Variational Iteration Method for q-Difference Equations of Second Order

Abstract

Recently, Liu extended He's variational iteration method to strongly nonlinear q-difference equations. In this study, the iteration formula and the Lagrange multiplier are given in a more accurate way. The q-oscillation equation of second order is approximately solved to show the new Lagrange multiplier's validness.

1. Introduction

Generally, applying the variational iteration method (VIM) [1, 2] in differential equations follows the three steps:(a)establishing the correction functional;(b)identifying the Lagrange multipliers;(c)determining the initial iteration.

Obviously, the step (b) is crucial and critical in the method.

For the strongly nonlinear  𝑞-difference equation, 𝑑²_𝑞𝑑_𝑞𝑡²()𝑑𝑥+2+𝜀𝑥^𝑞𝑑_𝑞𝑡𝑥+Ω²𝑥+𝑥²=0, (1.1) where 𝑑_𝑞/𝑑_𝑞𝑡 is the  𝑞-derivative [3], Liu [4] used the Lagrange multiplier 𝜆(𝑡,𝑠)=𝑠−𝑡, (1.2) which results in the iteration formula (see [4, (4.10) and (4.11)]): 𝑥_𝑛+1=𝑥_𝑛+^𝑡₀()𝑑𝑠−𝑡^2𝑞𝑑_𝑞𝑠²𝑥_𝑛+2+𝜀𝑥_𝑛𝑑_𝑞𝑑_𝑞𝑠𝑥_𝑛+Ω²𝑥_𝑛+𝑥²_𝑛𝑑_𝑞𝑠. (1.3)

In this paper, it is pointed out that the iteration formula (1.3) can be given in a more accurate way and a new Lagrange multiplier is explicitly identified.

2. Properties of 𝑞-Calculus

2.1. 𝑞-Calculus

Let 𝑓(𝑥)  be a real continuous function. The 𝑞-derivative is defined as 𝑑_𝑞𝑑_𝑞𝑥𝑓(𝑥)=𝑓𝑞𝑥−𝑓(𝑥)𝑞−1𝑥,𝑥≠0,0<𝑞<1, (2.1) and (𝑑_𝑞/𝑑_𝑞𝑥)𝑓(𝑥)|_𝑥=0=lim_𝑛→∞((𝑓(𝑞^𝑛)−𝑓(0))/𝑞^𝑛).

The partial 𝑞-derivative with respect to  𝑥  is 𝜕_𝑞𝜕_𝑞𝑥𝑓=𝑓𝑥;𝑦;…𝑞𝑥;𝑦;…−𝑓𝑥;𝑦;…𝑥.𝑞−1 (2.2)The corresponding 𝑞-integral [5] is ^𝑥₀𝑓(𝑡)𝑑_𝑞𝑡=1−𝑞𝑥^∞_𝑛=0𝑞^𝑛𝑓𝑞^𝑛𝑥. (2.3)

2.2. 𝑞-Leibniz Product Law

One has 𝑑_𝑞𝑑_𝑞𝑥𝑔(𝑥)𝑓(𝑥𝑑)=𝑔𝑞𝑥^𝑞𝑑_𝑞𝑥𝑓(𝑥(𝑥)𝑑)+𝑓^𝑞𝑑_𝑞𝑥𝑔(𝑥). (2.4)

2.3. 𝑞-Integration by Parts

One has ^𝑏_𝑎𝑔𝑑𝑞𝑡^𝑞𝑑_𝑞𝑡𝑓(𝑡)𝑑_𝑞𝑡=𝑓(𝑡)𝑔(𝑡)^𝑏_𝑎−^𝑏_𝑎𝑓(𝑡)𝑑_𝑞𝑑_𝑞𝑡𝑔(𝑡)𝑑_𝑞𝑡. (2.5)

The properties above are needed in the construction of the correction functional for 𝑞-difference equations. For more results and properties in 𝑞-calculus, readers are referred to the recent monographs [5–8].

3. A 𝑞-Analogue of Lagrange Multiplier

In order to identify the Lagrange multipliers of the 𝑞-difference equations, we first establish the correctional functional for (1.1) as 𝑥_𝑛+1=𝑥_𝑛+^𝑡₀𝜆𝑡,𝑞²𝑠𝑑²_𝑞𝑑_𝑞𝑠²𝑥_𝑛+2+𝜀𝑥_𝑛𝑑_𝑞𝑑_𝑞𝑠𝑥_𝑛+Ω²𝑥_𝑛+𝑥²_𝑛𝑑_𝑞𝑠. (3.1)

The correction functional here is different from the one in ordinary calculus since the parameter 𝑞 “disappears” after the integration by parts (2.5) each time. As a result, we use 𝜆(𝑡,𝑞²𝑠) in the above functional.

