Application of the Variational Iteration Method to Strongly Nonlinear 𝑞-Difference Equations

Abstract

The theory of approximate solution lacks development in the area of nonlinear 𝑞-difference equations. One of the difficulties in developing a theory of series solutions for the homogeneous equations on time scales is that formulas for multiplication of two 𝑞-polynomials are not easily found. In this paper, the formula for the multiplication of two 𝑞-polynomials is presented. By applying the obtained results, we extend the use of the variational iteration method to nonlinear 𝑞-difference equations. The numerical results reveal that the proposed method is very effective and can be applied to other nonlinear 𝑞-difference equations.

1. Introduction

A time scale is a nonempty closed subset of real numbers. Recently, much research activity has focused on the theory and application of the 𝑞-calculus. For example, the 𝑞-calculus has being given a financial meaning by Muttel [1] and is applied to pricing the financial derivatives. Many real world problems are now formulated as 𝑞-difference equations. Nonlinear 𝑞-difference equations, as well as their analytic and numerical solutions, play an important role in various fields of science and engineering, especially in nonlinear physical science, since their solutions can provide more inside into the physical aspects of the problems.

Solutions of linear differential equations on time scales have been studies and published during the past two decades. One area lacking in development is the theory of approximate solutions on nonlinear 𝑞-difference equations. Recent developments in the theory of approximate solution have aroused further interest in the discussion of nonlinear 𝑞-difference equations.

One of the difficulties in developing a theory of series solutions for linear or nonlinear homogeneous equations on time scales is that formulas for multiplication of two generalized polynomials are not easily found. Haile and Hall [2] provided an exact formula for the multiplication of two generalized polynomials if the time scale had constant graininess. Using the obtained results, the series solutions for linear dynamic equations were proposed on the time scales ℝ and 𝕋=ℎℤ (difference equations with step size ℎ). For generalized time scales, Mozyrska and Pawtuszewicz [3] presented the formula for the multiplication of the generalized polynomials of degree one and degree 𝑛∈ℕ.

The variational iteration method proposed by He [4] has been proved by many authors to be a powerful mathematical tool for analysing the nonlinear problems on ℝ (the set of real numbers). The advantages of this method include (i) that it can be applied directly to all types of difference equations, and (ii) that it reduces the size of computational work while maintaining the high accuracy of the numerical solution. For the nonlinear 𝑞-difference equations, the approximate solution obtained by using the variational iteration method may not yet been found.

In this paper, we presented a formula for the multiplication of two 𝑞-polynomials. The obtained results can be used to find a series solution of the 𝑞-difference equations. The aim is to extend the use of the variational iteration method to strongly nonlinear 𝑞-difference equations. Precisely, the equation is described as𝑥^ΔΔ(𝑡)+2𝛾+𝜀𝛾₁𝑥(𝑡)𝑥^Δ(𝑡)+Ω²𝑥(𝑡)+𝑥²(𝑡)=0,(1.1) where 𝑥^Δ=Δ𝑥/Δ𝑡 is the 𝑞-derivative as defined in Definition 2.1. In future studies, we intend to extend the use of the variational iteration method to the other nonlinear 𝑞-difference equations.

This paper is organized as follows: in Section 2 basic ideas on 𝑞-calculus are introduced; in Section 3, the multiplication of two 𝑞-polynomials is demonstrated; in Section 4, the variational iteration method is applied to find an approximate solution of strongly nonlinear damped 𝑞-difference equations; in Section 5, the numerical results and the approximate solutions, which were very close, are presented; finally, a concise conclusion is provided in Section 6.

2. Introduction to 𝑞-Calculus

Let 0<𝑞<1 and use the notations 𝑞^ℕ={𝑞^𝑛∣𝑛∈ℕ},𝑞^ℕ=𝑞^ℕ∪{0},(2.1) where ℕ denotes the set of positive integers.

Let 𝑎 and 𝑞 be real numbers such that 0<𝑞<1. The 𝑞-shift factorial [5] is defined by(𝑎;𝑞)₀=1,(𝑎;𝑞)_𝑛=^𝑛−1_𝑘=01−𝑎𝑞^𝑘,𝑛=1,2,…,𝑛.(2.2)

Definition 2.1. Assume that 𝑓∶𝑞^ℕ→ℝ is a function and 𝑡∈𝑞^ℕ. The 𝑞-derivative [6] at 𝑡 is defined by 𝑓^Δ(𝑡)=𝑓(𝑞𝑡)−𝑓(𝑡)𝑓(𝑞−1)𝑡_Δ,(0)=lim_𝑛→∞𝑓(𝑞^𝑛)−𝑓(0)𝑞^𝑛.(2.3)

A 𝑞-difference equation is an equation that contains 𝑞-derivatives of a function defined on 𝑞^ℕ.

