov system, Maccari system and Coupled Higgs equation in applied mathematics and mathematical physics. The exact solutions obtained by the proposed method indicate that the approach is easy to implement and computationally very attractive. Also we can see that when the parameters are assigned special values, solitary wave solutions can be obtained from the exact solutions. All calculations in this study have been made with the aid of the Maple packet program.

Conservation laws and Exact Solutions of Phi-Four (Phi-4) Equation via the (G′/G, 1/G)-Expansion Method

Abstract In this article, we constructed formal Lagrangian of Phi-4 equation, and then via this formal Lagrangian, we found adjoint equation. We investigated if the Lie point symmetries of the equation satisfy invariance condition or not. Then we used conservation theorem to find conservation laws of Phi-4 equation. Finally, the exact solutions of the equation were obtained through the (G′/G, 1/G)-expansion method.

Solving nonlinear evolution equation system using two different methods

Abstract This paper deals with constructing more general exact solutions of the coupled Higgs equation by using the (G0/G, 1/G)-expansion and (1/G0)-expansion methods. The obtained solutions are expressed by three types of functions: hyperbolic, trigonometric and rational functions with free parameters. It has been shown that the suggested methods are productive and will be used to solve nonlinear partial differential equations in applied mathematics and engineering. Throughout the paper, all the calculations are made with the aid of the Maple software.

Optical soliton solutions of the cubic-quintic non-linear Schrödinger’s equation including an anti-cubic term

Abstract A wide range of problems in different fields of the applied sciences especially non-linear optics is described by non-linear Schrödinger’s equations (NLSEs). In the present paper, a specific type of NLSEs known as the cubic-quintic non-linear Schrödinger’s equation including an anti-cubic term has been studied. The generalized Kudryashov method along with symbolic computation package has been exerted to carry out this objective. As a consequence, a series of optical soliton solutions have formally been retrieved. It is corroborated that the generalized form of Kudryashov method is a direct, effectual, and reliable technique to deal with various types of non-linear Schrödinger’s equations.

The Auto-Bäcklund transformations for the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation

In this work, the homogeneous balance method is used to construct Auto-Backlund transformation of the Boiti-Leon-Manna-Pempinelli (BLMP) equation. With the aid of the transformations founded in this paper and Maple packet programme, abundant exact and explicit solutions to the BLMP equation are constructed.

A novel generalized Kudryashov method for exact solutions of nonlinear evolution equations

Nonlinear evolution equation (NLEE) systems model the most essential topics in nonlinear sciences. Exact solutions of these equations play a major role in the suitable understanding of mechanisms of the various physical phenomena modelled by these NLEEs. For this reason, searching exact solutions of these equations have taken intense interest lately. In the present work, we have obtained new exact solutions of the and Kaup-Boussinesq systems with the generalized Kudryashov method.