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

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.

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 include 𝜃-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 𝑦^′′(𝑡)(𝑡)(𝛼(𝑡[],𝑦=𝑓𝑡,𝑦,𝑦)),𝑡∈0,𝑇_′(𝑡)=𝜑^′(𝑡)[],𝑦(𝑡)(𝑡)[],,𝑡∈−𝜏,0=𝜑,𝑡∈−𝜏,0 (1) 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 𝑓𝑡,𝑢₁,𝑣−𝑓𝑡,𝑢₂,𝑣≤𝛽₀𝑢₁−𝑢₂,𝑓𝑡,𝑢,𝑣₁−𝑓𝑡,𝑢,𝑣₂≤𝛽₁𝑣₁−𝑣₂, (2) where 𝛽₀,𝛽₁ are Lipschitz constants; ‖⋅‖ denotes the standard Euclidean norm.

Now the VIM for (1) can read 𝑦_𝑚+1(𝑡)=𝑦_𝑚(𝑡)+^𝑡₀𝜆(𝑡,𝜉)𝑦_𝑚^′′(𝜉)−𝑓𝜉,𝑦_𝑚(𝜉),𝜑(𝛼(𝜉))𝑑𝜉,0<𝑡<𝑡₁;𝑦_𝑚+1(𝑡)=𝑦_𝑚(𝑡)+^𝑡1₀𝜆(𝑡,𝜉)𝑦_𝑚^′′(𝜉)−𝑓𝜉,𝑦_𝑚(𝜉)(𝛼(𝜉+,𝜑))𝑑𝜉_𝑡𝑡1𝜆(𝑡,𝜉)𝑦_𝑚^′′(𝜉)−𝑓𝜉,𝑦_𝑚(𝜉),𝑦_𝑚(𝛼(𝜉))𝑑𝜉,𝑡>𝑡₁, (3)(4) where 𝑦_𝑚(𝑡)=𝜑(𝑡) for 𝑡∈[−𝜏,0]; 𝑓 denotes the restrictive variation, that is, 𝛿𝑓=0. Thus, we have 𝛿𝑦_𝑚+1(𝑡)=𝛿𝑦_𝑚(𝑡)+^𝑡₀𝛿𝜆(𝑡,𝜉)𝑦_𝑚^′′(𝜉)−𝑓𝜉,𝑦_𝑚(𝜉),𝜑(𝛼(𝜉))𝑑𝜉=𝛿𝑦_𝑚(𝑡)+^𝑡₀𝛿𝜆(𝑡,𝜉)𝑦_𝑚^′′(𝜉)𝑑𝜉,0<𝑡<𝑡₁. (5) Using integration by parts to (4), we have 𝛿𝑦_𝑚+1(𝑡)=𝛿𝑦_𝑚(𝑡)+^𝑡1₀𝛿𝜆(𝑡,𝜉)𝑦_𝑚^′′(𝜉)−𝑓𝜉,𝑦_𝑚(𝜉)(𝛼(𝜉+,𝜑))𝑑𝜉_𝑡𝑡1𝛿𝜆(𝑡,𝜉)𝑦_𝑚^′′(𝜉)−𝑓𝜉,𝑦_𝑚(𝜉),𝑦_𝑚(𝛼(𝜉))𝑑𝜉=𝛿𝑦_𝑚(𝑡)+𝛿𝜆(𝑡,𝜉)𝑦^′_𝑚(𝜉)_𝜉=𝑡−𝜕𝜆(𝑡,𝜉)𝜕𝜉𝛿𝑦_𝑚(𝜉)_𝜉=𝑡+^𝑡₀𝜕²𝜆(𝑡,𝜉)𝜕𝜉²𝛿𝑦_𝑚(𝜉)𝑑𝜉. (6) From the above formula, the stationary conditions are obtained as 𝜕²𝜆(𝑡,𝜉)𝜕𝜉²()=0,1−𝜕𝜆𝑡,𝜉𝜕𝜉_𝜉=𝑡𝜆(=0,𝑡,𝜉)_𝜉=𝑡=0. (7) Moreover, the general Lagrange multiplier 𝜆(𝑡,𝜉)=𝜉−𝑡 (8) can be readily identified by (7). Thus, the variational iteration formula can be written as 𝑦_𝑚+1(𝑡)=𝑦_𝑚(𝑡)+^𝑡₀(𝜉−𝑡)𝑦_𝑚^′′(𝜉)−𝑓𝜉,𝑦_𝑚(𝜉),𝜑(𝛼(𝜉))𝑑𝜉,0<𝑡<𝑡₁;𝑦_𝑚+1(𝑡)=𝑦_𝑚(𝑡)+^𝑡1₀(𝜉−𝑡)𝑦_𝑚^′′(𝜉)−𝑓𝜉,𝑦_𝑚(𝜉)(𝛼(𝜉+,𝜑))𝑑𝜉_𝑡𝑡1(𝜉−𝑡)𝑦_𝑚^′′(𝜉)−𝑓𝜉,𝑦_𝑚(𝜉),𝑦_𝑚(𝛼(𝜉))𝑑𝜉,𝑡>𝑡₁. (9)(10)