Definition 2.2. On the time scale 𝑞^ℕ, the 𝑞-polynomials ℎ_𝑘(⋅,𝑡₀)∶𝑞^ℕ→ℝ are defined recursively as follows: ℎ₀(𝑡,𝑠)=1,ℎ_𝑘+1=^𝑡𝑠ℎ_𝑘(𝜏,𝑠)Δ𝜏.(2.4) By computing the recurrence relation, the 𝑞-polynomials can be represented as ℎ_𝑘(𝑡,𝑠)=^𝑘−1_𝑣=0𝑡−𝑠𝑞^𝑣∑^𝑣𝑗=0𝑞^𝑗(2.5) on 𝑞^ℕ [6].

Hence, for each fixed 𝑠, the delta derivative of ℎ_𝑘 with respect to 𝑡 satisfiesℎ^▵𝑘(𝑡,𝑠)=ℎ_𝑘−1(𝑡−𝑠),𝑘≥1.(2.6)

Using 𝑞-polynomials, Agarwal and Bohner [7] gave a Taylor’s formula for functions on a general time scale. On 𝑞^ℕ Taylor’s formula is written follows.

Theorem 2.3. Let 𝑛∈ℕ. Suppose that 𝑓 is 𝑛 times differentiable on 𝑞^ℕ. Let 𝛼,𝑡∈𝑞^ℕ. Then one has 𝑓(𝑡)=^𝑛−1_𝑘=0ℎ_𝑘𝑓^Δ𝑘(𝛼)+^{𝜌𝑛−1𝛼(𝑡)}ℎ_𝑛−1(𝑡,𝜎(𝜏))𝑓^Δ𝑛(𝜏)Δ𝜏.(2.7)

3. Multiplication of Two 𝑞-Polynomials

The purpose of this section is to propose a production rule of two 𝑞-polynomials at 0 [8] which will be used to derive an approximate solution in the following section.

Theorem 3.1. Let ℎ_𝑖(𝑡,0) and ℎ_𝑗(𝑡,0) be two 𝑞-polynomials at zero. One has ℎ_𝑖(𝑡,0)ℎ_𝑗(𝑡,0)=𝑞^𝑖+1;𝑞_𝑗(𝑞;𝑞)_𝑗ℎ_𝑖+𝑗(𝑡,0).(3.1)

Proof. Since ℎ_𝑖+𝑗(𝑡,0)=^{𝑖+𝑗−1}_𝜈=0𝑡∑^𝜈𝜇=0𝑞^𝜇,(3.2) we have ℎ_𝑖+𝑗(𝑡,0)=^𝑖−1_𝜈=0𝑡∑^𝜈𝜇=0𝑞^𝜇^{𝑖+𝑗−1}_𝜈=𝑖𝑡=ℎ_𝑖∑^𝜈𝜇=0𝑞^𝜇(𝑡,0)∏^{𝑗−1𝜈=0}∑^𝜈𝜇=0𝑞^𝜇∏^{𝑗−1𝜈=0}∑^𝜈𝜇=0𝑞^𝜇𝑡^𝑗^{𝑖+𝑗−1}_𝜈=𝑖1=ℎ_𝑖∑^𝜈𝜇=0𝑞^𝜇(𝑡,0)^𝑗−1_𝜈=0𝑡∑^𝜈𝜇=0𝑞^𝜇^𝑗−1_𝜈𝜈=0_𝜇=0𝑞^𝜇^{𝑖+𝑗−1}_𝜈=𝑖1=ℎ_𝑖∑^𝜈𝜇=0𝑞^𝜇(𝑡,0)ℎ_𝑗(𝑡,0)^𝑗−1_𝜈=0∑^𝜈𝜇=0𝑞^𝜇∑^{𝜈+𝑖𝜇=0}𝑞^𝜇.(3.3) This implies that ℎ_𝑖(𝑡,0)ℎ_𝑗(𝑡,0)=^𝑗−1_𝜈=0∑^{𝜈+𝑖𝜇=0}𝑞^𝜇∑^𝜈𝜇=0𝑞^𝜇ℎ^𝑖+𝑗(𝑡,0)=^𝑗−1_𝜈=01−𝑞^𝜐+𝑖+11−𝑞^𝜐+1ℎ^𝑖+𝑗(𝑡,0)=𝑞^𝑖+1;𝑞_𝑗(𝑞;𝑞)_𝑗ℎ_𝑖+𝑗(𝑡,0).(3.4)

