# Convergence of Variational Iteration Method for Second-Order Delay Differential Equations

Authors: Hongliang Liu; Aiguo Xiao; Lihong Su
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

This paper employs the variational iteration method  to obtain analytical solutions of second-order delay differential equations. The corresponding convergence results are obtained, and an effective technique for choosing a reasonable Lagrange multiplier is designed in the solving process. Moreover, some illustrative examples are given to show the efficiency of this method.


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## Body

## 1. Introduction

The second-order delay differential equations often appear in the dynamical system, celestial mechanics, kinematics, and so forth. Some numerical methods for solving second-order delay differential equations have been discussed, which includeθ-method [1], trapezoidal method [2], and Runge-Kutta-Nyström method [3]. The variational iteration method (VIM) was first proposed by He [4, 5] and has been extensively applied due to its flexibility, convenience,  and efficiency. So far, the VIM is applied to autonomous ordinary differential systems [6], pantograph equations [7], integral equations [8], delay differential equations [9], fractional differential equations [10], the singular perturbation problems [11], and delay differential-algebraic equations [12]. Rafei et al. [13] and Marinca et al. [14] applied the VIM to oscillations. Tatari and Dehghan [15] consider the VIM for solving second-order initial value problems. For a more comprehensive survey on this method and its applications, the readers refer to the review articles [16–19] and the references therein. But the VIM for second-order delay differential equations has not been considered.The article apply the VIM to second-order delay differential equations to obtain the analytical or approximate analytical solutions. The corresponding convergence results are obtained. Some illustrative examples confirm the theoretical results.