Theorem 1. Suppose that the initial value problems (1) satisfy the condition (2), and 𝑦(𝑡),𝑦_𝑖(𝑡)∈𝐶²[0,𝑇], 𝑖=1,2,…. Then the sequence {𝑦_𝑚(𝑡)}^∞_𝑚=1 defined by (9) and (10) with 𝑦₀(𝑡) converges to the solution of (1).

Proof. From (1), we have 𝑦(𝑡)=𝑦(𝑡)+^𝑡₀(𝜉−𝑡)𝑦^′′(𝜉)−𝑓𝜉,𝑦(𝜉),𝜑(𝛼(𝜉))𝑑𝜉,0<𝑡<𝑡₁;𝑦(𝑡)=𝑦(𝑡)+^𝑡1₀(𝜉−𝑡)𝑦^′′(𝜉)(𝜉)(𝛼(𝜉+−𝑓𝜉,𝑦,𝜑))𝑑𝜉_𝑡𝑡1(𝜉−𝑡)𝑦^′′(𝜉)−𝑓𝜉,𝑦(𝜉),𝑦(𝛼(𝜉))𝑑𝜉,𝑡>𝑡₁. (11)(12) Let 𝐸_𝑖(𝑡)=𝑦_𝑖(𝑡)−𝑦(𝑡), 𝑖=0,1,…. If 𝑡⩽0, then 𝐸_𝑖(𝑡)=0,𝑖=0,1,…. From (9) and (11), we have 𝐸_𝑚+1(𝑡)=𝐸_𝑚(𝑡)+^𝑡₀(𝜉−𝑡)𝐸_𝑚^′′(𝜉)−𝑓𝜉,𝑦_𝑚(𝜉)(𝛼(𝜉(𝜉)(𝛼(𝜉,𝜑))−𝑓𝜉,𝑦,𝜑))𝑑𝜉,0<𝑡<𝑡₁. (13) From (10) and (12), we have 𝐸_𝑚+1(𝑡)=𝐸_𝑚(𝑡)+^𝑡1₀(𝜉−𝑡)𝐸_𝑚^′′(𝜉)−𝑓𝜉,𝑦_𝑚(𝜉)(𝛼(𝜉(𝜉)(𝛼(𝜉+,𝜑))−𝑓𝜉,𝑦,𝜑))𝑑𝜉_𝑡𝑡1(𝜉−𝑡)𝐸_𝑚^′′(𝜉)−𝑓𝜉,𝑦_𝑚(𝜉),𝑦_𝑚(𝛼(𝜉(𝜉)(𝛼(𝜉))−𝑓𝜉,𝑦,𝑦))𝑑𝜉,𝑡>𝑡₁. (14) Using integration by parts, we have 𝐸_𝑚+1(𝑡)=𝐸_𝑚(𝑡)+^𝑡₀(𝜉−𝑡)𝐸_𝑚^′′(𝜉)−𝑑𝜉_𝑡0(𝜉−𝑡)𝑓𝜉,𝑦_𝑚(𝜉)(𝛼(𝜉(𝜉)(𝛼(𝜉,𝜑))−𝑓𝜉,𝑦,𝜑))𝑑𝜉=−^𝑡₀(𝜉−𝑡)𝑓𝜉,𝑦_𝑚(𝜉)(𝛼(𝜉(𝜉)(𝛼(𝜉,𝜑))−𝑓𝜉,𝑦,𝜑))𝑑𝜉,0<𝑡<𝑡₁;𝐸_𝑚+1(𝑡)=