Proposition 3.2. Let ℎ_𝑖(𝑡,0) and ℎ_𝑗(𝑡,0) be any two 𝑞-polynomials. We have ℎ_𝑖(𝑡,0)ℎ_𝑗(𝑡,0)=ℎ_𝑗(𝑡,0)ℎ_𝑖(𝑡,0).(3.5)

Proof. By Theorem 3.1, it suffices to show that 𝑞^𝑖+1;𝑞_𝑗(𝑞,𝑞)_𝑗=𝑞^𝑗+1;𝑞_𝑖(𝑞,𝑞)_𝑖.(3.6) Suppose 𝑖>𝑗, one has 𝑞^𝑖+1;𝑞_𝑗(𝑞,𝑞)_𝑗−𝑞^𝑗+1;𝑞_𝑖(𝑞,𝑞)_𝑖=1−𝑞^𝑗+1⋯1−𝑞^𝑖+𝑗(1−𝑞)⋯(1−𝑞^𝑖)−1−𝑞^𝑖+1⋯1−𝑞^𝑖+𝑗=1−𝑞_𝑗+1(1−𝑞)⋯(1−𝑞^𝑗)⋯1−𝑞^𝑖+𝑗(1−𝑞)⋯(1−𝑞^𝑖)−1−𝑞^𝑖+1⋯1−𝑞^𝑖+𝑗1−𝑞^𝑗+1⋯1−𝑞^𝑖=0.(1−𝑞)⋯(1−𝑞^𝑗)1−𝑞^𝑗+1⋯(1−𝑞^𝑖)(3.7)

4. Variational Iteration Method

4.1. Basic Ideas of Variational Iteration Method

To clarify the ideas of the variational iteration method, we consider the following nonlinear equation:𝐿𝑥(𝑡)+𝑁𝑥(𝑡)=𝑔(𝑡),(4.1) where 𝐿 is a linear operator, 𝑁 is a nonlinear operator, and 𝑔 is an inhomogeneous term. According to the variational iteration method, we can construct a correction functional as follows: 𝑥_𝑛+1=𝑥_𝑛(𝑡)+^𝑡0𝜆𝐿𝑥_𝑛(𝑠)+𝑁𝑥̃_𝑛(𝑠)−𝑔(𝑠)𝑑𝑠,(4.2) where 𝜆 is a general Lagrange multiplier, 𝑢₀ is an initial approximation which must be chosen suitably, and 𝑥̃_𝑛 is considered a restricted variation; that is, 𝛿𝑥̃_𝑛=0. To find the optimal value of 𝜆, we make the above correction functional stationary with respect to 𝑥_𝑛, noticing that 𝛿𝑥_𝑛(0)=0, and have 𝛿𝑥_𝑛+1(𝑡)=𝛿𝑥_𝑛(𝑡)+𝛿^𝑡0𝜆𝐿𝑥(𝑠)𝑑𝑠=0.(4.3) Having obtained the optimal Lagrange multiplier, the successive approximations 𝑥_𝑛, 𝑛≥0, of the solution 𝑥 will be determined upon the initial function 𝑥₀. Therefore, the exact solution is obtained at the limit of the resulting successive approximations.

4.2. Approximate Solution to Nonlinear Damped 𝑞-Equations

In this section, we extend the use of the variational iteration method to strongly nonlinear damped 𝑞-difference equation as follows:𝑥^ΔΔ(𝑡)+2𝛾+𝜀𝛾₁𝑥(𝑡)𝑥^Δ(𝑡)+Ω²𝑥(𝑡)+𝑥²(𝑡)=0,𝑡∈𝑞^ℕ(4.4) with 𝑥(0)=𝑎 and 𝑥^Δ(0)=𝑏.

