Emine Misirli

Extended trial equation method to generalized nonlinear partial differential equations

In this article, we give the extended trial equation method for solving nonlinear partial differential equations with higher order nonlinearity. By use of this method, the exact traveling wave solutions including soliton solution, singular soliton solutions, rational function solution and elliptic integral function solution to one-dimensional general improved KdV (GIKdV) equation and R(m,n) equation are obtained. Also, a more general trial equation method is proposed.

The modified Kudryashov method for solving some fractional-order nonlinear equations

In this paper, the modified Kudryashov method is proposed to solve fractional differential equations, and Jumarie’s modified Riemann-Liouville derivative is used to convert nonlinear partial fractional differential equation to nonlinear ordinary differential equations. The modified Kudryashov method is applied to compute an approximation to the solutions of the space-time fractional modified Benjamin-Bona-Mahony equation and the space-time fractional potential Kadomtsev-Petviashvili equation. As a result, many analytical exact solutions are obtained including symmetrical Fibonacci function solutions, hyperbolic function solutions, and rational solutions. This method is powerful, efficient, and it can be used as an alternative to establish new solutions of different types of fractional differential equations applied in mathematical physics.

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.

Exp-function method for solving nonlinear evolution equations with higher order nonlinearity

Abstract In this paper, the Exp-function method is used to obtain generalized solitary solutions of the generalized Drinfel’d–Sokolov–Wilson (DSW) system and the generalized (2+1)-dimensional Burgers-type equation. Then, some of the solitary solutions are converted to periodic solutions or hyperbolic function solutions by a simple transformation. The results show that the Exp-function method is a powerful and convenient mathematical tool for solving nonlinear evolution equations with higher order nonlinearity.

Multiplicative Adams Bashforth---Moulton methods

The multiplicative version of Adams Bashforth–Moulton algorithms for the numerical solution of multiplicative differential equations is proposed. Truncation error estimation for these numerical algorithms is discussed. A specific problem is solved by methods defined in multiplicative sense. The stability properties of these methods are analyzed by using the standart test equation.

Exact solutions of the Drinfel’d–Sokolov–Wilson equation using the Exp-function method

Abstract The generalized solitary solutions of the classical Drinfel’d–Sokolov–Wilson equation (DSWE) are obtained using the Exp-function method. Then, some of these solutions are easily converted into kink-shaped solutions and blow-up solutions by a simple transformation.

Classification of exact solutions to the generalized Kadomtsev-Petviashvili equation

In this paper, we study the Kadomtsev–Petviashvili equation with generalized evolution and derive some new results using the approach called the trial equation method. The obtained results can be expressed by the soliton solutions, rational function solutions, elliptic function solutions and Jacobi elliptic function solutions. In the discussion, we give a new version of the trial equation method for nonlinear differential equations.

Multiplicative finite difference methods

Based on multiplicative calculus, the finite difference schemes for the numerical solution of multiplicative differential equations and Volterra differential equations are presented. Sample problems were solved using these new approaches.

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.