𝐸_𝑚(𝑡)+^𝑡₀(𝜉−𝑡)𝐸_𝑚^′′(𝜉)−𝑑𝜉_𝑡10(𝜉−𝑡)𝑓𝜉,𝑦_𝑚(𝜉)(𝛼(𝜉(𝜉)(𝛼(𝜉,𝜑))−𝑓𝜉,𝑦,𝜑))𝑑𝜉−^𝑡_𝑡1(𝜉−𝑡)𝑓𝜉,𝑦_𝑚(𝜉),𝑦_𝑚(𝛼(𝜉(𝜉)(𝛼(𝜉))−𝑓𝜉,𝑦,𝑦))𝑑𝜉=−^𝑡1₀(𝜉−𝑡)𝑓𝜉,𝑦_𝑚(𝜉)(𝛼(𝜉(𝜉)(𝛼(𝜉−,𝜑))−𝑓𝜉,𝑦,𝜑))𝑑𝜉_𝑡𝑡1(𝜉−𝑡)𝑓𝜉,𝑦_𝑚(𝜉),𝑦_𝑚(𝛼(𝜉(𝜉)(𝛼(𝜉))−𝑓𝜉,𝑦,𝑦))𝑑𝜉,𝑡>𝑡₁. (15) 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 𝐸_𝑚+1(𝑡)≤^𝑡₀𝑡−𝜉𝑓𝜉,𝑦_𝑚(𝜉)(𝛼(𝜉(𝜉)(𝛼(𝜉≤,𝜑))−𝑓𝜉,𝑦,𝜑))𝑑𝜉_𝑡0𝑇𝛽₀𝑦_𝑚(𝜉)(𝜉=−𝑦)𝑑𝜉_𝑡0𝑇𝛽₀𝐸_𝑚(𝜉)𝑑𝜉,0<𝑡<𝑡₁;𝐸_𝑚+1(𝑡)≤^𝑡1₀𝑡−𝜉𝑓𝜉,𝑦_𝑚(𝜉)(𝛼(𝜉(𝜉)(𝛼(𝜉+,𝜑))−𝑓𝜉,𝑦,𝜑))𝑑𝜉_𝑡𝑡1𝑡−𝜉𝑓𝜉,𝑦_𝑚(𝜉),𝑦_𝑚(𝛼(𝜉(𝜉)(𝛼(𝜉))−𝑓𝜉,𝑦,𝑦))𝑑𝜉≤^𝑡1₀𝑇𝛽₀𝑦_𝑚(𝜉)(𝜉+−𝑦)𝑑𝜉_𝑡𝑡1𝑇𝛽₀𝐸_𝑚(𝜉)+𝛽₁𝐸_𝑚(𝛼(𝜉=))𝑑𝜉_𝑡0𝑇𝛽₀𝐸_𝑚(𝜉)𝑑𝜉+^𝑡_𝑡1𝑇𝛽₁𝐸_𝑚(𝛼(𝜉))𝑑𝜉=^𝑡₀𝑇𝛽₀𝐸_𝑚(𝜉+)𝑑𝜉_{𝛼𝑡𝛼(𝑡)1}𝑇𝛽₁𝐸_𝑚(𝜉)𝛼⁻¹(𝜉)^′𝑑𝜉≤𝑇𝑀𝛽^𝑡₀𝐸_𝑚(𝜉)𝑑𝜉,𝑡>𝑡₁, (16)(17) where 𝛽=max𝛽_𝑖, 𝑖=1,2. Moreover, 𝐸_𝑚+1(𝑡)≤𝑇𝑀𝛽^2𝑡₀^𝑠1₀𝐸_𝑚−1𝑠₂𝑑𝑠₂𝑑𝑠₁≤𝑇𝑀𝛽^3𝑡₀^𝑠1₀^𝑠2₀𝐸_𝑚−2𝑠₃𝑑𝑠₃𝑑𝑠₂𝑑𝑠₁≤𝑇𝑀𝛽^4𝑡₀^𝑠1₀^𝑠2₀^𝑠3₀𝐸_𝑚−3𝑠₄𝑑𝑠₄𝑑𝑠₃𝑑𝑠₂𝑑𝑠₁⋯≤𝑇𝑀𝛽^𝑚+1𝑡₀^𝑠1₀^𝑠2₀⋯^𝑠𝑚₀𝐸₀𝑠_𝑚+1𝑑𝑠_𝑚+1⋯𝑑𝑠₃𝑑𝑠₂𝑑𝑠₁, (18) where ‖𝐸₀(𝑡)‖ is constant. Therefore, we have 𝐸_𝑚+1(𝑡)≤𝐸₀(𝑡)𝑇𝑀𝛽^𝑚+1()!().𝑚+1→0,𝑚→∞ (19)