First of all, we illustrate the main idea of the variational iteration method. The basic character of the method is to construct a correction functional for the system (4.4), which reads 𝑥_𝑛+1(𝑡)=𝑥_𝑛(𝑡)+^𝑡𝑡₀𝜆(𝑠)𝐿𝑥_𝑛(𝑠)+𝑁𝑥̃_𝑛(𝑠)Δ𝑠,(4.5) where 𝐿 is a linear operator, 𝑁 is a nonlinear operator, 𝜆 is a Lagrange multiplier which can be identified optimally by variational theory, 𝑥_𝑛 is the 𝑛th approximation, and 𝑥̃_𝑛 denotes a restricted variation, that is, 𝛿𝑥̃_𝑛=0.

In this work, the linear operator 𝐿 is selected as 𝐿𝑥=𝑥^ΔΔ,(4.6) and the nonlinear operator 𝑁 is selected as𝑁𝑥=2𝛾+𝜀𝛾₁𝑥𝑥^Δ+Ω²𝑥+𝑥².(4.7) Making the above correction functional stationary with respect to 𝑥_𝑛𝛿𝑥_𝑛+1(𝑡)==𝛿1𝑥_𝑛−(𝑡𝜆)_Δ+𝛿^𝑡0𝜆(𝑠)𝑢^𝑛ΔΔ+𝑁𝑢_𝑛̃(𝑠)Δ𝑠𝛿𝑥_𝑛(𝑡)+𝜆(𝑡)𝛿𝑥^Δ(𝑡)+^𝑡0𝜆^ΔΔ(𝑠)𝛿𝑥_𝑛(𝜎(𝑠))Δ𝑠,(4.8) we, therefore, have the following stationary conditions:1𝜆−𝜆𝜆(𝑡^Δ)(=𝑡)0=,_ΔΔ0,(𝑠)=0.(4.9) The Lagrange multiplier can be readily identified: 𝜆(𝑠)=𝑠−𝑡=ℎ₁(𝑠)−ℎ₁(𝑡).(4.10)

As a result, we obtain the variational iteration formula: 𝑥_𝑛+1(𝑡)=𝑥_𝑛(𝑡)+^𝑡0ℎ₁(𝑠)−ℎ₁(𝑡)⎡⎢⎢⎣𝑥^𝑛ΔΔ(𝑠)++Ω₂2𝛾+𝜀𝛾₁𝑥_𝑛(𝑠)𝑥^Δ𝑛(𝑠)𝑥_𝑛(𝑠)+𝑥^2𝑛(𝑠)⎤⎥⎥⎦Δ𝑠.(4.11) According to the initial condition, we begin with the following initial approximation:𝑥₀(𝑡)=𝑎+𝑏ℎ₁(𝑡).(4.12) According to the variational iteration formula, we have𝑥₁(𝑡)=𝑎+𝑏ℎ₁(𝑡)+𝐴₁ℎ₂(𝑡)+𝐵₁ℎ₃(𝑡)+𝐶₁ℎ₄(𝑡),(4.13) where 𝐴𝐵₁=2𝛾+𝜀𝛾₁𝑎𝑏+𝑎Ω²+𝑎²(1−𝐻(1,1)),₁𝐶=𝜀𝛾₁𝑏²+Ω²𝑏+2(𝑎𝑏)[𝐻(1,1)−𝐻(2,1)₁],=𝑏²𝐻(1,1)[𝐻(1,2)−𝐻(1,3)],(4.14) and 𝐻(𝑖,𝑗)=(𝑞^𝑖+1;𝑞)_𝑗/(𝑞;𝑞)_𝑗.

5. Numerical Method

By Definition 2.1, the derivative of 𝑥(𝑡) at 0 is defined as 𝑥^Δ(0)=lim_𝑛→∞𝑥(𝑞^𝑛)−𝑥(0)𝑞^𝑛,if𝑞<1.(5.1)

Let 𝑁₀>0 be a nonnegative integer. To obtain an approximation for the derivative of 𝑥(𝑡) at 𝑡=0, we use 𝑥𝑞^𝑁0=𝑥(0)+𝑞^𝑁0𝑥^Δ(0)+𝑞^2𝑁02𝑥^Δ2(0)+⋯.(5.2) Rearrangement leads to𝑥^Δ(0)≈𝑥𝑞^𝑁0−𝑥(0)𝑞^𝑁0−𝑞^𝑁02𝑥^Δ2=𝑥𝑞(0)_𝑁0−𝑥(0)𝑞^𝑁0+𝑂𝑞^𝑁0,(5.3) where the dominant term in the truncation error is 𝑂(𝑞^𝑁0).