## 2. Convergence

### 2.1. The First Kind of Second-Order Delay Differential Equations

Consider the initial value problems of second-order delay differential equations(1)y′′(t)=f(t,y(t),y(α(t))),t∈[0,T],y′(t)=φ′(t),t∈[-τ,0],y(t)=φ(t),t∈[-τ,0],
where φ(t) is a differentiable function, α(t)∈C1[0,T] is a strictly monotone increasing function and satisfies that -τ≤α(t)≤t and α(0)=-τ, there exists t1∈[0,T] such that α(t1)=0, and f:D=[0,T]×R×R→R is a given continuous mapping and satisfies the Lipschitz condition
(2)∥f(t,u1,v)-f(t,u2,v)∥≤β0∥u1-u2∥,∥f(t,u,v1)-f(t,u,v2)∥≤β1∥v1-v2∥,
where β0,β1 are Lipschitz constants; ∥·∥ denotes the standard Euclidean norm.Now the VIM for (1) can read
(3)ym+1(t)=ym(t)+∫0t‍λ(t,ξ)[ym′′(ξ)-f~(ξ,ym(ξ),φ(α(ξ)))]dξ,×λ(t,ξ)dξ,f~(ξ,ym(ξ),φ(α(ξ))),0<t<t1;(4)ym+1(t)=ym(t)+∫0t1‍λ(t,ξ)[ym′′(ξ)-f~(ξ,ym(ξ),φ(α(ξ)))]dξ+∫t1t‍λ(t,ξ)[ym′′(ξ)-f~(ξ,ym(ξ),ym(α(ξ)))]dξ,[ym′′(ξ)-+∫t1t‍λ(t,ξ)f~(ξ,ym(ξ)),ym(α(ξ)))]t>t1,
where ym(t)=φ(t) for t∈[-τ,0]; f~ denotes the restrictive variation, that is, δf~=0. Thus, we have
(5)δym+1(t)=δym(t)+∫0tδλ(t,ξ)[ym′′(ξ)-f~(ξ,ym(ξ),φ(α(ξ)))]dξ=δym(t)+∫0tδλ(t,ξ)ym′′(ξ)dξ,0<t<t1.
Using integration by parts to (4), we have
(6)δym+1(t)=δym(t)+∫0t1‍δλ(t,ξ)[ym′′(ξ)-f~(ξ,ym(ξ),φ(α(ξ)))]dξ+∫t1t‍δλ(t,ξ)[ym′′(ξ)-f~(ξ,ym(ξ),ym(α(ξ)))]dξ=δym(t)+δλ(t,ξ)ym′(ξ)|ξ=t-∂λ(t,ξ)∂ξδym(ξ)|ξ=t+∫0t‍∂2λ(t,ξ)∂ξ2δym(ξ)dξ.
From the above formula, the stationary conditions are obtained as
(7)∂2λ(t,ξ)∂ξ2=0,1-∂λ(t,ξ)∂ξ|ξ=t=0,λ(t,ξ)|ξ=t=0.
Moreover, the general Lagrange multiplier
(8)λ(t,ξ)=ξ-t
can be readily identified by (7). Thus, the variational iteration formula can be written as
(9)ym+1(t)=ym(t)+∫0t‍(ξ-t)[ym′′(ξ)-f(ξ,ym(ξ),φ(α(ξ)))]dξ,[y′′m(ξ)-f(ξ,ym(ξ),φ(α(ξ)))]0<t<t1;(10)ym+1(t)=ym(t)+∫0t1‍(ξ-t)[ym′′(ξ)-f(ξ,ym(ξ),φ(α(ξ)))]dξ+∫t1t‍(ξ-t)[ym′′(ξ)-f(ξ,ym(ξ),ym(α(ξ)))]dξ,×dξ,[y′′m(ξ)-f(ξ,ym(ξ),φ(α(ξ)))]t>t1.Theorem 1.
Suppose that the initial value problems (1) satisfy the condition (2), and y(t),yi(t)∈C2[0,T], i=1,2,…. Then the sequence {ym(t)}m=1∞ defined by (9) and (10) with y0(t) converges to the solution of (1).Proof.
From (1), we have
(11)y(t)=y(t)+∫0t‍(ξ-t)[y′′(ξ)-f(ξ,y(ξ),φ(α(ξ)))]dξ,[y′′(ξ)-f(ξ,y(ξ),φ(α(ξ)))]0<t<t1;(12)y(t)=y(t)+∫0t1‍(ξ-t)[y′′(ξ)-f(ξ,y(ξ),φ(α(ξ)))]dξ+∫t1t‍(ξ-t)[y′′(ξ)-f(ξ,y(ξ),y(α(ξ)))]dξ,[y′′(ξ)-f(ξ,y(ξ),y(α(ξ)))]dξ,t>t1.
Let Ei(t)=yi(t)-y(t), i=0,1,…. If t⩽0, then Ei(t)=0,i=0,1,…. From (9) and (11), we have
(13)Em+1(t)=Em(t)+∫0t‍(ξ-t)[Em′′(ξ)-(f(ξ,ym(ξ),φ(α(ξ)))+∫0t1‍Em′′(ξ)-(ξ-t)-f(ξ,y(ξ),φ(α(ξ)))){Em′′}]dξ,+∫0t‍×ξ,(ξ,ym(ξ),φ(α(ξ)))0<t<t1.
From (10) and (12), we have
(14)Em+1(t)=Em(t)+∫0t1(ξ-t)[Em′′(ξ)-(f(ξ,ym(ξ),φ(α(ξ)))+∫0t1‍Em′′(ξ)-(ξ-t)-f(ξ,y(ξ),φ(α(ξ))))Em′′]dξ+∫t1t(ξ-t)[Em′′(ξ)-(f(ξ,ym(ξ),ym(α(ξ)))+∫0t1‍Em′′(ξ)-(ξ-t)-f(ξ,y(ξ),y(α(ξ))))Em′′]dξ,∫0t1‍(ξ-t)[Em′′(ξ)-(fdξ,d(f(ξ,ym(ξ),φ(α(ξ)))t>t1.
Using integration by parts, we have
(15)Em+1(t)=Em(t)+∫0t‍(ξ-t)Em′′(ξ)dξ-∫0t‍(ξ-t)[f(ξ,ym(ξ),φ(α(ξ)))-∫0t‍(ξ-t)-f(ξ,y(ξ),φ(α(ξ)))]dξ=-∫0t‍(ξ-t)[f(ξ,ym(ξ),φ(α(ξ)))-∫0t‍(ξ-t)-f(ξ,y(ξ),φ(α(ξ)))]dξ,+∫0t‍(ξ,ym(ξ),φ(α(ξ)))dξ,0<t<t1;Em+1(t)=Em(t)+∫0t‍(ξ-t)Em′′(ξ)dξ-∫0t1‍(ξ-t)[f(ξ,ym(ξ),φ(α(ξ)))-∫0t‍(ξ-t)-f(ξ,y(ξ),φ(α(ξ)))]dξ-∫t1t‍(ξ-t)[f(ξ,ym(ξ),ym(α(ξ)))-∫0t‍(ξ-t)-f(ξ,y(ξ),y(α(ξ)))]dξ=-∫0t1‍(ξ-t)[f(ξ,ym(ξ),φ(α(ξ)))-∫0t‍(ξ-t)-f(ξ,y(ξ),φ(α(ξ)))]dξ-∫t1t‍(ξ-t)[f(ξ,ym(ξ),ym(α(ξ)))-∫0t‍(ξ-t)-f(ξ,y(ξ),y(α(ξ)))]dξ,+∫0t‍(ξ,ym(ξ),ym(α(ξ)))(α(ξ))t>t1.
Since [α-1(t)]′ is bounded, M=max-τ≤ξ≤α(T)(α-1(ξ))′ is bounded. Moreover, it follows from (2) and the inequality |ξ-t|≤T that
(16)∥Em+1(t)∥≤∫0t‍|t-ξ|∥f(ξ,ym(ξ),φ(α(ξ)))∫0t‍|t-ξ|-f(ξ,y(ξ),φ(α(ξ)))∥dξ≤∫0t‍Tβ0∥ym(ξ)-y(ξ)∥dξ=∫0t‍Tβ0∥Em(ξ)∥dξ,0<t<t1;(17)∥Em+1(t)∥≤∫0t1‍|t-ξ|∥f(ξ,ym(ξ),φ(α(ξ)))∫0t‍|t-ξ|-f(ξ,y(ξ),φ(α(ξ)))∥dξ+∫t1t‍|t-ξ|∥f(ξ,ym(ξ),ym(α(ξ)))∫0t‍|t-ξ|-f(ξ,y(ξ),y(α(ξ)))∥dξ≤∫0t1‍Tβ0∥ym(ξ)-y(ξ)∥dξ+∫t1t‍T(β0∥Em(ξ)∥+β1∥Em(α(ξ))∥)dξ=∫0t‍Tβ0∥Em(ξ)∥dξ+∫t1t‍Tβ1∥Em(α(ξ))∥dξ=∫0t‍Tβ0∥Em(ξ)∥dξ+∫α(t1)α(t)‍Tβ1∥Em(ξ)∥(α-1(ξ))′dξ≤TMβ∫0t‍∥Em(ξ)∥dξ,t>t1,
where β=maxβi, i=1,2. Moreover,
(18)∥Em+1(t)∥≤(TMβ)2∫0t‍∫0s1‍∥Em-1(s2)∥ds2ds1≤(TMβ)3∫0t‍∫0s1‍∫0s2‍∥Em-2(s3)∥ds3ds2ds1≤(TMβ)4∫0t‍∫0s1‍∫0s2‍∫0s3‍∥Em-3(s4)∥ds4ds3ds2ds1⋯≤(TMβ)m+1∫0t‍∫0s1‍∫0s2‍⋯∫0sm‍∥E0(sm+1)∥dsm+1(TMβ)m+1∫0t‍∫0s1‍∫0s2‍⋯∫0sm‍⋯ds3ds2ds1,
where ∥E0(t)∥ is constant. Therefore, we have
(19)∥Em+1(t)∥≤∥E0(t)∥(TMβ)m+1(m+1)!→0,(m→∞).