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

Consider the initial value problems of second-order delay oscillation differential equations 𝑦^′′(𝑡)=−𝜔²𝑦(𝑡)(𝑡)(𝛼(𝑡[],𝑦−𝑓𝑡,𝑦,𝑦)),𝑡∈0,𝑇_′(𝑡)=𝜑^′(𝑡)[],𝑦(𝑡)(𝑡)[],,𝑡∈−𝜏,0=𝜑,𝑡∈−𝜏,0 (20) 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 𝑓𝑡,𝑢₁,𝑣−𝑓𝑡,𝑢₂,𝑣≤𝜅₀𝑢₁−𝑢₂,𝑓𝑡,𝑢,𝑣₁−𝑓𝑡,𝑢,𝑣₂≤𝜅₁𝑣₁−𝑣₂, (21) where 𝜅₀,𝜅₁ are Lipschitz constants.

Now the VIM for (20) can read 𝑦_𝑚+1(𝑡)=𝑦_𝑚(𝑡)+^𝑡₀𝜆(𝑡,𝜉)𝑦_𝑚^′′(𝜉)+𝜔²𝑦_𝑚(𝜉)+𝑓𝜉,𝑦_𝑚(𝜉),𝜑_𝑚(𝛼(𝜉))𝑑𝜉,0<𝑡<𝑡₁;𝑦_𝑚+1(𝑡)=𝑦_𝑚(𝑡)+^𝑡1₀𝜆(𝑡,𝜉)𝑦_𝑚^′′(𝜉)+𝜔²𝑦_𝑚(𝜉)+𝑓𝜉,𝑦_𝑚(𝜉),𝜑_𝑚(𝛼(𝜉+))𝑑𝜉_𝑡𝑡1𝜆(𝑡,𝜉)𝑦_𝑚^′′(𝜉)+𝜔²𝑦_𝑚(𝜉)+𝑓𝜉,𝑦_𝑚(𝜉),𝑦_𝑚(𝛼(𝜉))𝑑𝜉,𝑡>𝑡₁, (22)(23) where 𝑦_𝑚(𝑡)=𝜑(𝑡) for 𝑡∈[−𝜏,0]; 𝑓 denotes the restrictive variation, that is, 𝛿𝑓=0. Thus, we have 𝛿𝑦_𝑚+1(𝑡)=𝛿𝑦_𝑚(𝑡)+^𝑡₀𝛿𝜆(𝑡,𝜉)𝑦_𝑚^′′(𝜉)+𝜔²𝑦_𝑚(𝜉)+𝑓𝜉,𝑦_𝑚(𝜉),𝜑_𝑚(𝛼(𝜉))𝑑𝜉=𝛿𝑦_𝑚(𝑡)+^𝑡₀𝛿𝜆(𝑡,𝜉)𝑦_𝑚^′′(𝜉)+𝜔²𝑦_𝑚(𝜉)𝑑𝜉,0<𝑡<𝑡₁. (24) Using integration by parts to (23), we have 𝛿𝑦_𝑚+1(𝑡)=𝛿𝑦_𝑚(𝑡)+^𝑡1₀𝛿𝜆(𝑡,𝜉)𝑦_𝑚^′′(𝜉)+𝜔²𝑦_𝑚(𝜉)+𝑓𝜉,𝑦_𝑚(𝜉),𝜑_𝑚(𝛼(𝜉+))𝑑𝜉_𝑡𝑡1𝛿𝜆(𝑡,𝜉)𝑦_𝑚^′′(𝜉)+𝜔²𝑦_𝑚(𝜉)+𝑓𝜉,𝑦_𝑚(𝜉),𝑦_𝑚(𝛼(𝜉))𝑑𝜉=𝛿𝑦_𝑚(𝑡)+^𝑡₀𝛿𝜆(𝑡,𝜉)𝑦_𝑚^′′(𝜉)+𝜔²𝑦_𝑚(𝜉)𝑑𝜉=𝛿𝑦_𝑚(𝑡)−𝜕𝜆(𝑡,𝜉)𝜕𝜉𝛿𝑦_𝑚(𝜉)_𝜉=𝑡+𝜆(𝑡,𝜉)𝛿𝑦^′_𝑚(𝜉)_𝜉=𝑡+^𝑡₀𝜔²𝜆()+𝜕𝑡,𝜉²𝜆(𝑡,𝜉)𝜕𝜉²𝛿𝑦_𝑚(𝜉)𝑑𝜉. (25) From the above formula, the stationary conditions are obtained as 𝜔²𝜆()+𝜕𝑡,𝜉²𝜆(𝑡,𝜉)𝜕𝜉²()=0,1−𝜕𝜆𝑡,𝜉𝜕𝜉_𝜉=𝑡=0,𝜆(𝑡,𝜉)_𝜉=𝑡=0. (26) Moreover, the general Lagrange multiplier 𝜆()=1𝑡,𝜉𝜔sin𝜔(𝜉−𝑡) (27) can be readily identified by (26). Thus, the variational iteration formula can be written as 𝑦_𝑚+1(𝑡)=𝑦_𝑚(𝑡)+^𝑡₀1𝜔sin𝜔(𝜉−𝑡)𝑦_𝑚^′′(𝜉)+𝜔²𝑦_𝑚(𝜉)+𝑓𝜉,𝑦_𝑚(𝜉),𝜑(𝛼(𝜉))𝑑𝜉,0<𝑡<𝑡₁;𝑦_𝑚+1(𝑡)=𝑦_𝑚(𝑡)+^𝑡1₀1𝜔sin𝜔(𝜉−𝑡)𝑦_𝑚^′′(𝜉)+𝜔²𝑦_𝑚(𝜉)+𝑓𝜉,𝑦_𝑚(𝜉)(𝛼(𝜉+,𝜑))𝑑𝜉_𝑡𝑡11𝜔sin𝜔(𝜉−𝑡)𝑦_𝑚^′′(𝜉)+𝜔²𝑦_𝑚(𝜉)+𝑓𝜉,𝑦_𝑚(𝜉),𝑦_𝑚(𝛼(𝜉))𝑑𝜉,𝑡>𝑡₁. (28)

