Variational Iteration Method for Singular Perturbation Initial Value Problems with Delays

Abstract

The variational iteration method (VIM) is applied to solve singular perturbation initial value problems with delays (SPIVPDs). Some convergence results of VIM for solving SPIVPDs are given. The obtained sequence of iterates is based on the use of general Lagrange multipliers; the multipliers in the functionals can be identified by the variational theory. Moreover, the numerical examples show the efficiency of the method.

1. Introduction

Singular perturbation initial value problems with delays play an important role in the research of various applied sciences, such as control theory, population dynamics, medical science, environment science, biology, and economics [1, 2]. These problems are characterized by a small parameter 𝜖 multiplying the highest derivatives, and the state variables depend not only on the value of the time, but also on the value prior to the time. Because the classical Lipschitz constant and one-sided Lipschitz constant are generally of size 𝒪(𝜖⁻¹)(0<𝜖≪1), the classical convergence theory, B-convergence theory, and D-convergence theory cannot be directly applied to SPIVPDs.

Starting from pioneering ideas going back to Inokuti-Sekine-Mura method [3], the variational iteration method was first proposed in later 1990s by He [4–6]. By recent years, this method has been extensively applied to various ODEs, integral equations, delay differential equations and fractional differential equations, two-point boundary value problems, oscillations, and stiff ODEs, notably, He [7, 8], Wazwaz [9, 10], Draganescu et al. [11, 12], Saadatmandi and Dehghan [13], Salkuyeh [14], Lu [15], Xu [16], Rafei et al. [17], Darvishi et al. [18], Tatari and Dehghan [19], Mamode [20], Saadati and Dehghan [21], Yu [22], Marinca et al. [23], Yang and Dumitru [24], and Wu [25] to mention only a few. Recently, Zhao and Xiao [26] have applied this method for solving singular perturbation initial value problems. For more comprehensive survey on this method and its applications, the reader is referred to the review articles [26–28] and the references therein.

In this paper, we apply the VIM to SPIVPDs to obtain the analytical or approximate analytical solutions. The convergence results of VIM for solving SPIVPDs are obtained. Some illustrative examples confirm the theoretical results.

In the remaining parts of the text, we denote 𝑥(𝑡)=𝑥(𝑡,𝜖),𝑦(𝑡)=𝑦(𝑡,𝜖) for simplicity; here, 𝜖 is the singular perturbation parameter. The vectors (𝑎₁,𝑎₂,…,𝑎_𝑛)^𝑇=𝑎≤𝑏=(𝑏₁,𝑏₂,…,𝑏_𝑛)^𝑇 mean that each component 𝑎_𝑖≤𝑏_𝑖  (𝑖=1,2,…,𝑛). ‖⋅‖ denotes the standard Euclidean norm of a vector.

2. Convergence Analysis

2.1. Case 1

Consider the following singular perturbation initial value problem with delays: 𝑥^′(𝑡)=𝑓𝑡,𝑥(𝑡),𝑥(𝑡−𝜏),𝑦(𝑡),𝑦(𝑡−𝜏),0≤𝑡≤𝑇,𝜖𝑦^′(𝑡)(𝑡)()(𝑡)(),𝑥(𝑡)(𝑡)(𝑡)(𝑡)=𝑔𝑡,𝑥,𝑥𝑡−𝜏,𝑦,𝑥𝑡−𝜏0<𝜖≪1,=𝜑,𝑦=𝜓,𝑡≤0, (1) where 𝑥∈𝑅^𝑛1 and 𝑦∈𝑅^𝑛2 are the state variables and  𝜖 is the singular perturbation parameter. 𝑓∶[−𝜏,𝑇]×𝑅^𝑛1×𝑅^𝑛1×𝑅^𝑛2×𝑅^𝑛2→𝑅^𝑛1, 𝑔∶[−𝜏,𝑇]×𝑅^𝑛1×𝑅^𝑛1×𝑅^𝑛2×𝑅^𝑛2→𝑅^𝑛2 are given continuous mappings which satisfy the following Lipschitz conditions: 𝑓𝑡,𝑥₁,𝑢₁,𝑦₁,𝑣₁−𝑓𝑡,𝑥₂,𝑢₂,𝑦₂,𝑣₂≤𝑙₁(𝑡)𝑥₁−𝑥₂+𝑙₂(𝑡)𝑢₁−𝑢₂+𝑙₃(𝑡)𝑦₁−𝑦₂+𝑙₄(𝑡)𝑣₁−𝑣₂,𝑔𝑡,𝑥₁,𝑢₁,𝑦₁,𝑣₁−𝑔𝑡,𝑥₂,𝑢₂,𝑦₂,𝑣₂≤𝑘₁(𝑡)𝑥₁−𝑥₂+𝑘₂(𝑡)𝑢₁−𝑢₂+𝑘₃(𝑡)𝑦₁−𝑦₂+𝑘₄(𝑡)𝑣₁−𝑣₂, (2a)(2b)where 𝑙_𝑖(𝑡),  𝑘_𝑖(𝑡)  (𝑖=1,…,4) are continuous bounded functions.

