Time Fractional KdV Equation Research Articles

Lie Symmetry Analysis and Explicit Solutions of the Time Fractional Fifth-Order KdV Equation

In this paper, using the Lie group analysis method, we study the invariance properties of the time fractional fifth-order KdV equation. A systematic research to derive Lie point symmetries to time fractional fifth-order KdV equation is performed. In the sense of point symmetry, all of the vector fields and the symmetry reductions of the fractional fifth-order KdV equation are obtained. At last, by virtue of the sub-equation method, some exact solutions to the fractional fifth-order KdV equation are provided.

Symmetry properties and explicit solutions of the nonlinear time fractional KdV equation

The time fractional KdV equation in the sense of the Riemann-Liouville derivatives is considered. The symmetry properties of the time fractional KdV equation is investigated by using the Lie group analysis method. On the basis of the point symmetry, the vector fields of the time fractional KdV equation are presented. And then, the symmetry reductions are constructed. By right of the obtained Lie point symmetries, it is shown that this equation could transform into a nonlinear ordinary differential equation of fractional order with the new independent variable . The derivative is an Erdelyi-Kober derivative depending on a parameter α. At last, by means of the sub-equation method, some exact and explicit solutions of the time fractional KdV equation are constructed. MSC:22E70, 26A33.

A New Coupled Fractional Reduced Differential Transform Method for Solving Time Fractional Coupled KdV Equations

Abstract In the present paper, a very new technique, Coupled Fractional Reduced Differential Transform, has been implemented to obtain the numerical approximate solution of coupled time fractional kdV equations. The fractional derivatives are described in the Caputo sense. By using the present method we can solve many linear and nonlinear coupled fractional differential equations. The obtained results are compared with the exact solutions. Numerical solutions are presented graphically to show the reliability and efficiency of the method.

Improved ()-Expansion Method for the Space and Time Fractional Foam Drainage and KdV Equations

The fractional complex transformation is used to transform nonlinear partial differential equations to nonlinear ordinary differential equations. The improved ()-expansion method is suggested to solve the space and time fractional foam drainage and KdV equations. The obtained results show that the presented method is effective and appropriate for solving nonlinear fractional differential equations.

Time-fractional KdV equation for plasma of two different temperature electrons and stationary ion

Using the time-fractional KdV equation, the nonlinear properties of small but finite amplitude electron-acoustic solitary waves are studied in a homogeneous system of unmagnetized collisionless plasma. This plasma consists of cold electrons fluid, non-thermal hot electrons, and stationary ions. Employing the reductive perturbation technique and the Euler-Lagrange equation, the time-fractional KdV equation is derived and it is solved using variational method. It is found that the time-fractional parameter significantly changes the soliton amplitude of the electron-acoustic solitary waves. The results are compared with the structures of the broadband electrostatic noise observed in the dayside auroral zone.

Ion-acoustic waves in plasma of warm ions and isothermal electrons using time-fractional KdV equation

The ion-acoustic solitary wave in collisionless unmagnetized plasma consisting of warm ions-fluid and isothermal electrons is studied using the time fractional KdV equation. The reductive perturbation method has been employed to derive the Korteweg-de Vries equation for small but finite amplitude ion-acoustic wave in warm plasma. The Lagrangian of the time fractional KdV equation is used in a similar form to the Lagrangian of the regular KdV equation with fractional derivative for the time differentiation. The variation of the functional of this Lagrangian leads to the Euler—Lagrange equation that gives the time fractional KdV equation. The variational-iteration method is used to solve the derived time fractional KdV equation. The calculations of the solution are carried out for different values of the time fractional order. These calculations show that the time fractional can be used to modulate the electrostatic potential wave instead of adding a higher order dissipation term to the KdV equation. The results of the present investigation may be applicable to some plasma environments, such as the ionosphere plasma.

Time-fractional KdV equation for electron-acoustic waves in plasma of cold electron and two different temperature isothermal ions

The time fractional KdV equation is derived for small but finite amplitude electron-acoustic solitary waves in plasma of cold electron fluid with two different temperature isothermal ions. The effects of the time fractional parameter on the electrostatic solitary structures are presented. It is shown that the effect of time fractional parameter can be used to modify the amplitude of the electrostatic waves (viz. the amplitude, width and electric field) of the electron-acoustic solitary waves. The model may provide a possible explanation for the low-frequency component of the broadband electrostatic noise in the plasma sheet boundary layer of the Earth’s magnetotail where the electron beams are not present.

Fractional green function for linear time-fractional inhomogeneous partial differential equations in fluid mechanics

This paper deals with the solutions of linear inhomogeneous time-fractional partial differential equations in applied mathematics and fluid mechanics. The fractional derivatives are described in the Caputo sense. The fractional Green function method is used to obtain solutions for time-fractional wave equation, linearized time-fractional Burgers equation, and linear time-fractional KdV equation. The new approach introduces a promising tool for solving fractional partial differential equations.