Theorem 2. Suppose that the initial value problems (20) satisfy the condition (21), and 𝑦(𝑡),𝑦_𝑖(𝑡)∈𝐶²[0,𝑇], 𝑖=1,2,…. Then the sequence {𝑦_𝑚(𝑡)}^∞_𝑚=1 defined by (28) with 𝑦₀(𝑡) converges to the solution of (20).

Proof. The proof process is similar that in Theorem 1.

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 𝑦^′′(𝑡)+𝑎(𝑡)𝑦^′(𝑡)(𝑡)𝑦(𝑡)(𝑡)(𝛼(𝑡[],𝑦+𝑏+𝑁𝑡,𝑦,𝑦))=0,𝑡∈0,𝑇_′(𝑡)=𝜑^′(𝑡)[],𝑦(𝑡)(𝑡)[],,𝑡∈−𝜏,0=𝜑,𝑡∈−𝜏,0 (29) 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 𝑁𝑡,𝑢₁,𝑣−𝑁𝑡,𝑢₂,𝑣≤𝛾₀𝑢₁−𝑢₂,𝑁𝑡,𝑢,𝑣₁−𝑁𝑡,𝑢,𝑣₂≤𝛾₁𝑣₁−𝑣₂, (30) where 𝛾₀, 𝛾₁ are Lipschitz constants.

