Haci Mehmet Baskonus

The Modified Trial Equation Method for Fractional Wave Equation and Time Fractional Generalized Burgers Equation

The fractional partial differential equations stand for natural phenomena all over the world from science to engineering. When it comes to obtaining the solutions of these equations, there are many various techniques in the literature. Some of these give to us approximate solutions; others give to us analytical solutions. In this paper, we applied the modified trial equation method (MTEM) to the one-dimensional nonlinear fractional wave equation (FWE) and time fractional generalized Burgers equation. Then, we submitted 3D graphics for different value of .

Active Control of a Chaotic Fractional Order Economic System

In this paper, a fractional order economic system is studied. An active control technique is applied to control chaos in this system. The stabilization of equilibria is obtained by both theoretical analysis and the simulation result. The numerical simulations, via the improved Adams–Bashforth algorithm, show the effectiveness of the proposed controller.

Exponential prototype structures for (2+1)-dimensional Boiti-Leon-Pempinelli systems in mathematical physics

Abstract In this study, a new method called improved Bernoulli sub-equation function method has been proposed. This method is based on the Bernoulli sub-ODE method. After we mention the general properties of proposed method, we apply this algorithm to the (2 + 1)-dimensional Boiti–Leon–Pempinelli equation system. This gives us some new prototype solutions such as exponential and rational function solutions. Then, we have plotted two- and three-dimensional surfaces of analytical solutions. Finally, we have submitted a comprehensive conclusion.

The Analytical Solution of Some Fractional Ordinary Differential Equations by the Sumudu Transform Method

We introduce the rudiments of fractional calculus and the consequent applications of the Sumudu transform on fractional derivatives. Once this connection is firmly established in the general setting, we turn to the application of the Sumudu transform method (STM) to some interesting nonhomogeneous fractional ordinary differential equations (FODEs). Finally, we use the solutions to form two-dimensional (2D) graphs, by using the symbolic algebra package Mathematica Program 7.

On the complex structures of Kundu-Eckhaus equation via improved Bernoulli sub-equation function method

In this study, we obtain some new complex analytical solutions to the Kundu–Eckhaus equation which seems in the quantum field theory, weakly nonlinear dispersive water waves and nonlinear optics using improved Bernoulli sub-equation function method. After we have mentioned the general structure of improved Bernoulli sub-equation function method, we have successfully applied this method and then obtained some new complex hyperbolic and complex trigonometric function solutions. Two- and three-dimensional surfaces of analytical solutions have been plotted via wolfram Mathematica 9 version. At the end of this article, a conclusion has been submitted by mentioning important points founded in this study.

On the numerical solutions of some fractional ordinary differential equations by fractional Adams-Bashforth-Moulton method

Abstract In this paper, we apply the Fractional Adams-Bashforth-Moulton Method for obtaining the numerical solutions of some linear and nonlinear fractional ordinary differential equations. Then, we construct a table including numerical results for both fractional differential equations. Then, we draw two dimensional surfaces of numerical solutions and analytical solutions by considering the suitable values of parameters. Finally, we use the L2 nodal norm and L∞ maximum nodal norm to evaluate the accuracy of method used in this paper.

New wave behaviors of the system of equations for the ion sound and Langmuir Waves

ABSTRACT This manuscript focuses on the new wave behaviors of the system of equations for the ion sound wave under the action of the ponderomotive force which results from a nonlinear force that a charged particle experiences in an inhomogeneous oscillating electromagnetic field due to high-frequency field. The sine-Gordon expansion method which is one of the powerful methods has been considered for finding traveling wave solutions of the system of equations for the ion sound wave under the pressure of the Langmuir wave. This algorithm yields new complex hyperbolic function solutions to the system considered in this paper. Wolfram Mathematica 9 has been successfully used throughout the paper for mathematical calculations.

Investigation of various soliton solutions to the Heisenberg ferromagnetic spin chain equation

Abstract This study uses two mathematical approaches in constructing dark, bright, kink-type and singular soliton solutions to the Heisenberg ferromagnetic spin chain equation. The (2+1)-dimensional Heisenberg ferromagnetic spin chain equation describes nonlinear dynamics of magnets. The acquired results in this study may help in explaining some physical meanings of some nonlinear physical models arising in electromagnetic waves. For instance, the hyperbolic tangent arises in the calculation of magnetic moment and rapidity of special relativity, the hyperbolic cotangent arises in the Langevin function for magnetic polarization. The 2-, 3-dimensional and the contour plots of all the acquired solutions are presented.

On the Complex and Hyperbolic Structures for the (2 + 1)-Dimensional Boussinesq Water Equation

In this study, we have applied the modified exp(−Ω(ξ))-expansion function method to the (2 + 1)-dimensional Boussinesq water equation. We have obtained some new analytical solutions such as exponential function, complex function and hyperbolic function solutions. It has been observed that all analytical solutions have been verified to the (2 + 1)-dimensional Boussinesq water equation by using Wolfram Mathematica 9. We have constructed the two- and three-dimensional surfaces for all analytical solutions obtained in this paper using the same computer program.

Symmetrical hyperbolic Fibonacci function solutions of generalized Fisher equation with fractional order

The main goal of the present research, the Modified Kudryashov Method has been used to obtain the exact solutions of Generalized Fisher Equation with fractional order. This approach has been newly submitted to the literature to obtain analytical solutions of nonlinear partial differential equations. In recent years, it has been aroused deep and comprehensive interest among scientists. This trend is largely because it is obtaining analytical solutions such as soliton solutions, hyperbolic function solutions and elliptic function solutions of fractional differential equations. When it comes to this study, we have applied Modified Kudryashov Method to obtain exact solution of Generalized Fisher Equation with fractional order in that it is hyperbolic Fibonacci function. The results obtained via this technique have been given for interpretation of the solutions which is modeling of equations.