### 2.2. The Second Kind of Second-Order Delay Differential Equations

Consider the initial value problems of second-order delay oscillation differential equations(20)y′′(t)=-ω2y(t)-f(t,y(t),y(α(t))),t∈[0,T],y′(t)=φ′(t),t∈[-τ,0],y(t)=φ(t),t∈[-τ,0],
where φ(t) is a differentiable function, α(t)∈C1[0,T] is a strictly monotone increasing function and satisfies that -τ≤α(t)≤t and α(0)=-τ, there exists t1∈[0,T] such that α(t1)=0, ω is a constant, and f:D=[0,T]×R×R→R is a given continuous mapping and satisfies the Lipschitz condition
(21)∥f(t,u1,v)-f(t,u2,v)∥≤κ0∥u1-u2∥,∥f(t,u,v1)-f(t,u,v2)∥≤κ1∥v1-v2∥,
where κ0,κ1 are Lipschitz constants.Now the VIM for (20) can read
(22)ym+1(t)=ym(t)+∫0t‍λ(t,ξ)[ym′′(ξ)+ω2ym(ξ)+∫0t‍λ(t,ξ)+f~(ξ,ym(ξ),φm(α(ξ))){ym′′}]dξ,f~(ξ,ym(ξ),φm(α(ξ)))+λ(t,ξ)0<t<t1;(23)ym+1(t)=ym(t)+∫0t1‍λ(t,ξ)[ym′′(ξ)+ω2ym(ξ)+∫0t‍λ(t,ξ)+f~(ξ,ym(ξ),φm(α(ξ))){ym′′}]dξ+∫t1t‍λ(t,ξ)[ym′′(ξ)+ω2ym(ξ)+∫0t‍λ(t,ξ)+f~(ξ,ym(ξ),ym(α(ξ))){ym′′}]dξ,(ξ,ym(ξ),ym(α(ξ)))dξ,+∫0t‍λ(t,ξ)t>t1,
where ym(t)=φ(t) for t∈[-τ,0]; f~ denotes the restrictive variation, that is, δf~=0. Thus, we have
(24)δym+1(t)=δym(t)+∫0t‍δλ(t,ξ)[ym′′(ξ)+ω2ym(ξ)+∫0t‍λ(t,ξ)+f~(ξ,ym(ξ),φm(α(ξ)))]dξ=δym(t)+∫0t‍δλ(t,ξ)[ym′′(ξ)+ω2ym(ξ)]dξ,dξ,ω2ym(ξ)[y′′m(ξ)+ω2ym(ξ)]dξ,0<t<t1.
Using integration by parts to (23), we have
(25)δym+1(t)=δym(t)+∫0t1‍δλ(t,ξ)[ym′′(ξ)+ω2ym(ξ)+∫0t‍λ(t,ξ)+f~(ξ,ym(ξ),φm(α(ξ)))]dξ+∫t1t‍δλ(t,ξ)[ym′′(ξ)+ω2ym(ξ)+∫0t‍λ(t,ξ)+f~(ξ,ym(ξ),ym(α(ξ)))]dξ=δym(t)+∫0t‍δλ(t,ξ)[ym′′(ξ)+ω2ym(ξ)]dξ=δym(t)-∂λ(t,ξ)∂ξδym(ξ)|ξ=t+λ(t,ξ)δym′(ξ)|ξ=t+∫0t‍[ω2λ(t,ξ)+∂2λ(t,ξ)∂ξ2]δym(ξ)dξ.
From the above formula, the stationary conditions are obtained as
(26)ω2λ(t,ξ)+∂2λ(t,ξ)∂ξ2=0,1-∂λ(t,ξ)∂ξ|ξ=t=0,λ(t,ξ)|ξ=t=0.
Moreover, the general Lagrange multiplier
(27)λ(t,ξ)=1ωsinω(ξ-t)
can be readily identified by (26). Thus, the variational iteration formula can be written as
(28)ym+1(t)=ym(t)+∫0t‍1ωsinω(ξ-t)[ym′′(ξ)+ω2ym(ξ)+∫0t‍1ωsinω(ξ-t)+f(ξ,ym(ξ),φ(α(ξ)))ym′′]dξ,(ξ,ym(ξ),ym(α(ξ)))(dξ,ξ,ym(ξ),ym(α(ξ)))0<t<t1;ym+1(t)=ym(t)+∫0t11ωsinω(ξ-t)[ym′′(ξ)+ω2ym(ξ)∫t1t1ωsinω(ξ-t)+f(ξ,ym(ξ),φ(α(ξ)))ym′′]dξ+∫t1t1ωsinω(ξ-t)[ym′′(ξ)+ω2ym(ξ)∫t1t1ωsinω(ξ-t)+f(ξ,ym(ξ),ym(α(ξ)))ym′′]dξ,(ξ,ym(ξ),ym(α(ξ)))(ym(α(ξ))ξ,ym(ξ),ym(α(ξ)))t>t1.Theorem 2.
Suppose that the initial value problems (20) satisfy the condition (21), and y(t),yi(t)∈C2[0,T], i=1,2,…. Then the sequence {ym(t)}m=1∞ defined by (28) with y0(t) converges to the solution of (20).Proof.
The proof process is similar that in Theorem1.