Now the VIM for (2.19) can read 𝑦_𝑚+1(𝑡)=𝑦_𝑚(𝑡)+^𝑡₀𝜆(𝑡,𝜉)𝑦_𝑚^′′(𝜉)+𝑎(𝜉)𝑦^′_𝑚(𝜉)+𝑏(𝜉)𝑦_𝑚(𝜉)+𝑁𝜉,𝑦_𝑚(𝜉),𝜑(𝛼(𝜉))𝑑𝜉,0<𝑡<𝑡₁;𝑦_𝑚+1(𝑡)=𝑦_𝑚(𝑡)+^𝑡1₀𝜆(𝑡,𝜉)𝑦_𝑚^′′(𝜉)+𝑎(𝜉)𝑦^′_𝑚(𝜉)+𝑏(𝜉)𝑦_𝑚(𝜉)+𝑁𝜉,𝑦_𝑚(𝜉)(𝛼(𝜉+,𝜑))𝑑𝜉_𝑡𝑡1𝜆(𝑡,𝜉)𝑦_𝑚^′′(𝜉)+𝑎(𝜉)𝑦^′_𝑚(𝜉)+𝑏(𝜉)𝑦_𝑚(𝜉)+𝑁𝜉,𝑦_𝑚,𝑦_𝑚(𝛼(𝜉))𝑑𝜉,𝑡>𝑡₁, (31)(32) where 𝑦_𝑚(𝑡)=𝜑(𝑡) for 𝑡∈[−𝜏,0]; 𝑁 denotes the restrictive variation, that is, 𝛿𝑁=0. Thus, we have 𝛿𝑦_𝑚+1(𝑡)=𝛿𝑦_𝑚(𝑡)+^𝑡₀𝛿𝜆(𝑡,𝜉)𝑦_𝑚^′′(𝜉)+𝑎(𝜉)𝑦^′_𝑚(𝜉)+𝑏(𝜉)𝑦_𝑚(𝜉)+𝑁𝜉,𝑦_𝑚(𝜉),𝜑(𝛼(𝜉))𝑑𝜉=𝛿𝑦_𝑚(𝑡)+^𝑡₀𝛿𝜆(𝑡,𝜉)𝑦_𝑚^′′(𝜉)+𝑎(𝜉)𝑦^′_𝑚(𝜉)+𝑏(𝜉)𝑦_𝑚(𝜉)𝑑𝜉,0<𝑡<𝑡₁. (33) Using integration by parts to (32), we have 𝛿𝑦_𝑚+1(𝑡)=𝛿𝑦_𝑚(𝑡)+^𝑡1₀𝛿𝜆(𝑡,𝜉)𝑦_𝑚^′′(𝜉)+𝑎(𝜉)𝑦^′_𝑚(𝜉)+𝑏(𝜉)𝑦_𝑚(𝜉)+𝑁𝜉,𝑦_𝑚(𝜉)(𝛼(𝜉+,𝜑))𝑑𝜉_𝑡𝑡1𝛿𝜆(𝑡,𝜉)𝑦_𝑚^′′(𝜉)+𝑎(𝜉)𝑦^′_𝑚(𝜉)+𝑏(𝜉)𝑦_𝑚(𝜉)+𝑁𝜉,𝑦_𝑚(𝜉),𝑦_𝑚(𝛼(𝜉))𝑑𝜉=𝛿𝑦_𝑚(𝑡)+^𝑡₀𝛿𝜆(𝑡,𝜉)𝑦_𝑚^′′(𝜉)+𝑎(𝜉)𝑦^′_𝑚(𝜉)+𝑏(𝜉)𝑦_𝑚(𝜉)𝑑𝜉=𝛿𝑦_𝑚(𝑡)+^𝑡₀𝜆(𝑡,𝜉)𝛿𝑑𝑦^′_𝑚(𝜉)+^𝑡₀𝜆(𝑡,𝜉)𝑎(𝜉)𝛿𝑑𝑦_𝑚(𝜉)+^𝑡₀𝛿𝜆(𝑡,𝜉)𝑏(𝜉)𝑦_𝑚(𝜉)=()𝑑𝜉1−𝜕𝜆𝑡,𝜉(𝜉)𝜆(𝜉)𝜕𝜉+𝑎𝛿𝑦_𝑚(𝜉)_𝜉=𝑡+𝜆(𝑡,𝜉)𝛿𝑦^′_𝑚(𝜉)_𝜉=𝑡+^𝑡₀𝜕²𝜆(𝑡,𝜉)𝜕²(𝜉)−𝜕(𝑎(𝜉)𝜆(𝑡,𝜉))(𝜉)𝜆(𝜕𝜉+𝑏𝑡,𝜉)×𝛿𝑦_𝑚(𝜉)𝑑𝜉. (34) From the above formula, the stationary conditions are obtained as 𝜕²𝜆(𝑡,𝜉)𝜕²𝜉−𝜕(𝑎(𝜉)𝜆(𝑡,𝜉))(𝜉)𝜆()()𝜕𝜉+𝑏𝑡,𝜉=0,1−𝜕𝜆𝑡,𝜉(𝜉)𝜆(𝜕𝜉+𝑎𝑡,𝜉)_𝜉=𝑡𝜆(=0,𝑡,𝜉)_𝜉=𝑡=0. (35) We