Since 𝑥^Δ(0)=𝑏, we have𝑥𝑞^𝑁0−𝑥(0)𝑞^𝑁0=𝑏(5.4) which yields𝑥𝑞^𝑁0=𝑥(0)+𝑞^𝑁0𝑏=𝑎+𝑞^𝑁0𝑏.(5.5) Set 𝑡₀=0 and 𝑡₁=𝑞^𝑁0 and define𝑡_𝑖=𝑞^{𝑁0−(𝑖−1)},∀𝑖=2,…,𝑁₀+1.(5.6) Then the interval [0,𝑞] is partitioned into 𝑁₀ subintervals.

Now we denote𝑥_𝑖=𝑥𝑡_𝑖,∀𝑖=0,1,2,…,𝑁₀+1.(5.7) The Delta-derivative of 𝑥(𝑡) at 𝑡_𝑖 can be calculated as𝑥^Δ𝑖=𝑥^𝑖+1−𝑥_𝑖𝑡_𝑖+1−𝑡_𝑖𝑥_𝑖ΔΔ=𝐷_𝑖𝑡_𝑖+1−𝐷_𝑖𝑡_𝑖,=𝑥^Δ𝑖+1−𝑥^Δ𝑖𝑡_𝑖+1−𝑡_𝑖=𝐴_𝑖𝑥_𝑖+2−𝐵_𝑖𝑥_𝑖+1+𝐶_𝑖𝑥_𝑖,(5.8) where𝐴_𝑖=1𝑡_𝑖+1−𝑡_𝑖𝑡_𝑖+2−𝑡_𝑖+1,𝐵^𝑖=𝑡^𝑖+2−𝑡_𝑖𝐶_𝑖𝑡_𝑖+1−𝑡_𝑖²𝑡_𝑖+2−𝑡_𝑖+1,=1𝑡_𝑖+1−𝑡_𝑖²,𝐷_𝑖=1𝑡_𝑖+1−𝑡_𝑖.(5.9) Substituting (5.8) into (4.4) yields the following: 𝐴_𝑖𝑥_𝑖+2+2𝛾𝐷_𝑖−𝐵_𝑖𝑥_𝑖+1+Ω+𝐶_𝑖−2𝛾𝐷_𝑖𝑥_𝑖+𝜀𝐷_𝑖𝑥_𝑖𝑥_𝑖+1+1−𝜀𝐷_𝑖𝑥^2𝑖=0.(5.10) This implies that 𝑥_𝑖+2=−1𝐴_𝑖2𝛾𝐷_𝑖−𝐵_𝑖𝑥_𝑖+1+Ω+𝐶_𝑖−2𝛾𝐷_𝑖𝑥_𝑖+𝜀𝛾₁𝐷_𝑖𝑥_𝑖𝑥_𝑖+1+1−𝜀𝛾₁𝐷_𝑖𝑥^2𝑖.(5.11)

6. Numerical Results

The theoretical considerations introduced in previous sections are illustrated with examples, where the approximate solutions are compared with the numerical solutions.

The time scale 𝑞^ℕ is given as {0.9^𝑛∣𝑛∈ℕ}∪{0}={0.9,0.81,0.729,…,0}, where 0 is the cluster point of 𝑞^ℕ. For the numerical computations, the interval [0,0.9] is partitioned into 100 subintervals. The maximum error and the average error are defined as maximumerror=max||𝑥_𝑛(𝑡)−𝑥̂(𝑡)||∣𝑡∈𝑞^ℕ,𝑡≥0.9¹⁰⁰,averageerror=sum||𝑥_𝑛(𝑡)−𝑥̂(𝑡)||∣𝑡∈𝑞^ℕ,𝑡≥0.9¹⁰⁰100,(6.1) respectively, where 𝑥_𝑛 is the approximate solution with 𝑛 iterations and 𝑥̂ is the numerical solution obtained by (5.11).