### 2.3. The Third Kind of Second-Order Delay Differential Equations

In order to improve the iteration speed, we modify the above iterative formulas and reconstruct the Lagrange multiplier. Consider the initial value problems of second-order delay differential equations(29)y′′(t)+a(t)y′(t)+b(t)y(t)+N(t,y(t),y(α(t)))=0,y′′(t)+a(t)y′(t)+b(t)y(t)+N(t,y(t)),yt∈[0,T],y′(t)=φ′(t),t∈[-τ,0],y(t)=φ(t),t∈[-τ,0],
where φ(t) is a differentiable function, α(t)∈C1[0,T] is a strictly monotone increasing function and satisfies that -τ≤α(t)≤t and α(0)=-τ, there exists t1∈[0,T] such that α(t1)=0, a(t),b(t) are bounded functions, and N:D=[0,T]×R×R→R is a given continuous mapping and satisfies the Lipschitz condition
(30)∥N(t,u1,v)-N(t,u2,v)∥≤γ0∥u1-u2∥,∥N(t,u,v1)-N(t,u,v2)∥≤γ1∥v1-v2∥,
where γ0, γ1 are Lipschitz constants.Now the VIM for (2.19) can read(31)ym+1(t)=ym(t)+∫0t‍λ(t,ξ)[ym′′(ξ)+a(ξ)ym′(ξ)+b(ξ)ym(ξ)+∫0t‍λ(t,ξ)+N~(ξ,ym(ξ),φ(α(ξ)))]dξ,×∥N(t,u,v1)-N(t,u,v2)∥≤γ10<t<t1;(32)ym+1(t)=ym(t)+∫0t1‍λ(t,ξ)[ym′′(ξ)+a(ξ)ym′(ξ)+b(ξ)ym(ξ)+∫0t‍λ(t,ξ)+N~(ξ,ym(ξ),φ(α(ξ)))]dξ+∫t1t‍λ(t,ξ)[ym′′(ξ)+a(ξ)ym′(ξ)+b(ξ)ym(ξ)+∫0t‍λ(t,ξ)+N~(ξ,ym,ym(α(ξ)))]dξ,(α(ξ))+N~(ξ,ym(ξ),φ(α(ξ)))]dξt>t1,
where ym(t)=φ(t) for t∈[-τ,0]; N~ denotes the restrictive variation, that is, δN~=0. Thus, we have
(33)δym+1(t)=δym(t)+∫0t‍δλ(t,ξ)[ym′′(ξ)+a(ξ)ym′(ξ)+b(ξ)ym(ξ)+∫0t‍λ(t,ξ)+N~(ξ,ym(ξ),φ(α(ξ)))ym′′]dξ=δym(t)+∫0t‍δλ(t,ξ)[ym′′(ξ)+a(ξ)ym′(ξ)+b(ξ)ym(ξ)]dξ,(,)[y′′m(ξ)+ay′m(ξ)dξ(ξ)+ξb(ξ)ym(ξ)]0<t<t1.
Using integration by parts to (32), we have
(34)δym+1(t)=δym(t)+∫0t1‍δλ(t,ξ)[ym′′(ξ)+a(ξ)ym′(ξ)+b(ξ)ym(ξ)+∫0t1‍δλ(t,ξ)+N~(ξ,ym(ξ),φ(α(ξ)))]dξ+∫t1t‍δλ(t,ξ)[ym′′(ξ)+a(ξ)ym′(ξ)+b(ξ)ym(ξ)+∫0t‍δλ(t,ξ)+N~(ξ,ym(ξ),ym(α(ξ)))]dξ=δym(t)+∫0t‍δλ(t,ξ)[ym′′(ξ)+a(ξ)ym′(ξ)+b(ξ)ym(ξ)]dξ=δym(t)+∫0t‍λ(t,ξ)δdym′(ξ)+∫0t‍λ(t,ξ)a(ξ)δdym(ξ)+∫0t‍δλ(t,ξ)b(ξ)ym(ξ)dξ=(1-∂λ(t,ξ)∂ξ+a(ξ)λ(ξ))δym(ξ)|ξ=t+λ(t,ξ)δym′(ξ)|ξ=t+∫0t‍[∂2λ(t,ξ)∂2(ξ)-∂(a(ξ)λ(t,ξ))∂ξ+b(ξ)λ(t,ξ)]×δym(ξ)dξ.
From the above formula, the stationary conditions are obtained as
(35)∂2λ(t,ξ)∂2ξ-∂(a(ξ)λ(t,ξ))∂ξ+b(ξ)λ(t,ξ)=0,1-∂λ(t,ξ)∂ξ+a(ξ)λ(t,ξ)|ξ=t=0,λ(t,ξ)|ξ=t=0.
We suppose that λ1(t,ξ), λ2(t,ξ) are the fundamental solutions of (35); the corresponding general solution of (35) is
(36)λ(t,ξ)=c1λ1(t,ξ)+c2λ2(t,ξ).
Using the initial conditions of (35), we have
(37)c1λ1(t,t)+c2λ2(t,t)=0,c1λ1′(t,t)+c2λ2′(t,t)=1.
Note that W(t,t)=|λ1(t,t)λ2(t,t)λ1′(t,t)λ2′(t,t)| is the Wronski determinant of λ1(t,ξ), λ2(t,ξ). We have
(38)λ(t,ξ)=-λ2(t,t)λ1(t,ξ)+λ1(t,t)λ2(t,ξ)W(t,t).
Using the Liouville formula, we have
(39)W(t,t)=W(t,0)e∫0t‍a(ξ)dξ.
So λ(t,ξ) can be expressed as
(40)λ(t,ξ)=-λ2(t,t)λ1(t,ξ)+λ1(t,t)λ2(t,ξ)λ1(t,0)λ2′(t,0)-λ1′(t,0)λ2(t,0)e-∫0t‍a(ξ)dξ.
Note that
(41)u1(ξ)=λ1(t,ξ)W(t,0),u2(ξ)=λ2(t,ξ)W(t,0).
Equation (40) can be expressed as
(42)λ(t,ξ)=-e-∫0t‍a(ξ)dξ[u1(ξ)u2(t)-u2(ξ)u1(t)].
Substituting (42) into (31) and (32), we obtain
(43)ym+1(t)=ym(t)+∫0t‍e-∫0t‍a(ξ)dξ[u1(ξ)u2(t)-u2(ξ)u1(t)]+∫0t‍×[ym′′(ξ)+a(ξ)ym′(ξ)+b(ξ)ym(ξ)+∫0t‍×+N(ξ,ym(ξ),φ(α(ξ)))ym′′]dξ,dξ,×N(ξ,ym(ξ),φ(α(ξ)))0<t<t1;(44)ym+1(t)=ym(t)+∫0t1‍e-∫0t‍a(ξ)dξ[u1(ξ)u2(t)-u2(ξ)u1(t)]+∫0t1‍×[ym′′(ξ)+a(ξ)ym′(ξ)+b(ξ)ym(ξ)+∫0t1‍+N(ξ,ym(ξ),φ(α(ξ)))ym′′]dξ+∫t1t‍e-∫0t‍a(ξ)dξ[u1(ξ)u2(t)-u2(ξ)u1(t)]+∫0t1‍×[ym′′(ξ)+a(ξ)ym′(ξ)+b(ξ)ym(ξ)+∫0t1‍+N(ξ,ym,ym(α(ξ)))ym′′]dξ,t>t1.Theorem 3.
Suppose that the initial value problems (2.19) satisfy the condition (30), and y(t),yi(t)∈C2[0,T], i=1,2,…. Then the sequence {ym(t)}m=1∞ defined by (43) and (44) with y0(t) converges to the solution of (2.19).Proof.
The proof process is similar to that in Theorem1.


