# Variational Iteration Method forq-Difference Equations of Second Order

Authors: Guo-Cheng Wu
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

Recently, Liu extended He's variational iteration method to strongly nonlinearq-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.


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## Body

## 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 nonlinearq-difference equation,
(1.1)dq2dqt2x+(2+εx)dqdqtx+Ω2x+x2=0,
where dq/dqt is the  q-derivative [3], Liu [4] used the Lagrange multiplier
(1.2)λ(t,s)=s-t,
which results in the iteration formula (see [4, (4.10) and (4.11)]):
(1.3)xn+1=xn+∫0t(s-t)(dq2dqs2xn+(2+εxn)dqdqsxn+Ω2xn+xn2)dqs.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 ofq-Calculus

### 2.1.q-Calculus

Letf(x)  be a real continuous function. The q-derivative is defined as
(2.1)dqdqxf(x)=f(qx)-f(x)(q-1)x,x≠0,0<q<1,
and (dq/dqx)f(x)|x=0=limn→∞((f(qn)-f(0))/qn).The partialq-derivative with respect to  x  is
(2.2)∂q∂qxf(x;y;…)=f(qx;y;…)-f(x;y;…)(q-1)x.The corresponding q-integral [5] is
(2.3)∫0xf(t)dqt=(1-q)x∑n=0∞qnf(qnx).


### 2.2.q-Leibniz Product Law

One has(2.4)dqdqx[g(x)f(x)]=g(qx)dqdqx[f(x)]+f(x)dqdqx[g(x)].


### 2.3.q-Integration by Parts

One has(2.5)∫abg(qt)dqdqtf(t)dqt=f(t)g(t)|ab-∫abf(t)dqdqtg(t)dqt.The properties above are needed in the construction of the correction functional forq-difference equations. For more results and properties in q-calculus, readers are referred to the recent monographs [5–8].


## 2.1.q-Calculus

Letf(x)  be a real continuous function. The q-derivative is defined as
(2.1)dqdqxf(x)=f(qx)-f(x)(q-1)x,x≠0,0<q<1,
and (dq/dqx)f(x)|x=0=limn→∞((f(qn)-f(0))/qn).The partialq-derivative with respect to  x  is
(2.2)∂q∂qxf(x;y;…)=f(qx;y;…)-f(x;y;…)(q-1)x.The corresponding q-integral [5] is
(2.3)∫0xf(t)dqt=(1-q)x∑n=0∞qnf(qnx).


## 2.2.q-Leibniz Product Law

One has(2.4)dqdqx[g(x)f(x)]=g(qx)dqdqx[f(x)]+f(x)dqdqx[g(x)].


## 2.3.q-Integration by Parts

One has(2.5)∫abg(qt)dqdqtf(t)dqt=f(t)g(t)|ab-∫abf(t)dqdqtg(t)dqt.The properties above are needed in the construction of the correction functional forq-difference equations. For more results and properties in q-calculus, readers are referred to the recent monographs [5–8].


## 3. Aq-Analogue of Lagrange Multiplier

In order to identify the Lagrange multipliers of theq-difference equations, we first establish the correctional functional for (1.1) as
(3.1)xn+1=xn+∫0tλ(t,q2s)(dq2dqs2xn+(2+εxn)dqdqsxn+Ω2xn+xn2)dqs.The correction functional here is different from the one in ordinary calculus since the parameterq “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(dq2/dqt2)x when other terms are restricted variations in (1.1)
(3.2)xn+1=xn+∫0tλ(t,q2s)(dq2dqs2xn+(2+εxn)dqdqsxn+Ω2xn+xn2)dqs.Through the integration by parts (2.5), we can have
(3.3)δxn+1=(1-q∂q∂qsλ(t,s)|s=t)δxn+λ(t,qs)|s=tδxn′-q∫0t∂q2∂qs2λ(t,s)δxndqs,
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 ofδxn:1-q(∂q/∂qs)λ(t,s)|s=t=0,the coefficient ofδxn′:λ(t,qs)|s=t=0,the coefficient ofδxn in the  q-integral : q(∂q2/∂qs2)λ(t,s)=0,from which we can get
(3.4)λ(t,s)=q-1(s-tq),
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 aboveq-analogue of Lagrange multiplier’s validness. For 0<q<1, let Tq be the time scale: Tq={qn:n∈Z}∪{0}, where Z is the set of positive integers. For the real continuous function u(t):Tq→R, a q-oscillator equation of second order is
(3.5)dq2dqt2u-u=0,u(0)=1,dqdqtu|t=0=1.From (3.4), the iteration formula can be given as
(3.6)un+1=un+∫0tq-1(q2s-tq)[dq2dqs2un(s)-un(s)]dqs.Starting from the initial iterationu0=1+t/[1]q!, the successive approximate solutions can be obtained as
(3.7)u0=1+t[1]q!,u1=1+t[1]q!+t2[2]q!+t3[3]q!,⋮un=∑k=02n+1tk[k]q!.
The limit u=limn→∞un=eq(t) is an exact solution of (3.5). Here  eq(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. Theq-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].



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