Example 6.1. Consider the underdamped cases with (i) 2𝛾=0.1, 𝛾₁=0.1, 𝜀=1, and Ω=1; and (ii) 2𝛾=0.1, 𝛾₁=2.5, 𝜀=1 and Ω=1. As the initial conditions are given as 𝑥(0)=1 and 𝑥^Δ(0)=0.5, we begin with the initial approximation 𝑥₀=1+0.5𝑡. By the variational iteration formula (4.11), we obtain the first few components of 𝑥_𝑛(𝑡). In the same manner the rest of the components of the iteration formula are obtained using the symbolic toolbox in the Matlab package.For Case (i)
The first two components of 𝑥_𝑛 are obtained as 𝑥₀=1+1𝑥2𝑡,₁=1+0.5ℎ₁(𝑡,0)−1.89ℎ₂(𝑡,0)−1.2353ℎ₃(𝑡,0)−0.3463ℎ₄(𝑡,0).(6.2) and so on. After 3 iterations, the maximum error is 0.0415 and the average error is 0.00356.

For case (ii)
The first two components of 𝑥_𝑛 are obtained as 𝑥₀=1+1𝑥2𝑡,₁=1+0.5ℎ₁(𝑡,0)−2.97ℎ₂(𝑡,0)−1.7213ℎ₃(𝑡,0)−0.3463ℎ₄(𝑡,0),(6.3) and so on. After 3 iterations, the maximum error is 0.0275 and the average error is 0.001167.
The responses of 𝑥(𝑡) are shown in Figures 1 and 2 for cases (i) and (ii), respectively.

Figure 1 

Time response for 𝑥ΔΔ+(0.1+0.1𝑥)𝑥Δ+𝑥+𝑥2=0 with 3 iterations.

Figure 2 

Time response for 𝑥ΔΔ+(0.1+2.5𝑥)𝑥Δ+𝑥+𝑥2=0 with 3 iterations.

Example 6.2. In this example, we consider the overdamped cases with (iii) 2𝛾=2.5, 𝛾₁=0.1, 𝜀=1, and Ω=1; and (iv) 2𝛾=2.5, 𝛾₁=2.5, 𝜀=1, and Ω=1. As the initial conditions are given as 𝑥(0)=1 and 𝑥^Δ(0)=0.5, we begin with the initial approximation 𝑥₀=1+0.5𝑡. By the variational iteration formula (4.11), we obtain the first few components of 𝑥_𝑛(𝑡). In the same manner the rest of the components of the iteration formula were obtained using the symbolic toolbox in the Matlab package.For Case (iii)
The first two components of 𝑥_𝑛 are obtained as 𝑥₀(𝑡)=1+1𝑥2𝑡,₁(𝑡)=1+0.5ℎ₁(𝑡,0)−2.97ℎ₂(𝑡,0)−1.2353ℎ₃(𝑡,0)−0.3463ℎ₄(𝑡,0),(6.4) and so on. After 3 iterations, the maximum error is 0.01609 and the average error is 0.001227.For Case (iv)
The first two components of 𝑥_𝑛 are obtained as 𝑥₀=1+1𝑥2𝑡,₁=1+0.5ℎ₁(𝑡,0)−4.05ℎ₂(𝑡,0)−1.7213ℎ₃(𝑡,0)−0.3463ℎ₄(𝑡,0)(6.5) and so on. After 7 iterations, the maximum error is 0.026 and the average error is 0.00105. At less than 7 iterations, the approximate solution is not close to the numerical solution.
The responses of 𝑥(𝑡) are shown in Figures 3 and 4 for cases (iii) and (iv), respectively.

Figure 3 

Time response for 𝑥ΔΔ+(2.5+0.1𝑥)𝑥Δ+𝑥+𝑥2=0 with 3 iterations.

Figure 4 

Time response for 𝑥ΔΔ+(2.5+2.5𝑥)𝑥Δ+𝑥+𝑥2=0 with 7 iterations.

These figures and the maximum/average errors indicate that the approximate solution is close to the numerical results.

7. Conclusion

In the area of 𝑞-calculus, the formula for the multiplication of two 𝑞-polynomials has long been in need of development. In this paper, we have presented the aforementioned formula, overcoming the previous difficulties in developing a theory of series solutions for the nonlinear 𝑞-difference equations. The goal of this paper was to extend the use of the variational iteration method to strongly nonlinear damped 𝑞-difference equations. The numerical results have demonstrated that the approximate solution obtained by the variational iteration method is very accurate. Therefore, the proposed method is very effective and can be applied to other nonlinear 𝑞-equations.