## 2.1. The First Kind of Second-Order Delay Differential Equations

Consider the initial value problems of second-order delay differential equations(1)y′′(t)=f(t,y(t),y(α(t))),t∈[0,T],y′(t)=φ′(t),t∈[-τ,0],y(t)=φ(t),t∈[-τ,0],
where φ(t) is a differentiable function, α(t)∈C1[0,T] is a strictly monotone increasing function and satisfies that -τ≤α(t)≤t and α(0)=-τ, there exists t1∈[0,T] such that α(t1)=0, and f:D=[0,T]×R×R→R is a given continuous mapping and satisfies the Lipschitz condition
(2)∥f(t,u1,v)-f(t,u2,v)∥≤β0∥u1-u2∥,∥f(t,u,v1)-f(t,u,v2)∥≤β1∥v1-v2∥,
where β0,β1 are Lipschitz constants; ∥·∥ denotes the standard Euclidean norm.Now the VIM for (1) can read
(3)ym+1(t)=ym(t)+∫0t‍λ(t,ξ)[ym′′(ξ)-f~(ξ,ym(ξ),φ(α(ξ)))]dξ,×λ(t,ξ)dξ,f~(ξ,ym(ξ),φ(α(ξ))),0<t<t1;(4)ym+1(t)=ym(t)+∫0t1‍λ(t,ξ)[ym′′(ξ)-f~(ξ,ym(ξ),φ(α(ξ)))]dξ+∫t1t‍λ(t,ξ)[ym′′(ξ)-f~(ξ,ym(ξ),ym(α(ξ)))]dξ,[ym′′(ξ)-+∫t1t‍λ(t,ξ)f~(ξ,ym(ξ)),ym(α(ξ)))]t>t1,
where ym(t)=φ(t) for t∈[-τ,0]; f~ denotes the restrictive variation, that is, δf~=0. Thus, we have
(5)δym+1(t)=δym(t)+∫0tδλ(t,ξ)[ym′′(ξ)-f~(ξ,ym(ξ),φ(α(ξ)))]dξ=δym(t)+∫0tδλ(t,ξ)ym′′(ξ)dξ,0<t<t1.
Using integration by parts to (4), we have
(6)δym+1(t)=δym(t)+∫0t1‍δλ(t,ξ)[ym′′(ξ)-f~(ξ,ym(ξ),φ(α(ξ)))]dξ+∫t1t‍δλ(t,ξ)[ym′′(ξ)-f~(ξ,ym(ξ),ym(α(ξ)))]dξ=δym(t)+δλ(t,ξ)ym′(ξ)|ξ=t-∂λ(t,ξ)∂ξδym(ξ)|ξ=t+∫0t‍∂2λ(t,ξ)∂ξ2δym(ξ)dξ.
From the above formula, the stationary conditions are obtained as
(7)∂2λ(t,ξ)∂ξ2=0,1-∂λ(t,ξ)∂ξ|ξ=t=0,λ(t,ξ)|ξ=t=0.
Moreover, the general Lagrange multiplier
(8)λ(t,ξ)=ξ-t
can be readily identified by (7). Thus, the variational iteration formula can be written as
(9)ym+1(t)=ym(t)+∫0t‍(ξ-t)[ym′′(ξ)-f(ξ,ym(ξ),φ(α(ξ)))]dξ,[y′′m(ξ)-f(ξ,ym(ξ),φ(α(ξ)))]0<t<t1;(10)ym+1(t)=ym(t)+∫0t1‍(ξ-t)[ym′′(ξ)-f(ξ,ym(ξ),φ(α(ξ)))]dξ+∫t1t‍(ξ-t)[ym′′(ξ)-f(ξ,ym(ξ),ym(α(ξ)))]dξ,×dξ,[y′′m(ξ)-f(ξ,ym(ξ),φ(α(ξ)))]t>t1.Theorem 1.
Suppose that the initial value problems (1) satisfy the condition (2), and y(t),yi(t)∈C2[0,T], i=1,2,…. Then the sequence {ym(t)}m=1∞ defined by (9) and (10) with y0(t) converges to the solution of (1).Proof.
From (1), we have
(11)y(t)=y(t)+∫0t‍(ξ-t)[y′′(ξ)-f(ξ,y(ξ),φ(α(ξ)))]dξ,[y′′(ξ)-f(ξ,y(ξ),φ(α(ξ)))]0<t<t1;(12)y(t)=y(t)+∫0t1‍(ξ-t)[y′′(ξ)-f(ξ,y(ξ),φ(α(ξ)))]dξ+∫t1t‍(ξ-t)[y′′(ξ)-f(ξ,y(ξ),y(α(ξ)))]dξ,[y′′(ξ)-f(ξ,y(ξ),y(α(ξ)))]dξ,t>t1.
Let Ei(t)=yi(t)-y(t), i=0,1,…. If t⩽0, then Ei(t)=0,i=0,1,…. From (9) and (11), we have
(13)Em+1(t)=Em(t)+∫0t‍(ξ-t)[Em′′(ξ)-(f(ξ,ym(ξ),φ(α(ξ)))+∫0t1‍Em′′(ξ)-(ξ-t)-f(ξ,y(ξ),φ(α(ξ)))){Em′′}]dξ,+∫0t‍×ξ,(ξ,ym(ξ),φ(α(ξ)))0<t<t1.
From (10) and (12), we have
(14)Em+1(t)=Em(t)+∫0t1(ξ-t)[Em′′(ξ)-(f(ξ,ym(ξ),φ(α(ξ)))+∫0t1‍Em′′(ξ)-(ξ-t)-f(ξ,y(ξ),φ(α(ξ))))Em′′]dξ+∫t1t(ξ-t)[Em′′(ξ)-(f(ξ,ym(ξ),ym(α(ξ)))+∫0t1‍Em′′(ξ)-(ξ-t)-f(ξ,y(ξ),y(α(ξ))))Em′′]dξ,∫0t1‍(ξ-t)[Em′′(ξ)-(fdξ,d(f(ξ,ym(ξ),φ(α(ξ)))t>t1.
Using integration by parts, we have
(15)Em+1(t)=Em(t)+∫0t‍(ξ-t)Em′′(ξ)dξ-∫0t‍(ξ-t)[f(ξ,ym(ξ),φ(α(ξ)))-∫0t‍(ξ-t)-f(ξ,y(ξ),φ(α(ξ)))]dξ=-∫0t‍(ξ-t)[f(ξ,ym(ξ),φ(α(ξ)))-∫0t‍(ξ-t)-f(ξ,y(ξ),φ(α(ξ)))]dξ,+∫0t‍(ξ,ym(ξ),φ(α(ξ)))dξ,0<t<t1;Em+1(t)=Em(t)+∫0t‍(ξ-t)Em′′(ξ)dξ-∫0t1‍(ξ-t)[f(ξ,ym(ξ),φ(α(ξ)))-∫0t‍(ξ-t)-f(ξ,y(ξ),φ(α(ξ)))]dξ-∫t1t‍(ξ-t)[f(ξ,ym(ξ),ym(α(ξ)))-∫0t‍(ξ-t)-f(ξ,y(ξ),y(α(ξ)))]dξ=-∫0t1‍(ξ-t)[f(ξ,ym(ξ),φ(α(ξ)))-∫0t‍(ξ-t)-f(ξ,y(ξ),φ(α(ξ)))]dξ-∫t1t‍(ξ-t)[f(ξ,ym(ξ),ym(α(ξ)))-∫0t‍(ξ-t)-f(ξ,y(ξ),y(α(ξ)))]dξ,+∫0t‍(ξ,ym(ξ),ym(α(ξ)))(α(ξ))t>t1.
Since [α-1(t)]′ is bounded, M=max-τ≤ξ≤α(T)(α-1(ξ))′ is bounded. Moreover, it follows from (2) and the inequality |ξ-t|≤T that
(16)∥Em+1(t)∥≤∫0t‍|t-ξ|∥f(ξ,ym(ξ),φ(α(ξ)))∫0t‍|t-ξ|-f(ξ,y(ξ),φ(α(ξ)))∥dξ≤∫0t‍Tβ0∥ym(ξ)-y(ξ)∥dξ=∫0t‍Tβ0∥Em(ξ)∥dξ,0<t<t1;(17)∥Em+1(t)∥≤∫0t1‍|t-ξ|∥f(ξ,ym(ξ),φ(α(ξ)))∫0t‍|t-ξ|-f(ξ,y(ξ),φ(α(ξ)))∥dξ+∫t1t‍|t-ξ|∥f(ξ,ym(ξ),ym(α(ξ)))∫0t‍|t-ξ|-f(ξ,y(ξ),y(α(ξ)))∥dξ≤∫0t1‍Tβ0∥ym(ξ)-y(ξ)∥dξ+∫t1t‍T(β0∥Em(ξ)∥+β1∥Em(α(ξ))∥)dξ=∫0t‍Tβ0∥Em(ξ)∥dξ+∫t1t‍Tβ1∥Em(α(ξ))∥dξ=∫0t‍Tβ0∥Em(ξ)∥dξ+∫α(t1)α(t)‍Tβ1∥Em(ξ)∥(α-1(ξ))′dξ≤TMβ∫0t‍∥Em(ξ)∥dξ,t>t1,
where β=maxβi, i=1,2. Moreover,
(18)∥Em+1(t)∥≤(TMβ)2∫0t‍∫0s1‍∥Em-1(s2)∥ds2ds1≤(TMβ)3∫0t‍∫0s1‍∫0s2‍∥Em-2(s3)∥ds3ds2ds1≤(TMβ)4∫0t‍∫0s1‍∫0s2‍∫0s3‍∥Em-3(s4)∥ds4ds3ds2ds1⋯≤(TMβ)m+1∫0t‍∫0s1‍∫0s2‍⋯∫0sm‍∥E0(sm+1)∥dsm+1(TMβ)m+1∫0t‍∫0s1‍∫0s2‍⋯∫0sm‍⋯ds3ds2ds1,
where ∥E0(t)∥ is constant. Therefore, we have
(19)∥Em+1(t)∥≤∥E0(t)∥(TMβ)m+1(m+1)!→0,(m→∞).