suppose that 𝜆₁(𝑡,𝜉), 𝜆₂(𝑡,𝜉) are the fundamental solutions of (35); the corresponding general solution of (35) is 𝜆(𝑡,𝜉)=𝑐₁𝜆₁(𝑡,𝜉)+𝑐₂𝜆₂(𝑡,𝜉). (36) Using the initial conditions of (35), we have 𝑐₁𝜆₁(𝑡,𝑡)+𝑐₂𝜆₂()𝑐𝑡,𝑡=0,₁𝜆^′₁(𝑡,𝑡)+𝑐₂𝜆^′₂(𝑡,𝑡)=1. (37) Note that 𝑊(𝑡,𝑡)=^{𝜆1(𝑡,𝑡)𝜆2}_𝜆^(𝑡,𝑡)_{′1(𝑡,𝑡)𝜆′2(𝑡,𝑡)} is the Wronski determinant of 𝜆₁(𝑡,𝜉), 𝜆₂(𝑡,𝜉). We have 𝜆(𝑡,𝜉)=−𝜆₂(𝑡,𝑡)𝜆₁(𝑡,𝜉)+𝜆₁(𝑡,𝑡)𝜆₂(𝑡,𝜉)𝑊().𝑡,𝑡 (38) Using the Liouville formula, we have 𝑊(𝑡,𝑡)=𝑊(𝑡,0)𝑒^{∫𝑡0𝑎(𝜉)𝑑𝜉}. (39) So 𝜆(𝑡,𝜉) can be expressed as 𝜆(𝑡,𝜉)=−𝜆₂(𝑡,𝑡)𝜆₁(𝑡,𝜉)+𝜆₁(𝑡,𝑡)𝜆₂(𝑡,𝜉)𝜆₁(𝑡,0)𝜆^′₂(𝑡,0)−𝜆^′₁(𝑡,0)𝜆₂()𝑒𝑡,0^{−∫𝑡0𝑎(𝜉)𝑑𝜉}. (40) Note that 𝑢₁(𝜉)=𝜆₁(𝑡,𝜉)𝑊(𝑡,0),𝑢₂(𝜉)=𝜆₂(𝑡,𝜉)𝑊().𝑡,0 (41) Equation (40) can be expressed as 𝜆(𝑡,𝜉)=−𝑒^{−∫𝑡0𝑎(𝜉)𝑑𝜉}𝑢₁(𝜉)𝑢₂(𝑡)−𝑢₂(𝜉)𝑢₁(𝑡). (42) Substituting (42) into (31) and (32), we obtain 𝑦_𝑚+1(𝑡)=𝑦_𝑚(𝑡)+^𝑡₀𝑒^{−∫𝑡0𝑎(𝜉)𝑑𝜉}𝑢₁(𝜉)𝑢₂(𝑡)−𝑢₂(𝜉)𝑢₁(𝑡×𝑦_𝑚′′)(𝜉)+𝑎(𝜉)𝑦^′_𝑚(𝜉)+𝑏(𝜉)𝑦_𝑚(𝜉)+𝑁𝜉,𝑦_𝑚(𝜉),𝜑(𝛼(𝜉))𝑑𝜉,0<𝑡<𝑡₁;𝑦_𝑚+1(𝑡)=𝑦_𝑚(𝑡)+^𝑡1₀𝑒^{−∫𝑡0𝑎(𝜉)𝑑𝜉}𝑢₁(𝜉)𝑢₂(𝑡)−𝑢₂(𝜉)𝑢₁(𝑡×𝑦_𝑚′′)(𝜉)+𝑎(𝜉)𝑦^′_𝑚(𝜉)+𝑏(𝜉)𝑦_𝑚(𝜉)+𝑁𝜉,𝑦_𝑚(𝜉)(𝛼(𝜉+,𝜑))𝑑𝜉_𝑡𝑡1𝑒^{−∫𝑡0𝑎(𝜉)𝑑𝜉}𝑢₁(𝜉)𝑢₂(𝑡)−𝑢₂(𝜉)𝑢₁(𝑡×𝑦_𝑚′′)(𝜉)+𝑎(𝜉)𝑦^′_𝑚(𝜉)+𝑏(𝜉)𝑦_𝑚(𝜉)+𝑁𝜉,𝑦_𝑚,𝑦_𝑚(𝛼(𝜉))𝑑𝜉,𝑡>𝑡₁. (43)(44)

