Solution of Time-Fractional Korteweg-de Vries Equation in warm Plasma

TL;DR

This paper derives and solves a time fractional Korteweg-de Vries equation for ion-acoustic waves in warm plasma using variational methods, with potential applications to plasma environments like the ionosphere.

Contribution

It introduces a fractional calculus approach to the KdV equation in plasma physics and applies the variational-iteration method for its solution.

Findings

Solution obtained for initial condition A0*sech(cx)^2

Method applicable to plasma environments like ionosphere

Extension of KdV equation with fractional derivatives

Abstract

The reductive perturbation method has been employed to derive the Korteweg-de Vries (KdV) equation for small but finite amplitude ion-acoustic 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. The result of the present investigation may be applicable to some plasma environments, such as ionosphere.