## 2.2. The Second Kind of Second-Order Delay Differential Equations

Consider the initial value problems of second-order delay oscillation differential equations(20)y′′(t)=-ω2y(t)-f(t,y(t),y(α(t))),t∈[0,T],y′(t)=φ′(t),t∈[-τ,0],y(t)=φ(t),t∈[-τ,0],
where φ(t) is a differentiable function, α(t)∈C1[0,T] is a strictly monotone increasing function and satisfies that -τ≤α(t)≤t and α(0)=-τ, there exists t1∈[0,T] such that α(t1)=0, ω is a constant, and f:D=[0,T]×R×R→R is a given continuous mapping and satisfies the Lipschitz condition
(21)∥f(t,u1,v)-f(t,u2,v)∥≤κ0∥u1-u2∥,∥f(t,u,v1)-f(t,u,v2)∥≤κ1∥v1-v2∥,
where κ0,κ1 are Lipschitz constants.Now the VIM for (20) can read
(22)ym+1(t)=ym(t)+∫0t‍λ(t,ξ)[ym′′(ξ)+ω2ym(ξ)+∫0t‍λ(t,ξ)+f~(ξ,ym(ξ),φm(α(ξ))){ym′′}]dξ,f~(ξ,ym(ξ),φm(α(ξ)))+λ(t,ξ)0<t<t1;(23)ym+1(t)=ym(t)+∫0t1‍λ(t,ξ)[ym′′(ξ)+ω2ym(ξ)+∫0t‍λ(t,ξ)+f~(ξ,ym(ξ),φm(α(ξ))){ym′′}]dξ+∫t1t‍λ(t,ξ)[ym′′(ξ)+ω2ym(ξ)+∫0t‍λ(t,ξ)+f~(ξ,ym(ξ),ym(α(ξ))){ym′′}]dξ,(ξ,ym(ξ),ym(α(ξ)))dξ,+∫0t‍λ(t,ξ)t>t1,
where ym(t)=φ(t) for t∈[-τ,0]; f~ denotes the restrictive variation, that is, δf~=0. Thus, we have
(24)δym+1(t)=δym(t)+∫0t‍δλ(t,ξ)[ym′′(ξ)+ω2ym(ξ)+∫0t‍λ(t,ξ)+f~(ξ,ym(ξ),φm(α(ξ)))]dξ=δym(t)+∫0t‍δλ(t,ξ)[ym′′(ξ)+ω2ym(ξ)]dξ,dξ,ω2ym(ξ)[y′′m(ξ)+ω2ym(ξ)]dξ,0<t<t1.
Using integration by parts to (23), we have
(25)δym+1(t)=δym(t)+∫0t1‍δλ(t,ξ)[ym′′(ξ)+ω2ym(ξ)+∫0t‍λ(t,ξ)+f~(ξ,ym(ξ),φm(α(ξ)))]dξ+∫t1t‍δλ(t,ξ)[ym′′(ξ)+ω2ym(ξ)+∫0t‍λ(t,ξ)+f~(ξ,ym(ξ),ym(α(ξ)))]dξ=δym(t)+∫0t‍δλ(t,ξ)[ym′′(ξ)+ω2ym(ξ)]dξ=δym(t)-∂λ(t,ξ)∂ξδym(ξ)|ξ=t+λ(t,ξ)δym′(ξ)|ξ=t+∫0t‍[ω2λ(t,ξ)+∂2λ(t,ξ)∂ξ2]δym(ξ)dξ.
From the above formula, the stationary conditions are obtained as
(26)ω2λ(t,ξ)+∂2λ(t,ξ)∂ξ2=0,1-∂λ(t,ξ)∂ξ|ξ=t=0,λ(t,ξ)|ξ=t=0.
Moreover, the general Lagrange multiplier
(27)λ(t,ξ)=1ωsinω(ξ-t)
can be readily identified by (26). Thus, the variational iteration formula can be written as
(28)ym+1(t)=ym(t)+∫0t‍1ωsinω(ξ-t)[ym′′(ξ)+ω2ym(ξ)+∫0t‍1ωsinω(ξ-t)+f(ξ,ym(ξ),φ(α(ξ)))ym′′]dξ,(ξ,ym(ξ),ym(α(ξ)))(dξ,ξ,ym(ξ),ym(α(ξ)))0<t<t1;ym+1(t)=ym(t)+∫0t11ωsinω(ξ-t)[ym′′(ξ)+ω2ym(ξ)∫t1t1ωsinω(ξ-t)+f(ξ,ym(ξ),φ(α(ξ)))ym′′]dξ+∫t1t1ωsinω(ξ-t)[ym′′(ξ)+ω2ym(ξ)∫t1t1ωsinω(ξ-t)+f(ξ,ym(ξ),ym(α(ξ)))ym′′]dξ,(ξ,ym(ξ),ym(α(ξ)))(ym(α(ξ))ξ,ym(ξ),ym(α(ξ)))t>t1.Theorem 2.
Suppose that the initial value problems (20) satisfy the condition (21), and y(t),yi(t)∈C2[0,T], i=1,2,…. Then the sequence {ym(t)}m=1∞ defined by (28) with y0(t) converges to the solution of (20).Proof.
The proof process is similar that in Theorem1.


