Time-Fractional KdV Equation for the plasma in auroral zone using Variational Methods

The reductive perturbation method has been employed to derive the Korteweg-de Vries (KdV) equation for small but finite amplitude electrostatic waves. The Lagrangian of the time fractional KdV equation is used in similar form to the Lagrangian of the regular KdV equation. The variation of the functional of this Lagrangian leads to the Euler-Lagrange equation that leads to the time fractional KdV equation. The Riemann-Liouvulle definition of the fractional derivative is used to describe the time fractional operator in the fractional KdV equation. The variational-iteration method given by He is used to solve the derived time fractional KdV equation. The calculations of the solution with initial condition A0*sech(cx)^2 are carried out. Numerical studies have been made using plasma parameters close to those values corresponding to the dayside auroral zone. The effects of the time fractional parameter on the electrostatic solitary structures are presented.Comment: 1 tex file + 5 eps figure

Time-Fractional KdV Equation: Formulation and Solution using Variational Methods

In this work, the semi-inverse method has been used to derive the Lagrangian of the Korteweg-de Vries (KdV) equation. Then, the time operator of the Lagrangian of the KdV equation has been transformed into fractional domain in terms of the left-Riemann-Liouville fractional differential operator. The variational of the functional of this Lagrangian leads neatly to Euler-Lagrange equation. Via Agrawal's method, one can easily derive the time-fractional KdV equation from this Euler-Lagrange equation. Remarkably, the time-fractional term in the resulting KdV equation is obtained in Riesz fractional derivative in a direct manner. As a second step, the derived time-fractional KdV equation is solved using He's variational-iteration method. The calculations are carried out using initial condition depends on the nonlinear and dispersion coefficients of the KdV equation. We remark that more pronounced effects and deeper insight into the formation and properties of the resulting solitary wave by additionally considering the fractional order derivative beside the nonlinearity and dispersion terms.Comment: The paper has been rewritten, 12 pages, 3 figure

Applying VIM to conformable partial differential equations

In this paper, we used new conformable variational iteration method, by the conformable derivative, for solving fractional heat-like and wave-like equations. This method is simple and very effective in the solution procedures of the fractional partial differential equations that have complicated solutions with classical fractional derivative definitions like Caputo, Riemann-Liouville and etc. The results show that conformable variational iteration method is usable and convenient for the solution of fractional partial differential equations. Obtained results are compared to the exact solutions and their graphics are plotted to demonstrate efficiency and accuracy of the method.Publisher's Versio

Some Problems on Variational Iteration Method

In this research project paper, I introduce some basic idea of Variational iteration method and its algorithm to solve the equations ODE & PDE, fractional differential equation, fractal differential equation and differential-difference equations. Also, some linear and nonlinear differential equations like Burger’s equation, Fisher’s equation,Wave equation and Schrodinger equation are solved by using Variational iteration method. Then I compare this method with Adomian decomposition method (ADM) and modified Variational iteration method (MVIM). The advantage of VIM, it does not require a small parameter in an equation as perturbation technique needs. The VIM is used to solve effectively, easily, and accurately a large class of non-linear problems with approximations which converge rapidly to accurate solutions. For linear problems, its exact solution can be obtained by only one iteration step due to the fact that the Lagrange multiplier can be exactly identified

Numerical Analytic Solution of SIR Model of Dengue Fever Disease in South Sulawesi using Homotopy Perturbation Method and Variational Iteration Method

In this research, the susceptible"“infected"“recovered (SIR) model of dengue fever is considered. We have implemented two analytical techniques, namely the variational iteration method (VIM) and the homotopy perturbation method (HPM) for solving the SIR model. The Lagrange multiplier was investigated for the VIM and He's polynomial approach for the HPM was used. In these schemes, the solution takes the form of a convergent series with easily computable components. The resultsshow thatthe VIM solution is more accurate than the HPM solution for short time intervals, whereasthe HPM is more accurate than the VIM for long time intervalswhencompared with the fourth-orderRunge-Kutta method (RK4).We found that the HPM and the RK4 were in excellent conformance