Theorem 3. Suppose that the initial value problems (2.19) satisfy the condition (30), and 𝑦(𝑡),𝑦_𝑖(𝑡)∈𝐶²[0,𝑇], 𝑖=1,2,…. Then the sequence {𝑦_𝑚(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 Theorem 1.

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 𝑦^′′(𝑡)𝑡=−𝑦2−𝑦²(𝑡)+sin⁴(𝑡)+sin²𝑡2𝜑+8,𝑡>0,_′(0)𝜑(0)=0,=2, (45) with the exact solution 𝑦(𝑡)=(5−cos2𝑡)/2. Using the VIM given in formulas (9) and (10), we construct the correction functional 𝑦_𝑚+1(𝑡)=𝑦_𝑚(𝑡)+^𝑡₀(𝜉−𝑡)𝑦_𝑚^′′(𝜉)+𝑦_𝑚𝜉2+𝑦²_𝑚(𝜉)−sin⁴(𝜉)−sin²(𝜉)−8𝑑𝜉,𝑚=1,2,…. (46) We take 𝑦₀(𝑡)=2 as the initial approximation and obtain that 𝑦₁(𝑡)=54+23𝑡16²+125cos𝑡+16cos²1𝑡−16cos⁴𝑦𝑡,₂(𝑡)=−989+512815𝑡512²+23𝑡1536⁴+121𝑡sin2𝑡+1−116𝑡sin𝑡1256𝑡sin𝑡cos𝑡+3cos2𝑡+11+16cos𝑡157512cos²1𝑡−16cos⁴𝑦𝑡,₃(𝑡)=−252541+20489781𝑡4096²+815𝑡49125⁴+23𝑡491520⁶1+14𝑡sin4𝑡+15116𝑡sin2𝑡+121+12048𝑡sin𝑡1256𝑡sin𝑡cos𝑡+120cos4𝑡+3981cos2𝑡+13932048cos𝑡+157512cos²𝑡116cos⁴1𝑡+𝑡2048²−1cos𝑡𝑡32²1cos2𝑡−12𝑡²1cos4𝑡,⋮ (47) 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 Example 4.

Example 5. Consider the second-order delay differential equation 𝑦^′′(𝑡)=−16𝑦(𝑡)+𝑦²𝑡4−sin²𝜑𝑡,𝑡>0,_′(0)𝜑(0)=4,=0. (48) Using the VIM given in formulas (28), we construct the correction functional 𝑦_𝑚+1(𝑡)=𝑦_𝑚(𝑡)+^𝑡₀14sin(4𝜉−4𝑡)𝑦_𝑚^′′(𝜉)+16𝑦_𝑚(𝜉)−𝑦²_𝑚𝜉4+sin²𝜉𝑑𝜉,𝑚=1,2,…. (49) We take 𝑦₀(𝑡)=4𝑡 as the initial approximation, and obtain that 𝑦₁(𝑡)=−0.0390625+0.0625𝑡²()()(),𝑦+sin4𝑡+0.04166666667cos2𝑡−0.002604166667cos4𝑡₂(𝑡)=0.00613912861−3.998216869𝑡+0.1805413564𝑡²−0.4340277778𝑡³+𝑜𝑡⁴+0.006666666667−0.5208333333𝑡+𝑜𝑡²+𝑡sin𝑡−0.1946373457−0.5555555556𝑡+𝑜²()⋮cos𝑡−0.1085069444sin2𝑡−2cos4𝑡, (50)

Example 6. Consider the second-order delay differential equation 𝑦^′′(𝑡)2=−𝑡𝑦^′(𝑡)+16𝑦²𝑡2+6−𝑡⁴𝜑,𝑡>0,_′(0)𝜑(0)=0,=0, (51) 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 𝑦_𝑚+1(𝑡)=𝑦_𝑚(𝑡)+^𝑡₀𝜉−𝜉+²𝑡𝑦_𝑚^′′(𝜉)+2𝜉𝑦^′_𝑚(𝜉)−16𝑦²_𝑚𝜉2−6+𝜉⁴𝑑𝜉,𝑚=1,2,…. (52) We take 𝑦₀(𝑡)=2𝑡 as the initial approximation and obtain that 𝑦₁(𝑡)1=−𝑡42⁶+𝑡²,𝑦₂(𝑡)=𝑡²+4𝑡21⁶−1𝑡1760¹⁰+1𝑡1935360¹⁴,⋮ (53) 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.
Table 1 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.

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.

Acknowledgments

The authors would like to thank the projects from NSF of China (11226322, 11271311), the Program for Changjiang Scholars and the Innovative Research Team in the University of China (IRT1179), the Aid Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province of China, and the Fund Project of Hunan Province Education Office (11C1120).