## 2.3. The Third Kind of Second-Order Delay Differential Equations

In order to improve the iteration speed, we modify the above iterative formulas and reconstruct the Lagrange multiplier. Consider the initial value problems of second-order delay differential equations(29)y′′(t)+a(t)y′(t)+b(t)y(t)+N(t,y(t),y(α(t)))=0,y′′(t)+a(t)y′(t)+b(t)y(t)+N(t,y(t)),yt∈[0,T],y′(t)=φ′(t),t∈[-τ,0],y(t)=φ(t),t∈[-τ,0],
where φ(t) is a differentiable function, α(t)∈C1[0,T] is a strictly monotone increasing function and satisfies that -τ≤α(t)≤t and α(0)=-τ, there exists t1∈[0,T] such that α(t1)=0, a(t),b(t) are bounded functions, and N:D=[0,T]×R×R→R is a given continuous mapping and satisfies the Lipschitz condition
(30)∥N(t,u1,v)-N(t,u2,v)∥≤γ0∥u1-u2∥,∥N(t,u,v1)-N(t,u,v2)∥≤γ1∥v1-v2∥,
where γ0, γ1 are Lipschitz constants.Now the VIM for (2.19) can read(31)ym+1(t)=ym(t)+∫0t‍λ(t,ξ)[ym′′(ξ)+a(ξ)ym′(ξ)+b(ξ)ym(ξ)+∫0t‍λ(t,ξ)+N~(ξ,ym(ξ),φ(α(ξ)))]dξ,×∥N(t,u,v1)-N(t,u,v2)∥≤γ10<t<t1;(32)ym+1(t)=ym(t)+∫0t1‍λ(t,ξ)[ym′′(ξ)+a(ξ)ym′(ξ)+b(ξ)ym(ξ)+∫0t‍λ(t,ξ)+N~(ξ,ym(ξ),φ(α(ξ)))]dξ+∫t1t‍λ(t,ξ)[ym′′(ξ)+a(ξ)ym′(ξ)+b(ξ)ym(ξ)+∫0t‍λ(t,ξ)+N~(ξ,ym,ym(α(ξ)))]dξ,(α(ξ))+N~(ξ,ym(ξ),φ(α(ξ)))]dξt>t1,
where ym(t)=φ(t) for t∈[-τ,0]; N~ denotes the restrictive variation, that is, δN~=0. Thus, we have
(33)δym+1(t)=δym(t)+∫0t‍δλ(t,ξ)[ym′′(ξ)+a(ξ)ym′(ξ)+b(ξ)ym(ξ)+∫0t‍λ(t,ξ)+N~(ξ,ym(ξ),φ(α(ξ)))ym′′]dξ=δym(t)+∫0t‍δλ(t,ξ)[ym′′(ξ)+a(ξ)ym′(ξ)+b(ξ)ym(ξ)]dξ,(,)[y′′m(ξ)+ay′m(ξ)dξ(ξ)+ξb(ξ)ym(ξ)]0<t<t1.
Using integration by parts to (32), we have
(34)δym+1(t)=δym(t)+∫0t1‍δλ(t,ξ)[ym′′(ξ)+a(ξ)ym′(ξ)+b(ξ)ym(ξ)+∫0t1‍δλ(t,ξ)+N~(ξ,ym(ξ),φ(α(ξ)))]dξ+∫t1t‍δλ(t,ξ)[ym′′(ξ)+a(ξ)ym′(ξ)+b(ξ)ym(ξ)+∫0t‍δλ(t,ξ)+N~(ξ,ym(ξ),ym(α(ξ)))]dξ=δym(t)+∫0t‍δλ(t,ξ)[ym′′(ξ)+a(ξ)ym′(ξ)+b(ξ)ym(ξ)]dξ=δym(t)+∫0t‍λ(t,ξ)δdym′(ξ)+∫0t‍λ(t,ξ)a(ξ)δdym(ξ)+∫0t‍δλ(t,ξ)b(ξ)ym(ξ)dξ=(1-∂λ(t,ξ)∂ξ+a(ξ)λ(ξ))δym(ξ)|ξ=t+λ(t,ξ)δym′(ξ)|ξ=t+∫0t‍[∂2λ(t,ξ)∂2(ξ)-∂(a(ξ)λ(t,ξ))∂ξ+b(ξ)λ(t,ξ)]×δym(ξ)dξ.
From the above formula, the stationary conditions are obtained as
(35)∂2λ(t,ξ)∂2ξ-∂(a(ξ)λ(t,ξ))∂ξ+b(ξ)λ(t,ξ)=0,1-∂λ(t,ξ)∂ξ+a(ξ)λ(t,ξ)|ξ=t=0,λ(t,ξ)|ξ=t=0.
We suppose that λ1(t,ξ), λ2(t,ξ) are the fundamental solutions of (35); the corresponding general solution of (35) is
(36)λ(t,ξ)=c1λ1(t,ξ)+c2λ2(t,ξ).
Using the initial conditions of (35), we have
(37)c1λ1(t,t)+c2λ2(t,t)=0,c1λ1′(t,t)+c2λ2′(t,t)=1.
Note that W(t,t)=|λ1(t,t)λ2(t,t)λ1′(t,t)λ2′(t,t)| is the Wronski determinant of λ1(t,ξ), λ2(t,ξ). We have
(38)λ(t,ξ)=-λ2(t,t)λ1(t,ξ)+λ1(t,t)λ2(t,ξ)W(t,t).
Using the Liouville formula, we have
(39)W(t,t)=W(t,0)e∫0t‍a(ξ)dξ.
So λ(t,ξ) can be expressed as
(40)λ(t,ξ)=-λ2(t,t)λ1(t,ξ)+λ1(t,t)λ2(t,ξ)λ1(t,0)λ2′(t,0)-λ1′(t,0)λ2(t,0)e-∫0t‍a(ξ)dξ.
Note that
(41)u1(ξ)=λ1(t,ξ)W(t,0),u2(ξ)=λ2(t,ξ)W(t,0).
Equation (40) can be expressed as
(42)λ(t,ξ)=-e-∫0t‍a(ξ)dξ[u1(ξ)u2(t)-u2(ξ)u1(t)].
Substituting (42) into (31) and (32), we obtain
(43)ym+1(t)=ym(t)+∫0t‍e-∫0t‍a(ξ)dξ[u1(ξ)u2(t)-u2(ξ)u1(t)]+∫0t‍×[ym′′(ξ)+a(ξ)ym′(ξ)+b(ξ)ym(ξ)+∫0t‍×+N(ξ,ym(ξ),φ(α(ξ)))ym′′]dξ,dξ,×N(ξ,ym(ξ),φ(α(ξ)))0<t<t1;(44)ym+1(t)=ym(t)+∫0t1‍e-∫0t‍a(ξ)dξ[u1(ξ)u2(t)-u2(ξ)u1(t)]+∫0t1‍×[ym′′(ξ)+a(ξ)ym′(ξ)+b(ξ)ym(ξ)+∫0t1‍+N(ξ,ym(ξ),φ(α(ξ)))ym′′]dξ+∫t1t‍e-∫0t‍a(ξ)dξ[u1(ξ)u2(t)-u2(ξ)u1(t)]+∫0t1‍×[ym′′(ξ)+a(ξ)ym′(ξ)+b(ξ)ym(ξ)+∫0t1‍+N(ξ,ym,ym(α(ξ)))ym′′]dξ,t>t1.Theorem 3.
Suppose that the initial value problems (2.19) satisfy the condition (30), and y(t),yi(t)∈C2[0,T], i=1,2,…. Then the sequence {ym(t)}m=1∞ defined by (43) and (44) with y0(t) converges to the solution of (2.19).Proof.
The proof process is similar to that in Theorem1.


