A modified Li-He’s variational principle for plasma

Abstract

Purpose

It is extremely difficult to establish a variational principle for plasma. Kalaawy obtained a variational principle by using the semi-inverse method in 2016, and Li and He suggested a modification in 2017. This paper aims to search for a generalized variational formulation with a free parameter.

Design/methodology/approach

The semi-inverse method is used by suitable construction of a trial functional with some free parameters.

Findings

A modification of Li-He’s variational principle with a free parameter is obtained.

Originality/value

This paper suggests a new approach to construction of a trial-functional with some free parameters.

Citation

He, J.-H. (2021), "A modified Li-He’s variational principle for plasma", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 31 No. 5, pp. 1369-1372. https://doi.org/10.1108/HFF-06-2019-0523

1. Introduction

In 2016, Kalaawy obtained the following equation for plasma (El-Kalaawy, 2016):

(1) ∂u∂t+∂∂x[23Au3/2−C∂u∂x]=0

and derived a variational principle by using the semi-inverse method (He, 2004; He, 2017; Wu and He, 2018) through introducing two special functions.

In 2017, Li and He found another variational principle, which reads (Li and He, 2017):

(2) JLi−He(u,Φ)=∬{12∂Φ∂x∂Φ∂t+23Au3/2∂Φ∂x−12C(∂Φ∂x)2−25Au5/2}dxdt

where Φ satisfies the following relations:

(3) ∂Φ∂x=u

(4) ∂Φ∂t=−23Au3/2+C∂u∂x

2. General variational principle with a free parameter

The semi-inverse method is widely used to search for variational principles directly from governing equations (El-Kalaawy, 2016; El-Kalaawy, 2017; Biswas et al., 2017). By the semi-inverse method (He, 2004; He, 2017; Wu and He, 2018), we can construct a trial-functional in the form:

(5) J(u,Φ)=∬{u∂Φ∂t+[23Au3/2−C∂u∂x]∂Φ∂x+F}dxdt

where F is a known function of u and its derivative, however we cannot identify F, so equation (5) has to be modified as:

(6) JNew(u,Φ)=∬{mu∂Φ∂t+n∂Φ∂x∂Φ∂t+aAu1/2(∂Φ∂x)2+bAu3/2∂Φ∂x−C∂u∂x∂Φ∂x+F}dxdt

where m, n, a and b are constants.

The stationary conditions of equation (6) are as follows:

(7) −m∂u∂t−2n∂2Φ∂x∂t−∂∂x[2aAu1/2∂Φ∂x+bAu3/2−C∂u∂x]=0

and

(8) m∂Φ∂t+12aAu−1/2(∂Φ∂x)2+32bAu1/2∂Φ∂x+C∂∂x(∂Φ∂x)+δFδu=0

In view of equations (3) and (4), we have:

(9) −(m+2n)∂u∂t−∂∂x[(2a+b)Au3/2−C∂u∂x]=0

(10) mC∂u∂x+(12a+32b−23m)Au3/2+C∂u∂x+δFδu=0

Equation (9) should be equivalent to equation (1), this requires:

(11) 2a+b=1

(12) m+2n=1

To identify F in equation (10), we set:

(13) m=−1

Equation (10) becomes:

(14) (12a+32b+23)Au3/2+δFδu=0

From equation (14), F can be identified as:

(15) F=−25(12a+32b+23)Au5/2=−25(136−52a)Au5/2

We, therefore, obtain the following variational principle:

JNew(u,Φ)=∬{∂Φ∂x∂Φ∂t−u∂Φ∂t+aAu1/2(∂Φ∂x)2+(1−2a)Au3/2∂Φ∂x−C∂u∂x∂Φ∂x−25(136−52a)Au5/2}dxdt

where a is a free parameter.

3. Conclusion

In this paper, we suggest a general construction of a trial functional with some parameters. The semi-inverse method is a powerful tool to establish variational principles from the governing equations. The variational principle is a foundation of the variational iteration method (Anjum and He, 2019; He, 2006), which is now widely applied in fractional calculus (Baleanu et al., 2018; Dogan Durgun and Konuralp, 2018; Inc et al., 2018; Jafari et al., 2018; Wang et al., 2018).