We only need to consider the leading term (𝑑²_𝑞/𝑑_𝑞𝑡²)𝑥 when other terms are restricted variations in (1.1) 𝑥_𝑛+1=𝑥_𝑛+^𝑡₀𝜆𝑡,𝑞²𝑠𝑑²_𝑞𝑑_𝑞𝑠²𝑥_𝑛+2+𝜀𝑥_𝑛𝑑_𝑞𝑑_𝑞𝑠𝑥_𝑛+Ω²𝑥_𝑛+𝑥²_𝑛𝑑_𝑞𝑠. (3.2)

Through the integration by parts (2.5), we can have 𝛿𝑥_𝑛+1=𝜕1−𝑞^𝑞𝜕_𝑞𝑠𝜆(𝑡,𝑠)_𝑠=𝑡𝛿𝑥_𝑛+𝜆𝑡,𝑞𝑠_𝑠=𝑡𝛿𝑥^′_𝑛−𝑞^𝑡₀𝜕²_𝑞𝜕_𝑞𝑠²𝜆(𝑡,𝑠)𝛿𝑥_𝑛𝑑_𝑞𝑠, (3.3) where 𝛿 is the variation operator and “′” denotes the  𝑞-derivative with respect to 𝑡. As a result, the system of the Lagrange multiplier can be obtained:  the coefficient of 𝛿𝑥_𝑛∶1−𝑞(𝜕_𝑞/𝜕_𝑞𝑠)𝜆(𝑡,𝑠)|_𝑠=𝑡=0, the coefficient of 𝛿𝑥^′_𝑛∶𝜆(𝑡,𝑞𝑠)|_𝑠=𝑡=0, the coefficient of 𝛿𝑥_𝑛 in the  𝑞-integral : 𝑞(𝜕²_𝑞/𝜕_𝑞𝑠²)𝜆(𝑡,𝑠)=0,from which we can get 𝜆(𝑡,𝑠)=𝑞⁻¹𝑠−𝑡𝑞, (3.4) instead of 𝜆(𝑡,𝑠)=𝑠−𝑡 in [4]. More introductions to the identification of various Lagrange multipliers of the VIM can be found in [9, 10].

We also can show the above 𝑞-analogue of Lagrange multiplier’s validness. For 0<𝑞<1, let 𝑇_𝑞 be the time scale: 𝑇_𝑞={𝑞^𝑛∶𝑛∈𝑍}∪{0}, where 𝑍 is the set of positive integers. For the real continuous function 𝑢(𝑡)∶𝑇_𝑞→𝑅, a 𝑞-oscillator equation of second order is 𝑑²_𝑞𝑑_𝑞𝑡²(0)𝑑𝑢−𝑢=0,𝑢=1,^𝑞𝑑_𝑞𝑡𝑢_𝑡=0=1. (3.5)

From (3.4), the iteration formula can be given as 𝑢_𝑛+1=𝑢_𝑛+^𝑡₀𝑞⁻¹𝑞²𝑑𝑠−𝑡𝑞^2𝑞𝑑_𝑞𝑠²𝑢_𝑛(𝑠)−𝑢_𝑛(𝑠)𝑑_𝑞𝑠. (3.6)

Starting from the initial iteration 𝑢₀=1+𝑡/[1]_𝑞!, the successive approximate solutions can be obtained as 𝑢₀𝑡=1+[1]_𝑞!,𝑢₁𝑡=1+[1]_𝑞!+𝑡²[2]_𝑞!+𝑡³[3]_𝑞!,⋮𝑢_𝑛=^2𝑛+1_𝑘=0𝑡^𝑘[𝑘]_𝑞!. (3.7) The limit 𝑢=lim_𝑛→∞𝑢_𝑛=𝑒_𝑞(𝑡) is an exact solution of (3.5). Here  𝑒_𝑞(𝑡)  is one of the  𝑞-exponential functions.

4. Conclusions

In the past ten years, the VIM has been one of the often used nonlinear methods. The  𝑞-derivative is a deformation of the classical derivative and it has played a crucial role in quantum mechanics and quantum calculus. In this study, the method is successfully extended to 𝑞 difference equations of second order. A 𝑞-analogue of Lagrange multiplier is presented. Readers who feel interested in the initial value problems of the 𝑞 difference equations are referred to [11–17].