## 3. Illustrative Examples

In this section, some illustrative examples are given to show the efficiency of the VIM for solving second-order delay differential equations.Example 4.
Consider the initial value problem of second-order differential equation with pantograph delay(45)y′′(t)=-y(t2)-y2(t)+sin4(t)+sin2(t2)+8,t>0,φ′(0)=0,φ(0)=2,
with the exact solution y(t)=(5-cos2t)/2. Using the VIM given in formulas (9) and (10), we construct the correction functional
(46)ym+1(t)=ym(t)+∫0t‍(ξ-t)(ym′′(ξ)+ym(ξ2)+ym2(ξ)-sin4(ξ)+∫0t‍(ξ-t)-sin2(ξ)-8(ξ2))dξ,m=1,2,….
We take y0(t)=2 as the initial approximation and obtain that
(47)y1(t)=54+2316t2+12cost+516cos2t-116cos4t,y2(t)=-989512+815512t2+231536t4+12tsin(12t)+116tsinty2(t)=-1256tsintcost+3cos(12t)+1116costy2(t)=+157512cos2t-116cos4t,y3(t)=-2525412048+97814096t2+81549125t4+23491520t6y2(t)=+14tsin(14t)+1516tsin(12t)+1212048tsinty2(t)=+1256tsintcost+120cos(14t)+398cos(12t)y2(t)=+13932048cost+157512cos2t116cos4t+12048t2costy2(t)=-132t2cos(12t)-12t2cos(14t),⋮
The exact and approximate solutions are plotted in Figure 1, which shows that the method gives a very good approximation to the exact solution.Figure 1
Results for Example4.Example 5.
Consider the second-order delay differential equation(48)y′′(t)=-16y(t)+y2(t4)-sin2t,t>0,φ′(0)=4,φ(0)=0.
Using the VIM given in formulas (28), we construct the correction functional
(49)ym+1(t)=ym(t)+∫0t‍14sin(4ξ-4t)(ym′′(ξ)+16ym(ξ)-ym2(ξ4)+∫0t‍14sin(4ξ-4t)+sin2ξ(ξ4))dξ,m=1,2,….
We take y0(t)=4t as the initial approximation, and obtain that
(50)y1(t)=-0.0390625+0.0625t2+sin(4t)y1(t)=+0.04166666667cos(2t)y1(t)=-0.002604166667cos(4t),y2(t)=0.00613912861-3.998216869t+0.1805413564t2y1(t)=-0.4340277778t3+o(t4)y1(t)=+(0.006666666667-0.5208333333t+o(t2))sinty1(t)=+(-0.1946373457-0.5555555556t+o(t2))costy1(t)=-0.1085069444sin(2t)-2cos4t,⋮Example 6.
Consider the second-order delay differential equation(51)y′′(t)=-2ty′(t)+16y2(t2)+6-t4,t>0,φ′(0)=0,φ(0)=0,
with the exact solution y(t)=t2. From (35), we can solve that λ(t,ξ)=-ξ+ξ2/t. Using the VIM given in formulas (43) and (44), we construct the correction functional
(52)ym+1(t)=ym(t)+∫0t‍(-ξ+ξ2t)(ym′′(ξ)+2ξym′(ξ)-16ym2(ξ2)+∫0t‍(-ξ+ξ2t)-6+ξ4(ξ2))dξ,m=1,2,….
We take y0(t)=2t as the initial approximation and obtain that
(53)y1(t)=-142t6+t2,y2(t)=t2+421t6-11760t10+11935360t14,⋮
We use the iterative formulas (9) and (43) for Example 6, respectively. When the iteration number n=2, the corresponding relative errors are showed in Table 1.
Table1 shows that the iteration speed of the iterative formula (43) for Example 6 is much faster than that of iterative formula (9). This demonstrates that it is important to choose a reasonable Lagrange multiplier.Table 1
The errors of the iteration solutions.
t
=
0.01
t
=
0.05
t
=
0.1
t
=
0.15
Iterative formula (43)
1.9048
E
-
09
1.1905
E
-
06
1.9048
E
-
05
906428
E
-
05
Iterative formula (9)
1.89278
E
-
05
1.0972
E
-
03
9.2763
E
-
03
3.8062
E
-
02


## 4. Conclusion

In this paper, we apply the VIM to obtain the analytical or approximate analytical solutions of second-order delay differential equations. Some illustrative examples show that this method gives a very good approximation to the exact solution. The VIM is a promising method for second-order delay differential equations.



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*Source: 634670-2013-02-10.xml*