According to VIM, we can construct the correction functionals as follows:𝑥_𝑛+1(𝑡)=𝑥_𝑛(𝑡)+^𝑡₀𝜆₁()×𝑥𝑠,𝑡_′𝑛(𝑠)−𝑓𝑠,𝑥_𝑛(𝑠),𝑥_𝑛(𝑠−𝜏),𝑦_𝑛(𝑠),𝑦_𝑛()𝑦𝑠−𝜏𝑑𝑠,_𝑛+1(𝑡)=𝑦_𝑛(𝑡)+^𝑡₀𝜆₂()×𝑦𝑠,𝑡_′𝑛(𝑠)−1𝜖𝑔𝑠,𝑥_𝑛(𝑠),𝑥_𝑛(),𝑦𝑠−𝜏_𝑛(𝑠),𝑦_𝑛(𝑠−𝜏)𝑑𝑠, (3a)(3b)where 𝜆₁(𝑠,𝑡),  𝜆₂(𝑠,𝑡) are general Lagrange multipliers, which can be defined optimally via variational theory, and 𝑓,𝑔 denote the restrictive variation; that is, 𝛿𝑓=𝛿𝑔=𝟎. Thus, we have 𝛿𝑥_𝑛+1(𝑡)=𝛿𝑥_𝑛(𝑡)+^𝑡₀𝜆₁(𝑠,𝑡)𝛿𝑥^′_𝑛(𝑠)𝑑𝑠,𝛿𝑦_𝑛+1(𝑡)=𝛿𝑦_𝑛(𝑡)+^𝑡₀𝜆₂(𝑠,𝑡)𝛿𝑦^′_𝑛(𝑠)𝑑𝑠, (4) and the stationary conditions are obtained as 1+𝜆₁(𝑠,𝑡)_𝑠=𝑡=0,𝜕𝜆₁(𝑠,𝑡)𝜕𝑠=0,1+𝜆₂(𝑠,𝑡)_𝑠=𝑡=0,𝜕𝜆₂(𝑠,𝑡)𝜕𝑠=0. (5) Moreover, the general Lagrange multiplier can be readily identified by 𝜆₁(𝑠,𝑡)=𝜆₂(𝑠,𝑡)=−1. (6) Therefore, the variational iteration formulas can be written as𝑥_𝑛+1(𝑡)=𝑥_𝑛(𝑡)−^𝑡₀𝑥^′_𝑛(𝑠)−𝑓𝑠,𝑥_𝑛(𝑠),𝑥_𝑛(),𝑦𝑠−𝜏_𝑛(𝑠),𝑦_𝑛()𝑦𝑠−𝜏𝑑𝑠,_𝑛+1(𝑡)=𝑦_𝑛(𝑡)−^𝑡₀𝑦^′_𝑛(𝑠)−1𝜖𝑔𝑠,𝑥_𝑛(𝑠),𝑥_𝑛(𝑠−𝜏),𝑦_𝑛(𝑠),𝑦_𝑛(𝑠−𝜏)𝑑𝑠. (7a)(7b) Now, we show that the iterative sequences {𝑥_𝑛(𝑡)}^∞_𝑛=1,  {𝑦_𝑛(𝑡)}^∞_𝑛=1 defined by (7a) and (7b) with 𝑥₀(𝑡)=𝜑(𝑡),  𝑦₀(𝑡)=𝜓(𝑡) converge to the solution of (1).

Theorem 1. Let 𝑥(𝑡),  𝑥_𝑖(𝑡)∈(𝐶¹[0,𝑇])^𝑛1,𝑦(𝑡),𝑦_𝑖(𝑡)∈(𝐶¹[0,𝑇])^𝑛2, 𝑖=0,1,…. The sequences defined by (7a) and (7b) with 𝑥₀(𝑡)=𝜑(𝑡),  𝑦₀(𝑡)=𝜓(𝑡) converge to the solution of (1).

Proof. Obviously from system (1), we have𝑥(𝑡)(𝑡)−=𝑥_𝑡0𝑥^′(𝑠)(𝑠)()(𝑠)()𝑦(𝑡)(𝑡)−𝑓𝑠,𝑥,𝑥𝑠−𝜏,𝑦,𝑦𝑠−𝜏𝑑𝑠,=𝑦−^𝑡₀𝑦^′(𝑠)−1𝜖𝑔𝑠,𝑥(𝑠),𝑥(𝑠−𝜏),𝑦(𝑠),𝑦(𝑠−𝜏)𝑑𝑠. (8a)(8b)Introduce 𝐸_𝑖𝑥(𝑡)=𝑥_𝑖(𝑡)−𝑥(𝑡),  𝐸_𝑖𝑦(𝑡)=𝑦_𝑖(𝑡)−𝑦(𝑡),  𝐸_𝑖𝑥(𝑡−𝜏)=𝑥_𝑖(𝑡−𝜏)−𝑥(𝑡−𝜏),  𝐸_𝑖𝑦(𝑡−𝜏)=𝑦_𝑖(𝑡−𝜏)−𝑦(𝑡−𝜏),  𝑖=0,1,…, where 𝐸_𝑗𝑥(0)=𝐸_𝑗𝑦(0)=0,  𝑗=0,1,…. Now from (7a), (7b)-(8a), and (8b) we obtain 𝐸_𝑛+1𝑥(𝑡)=𝐸_𝑛𝑥(𝑡)−^𝑡₀𝐸_𝑛𝑥^′(𝑠)−𝑓𝑠,𝑥_𝑛(𝑠),𝑥_𝑛(),𝑦𝑠−𝜏_𝑛(𝑠),𝑦_𝑛(((𝑠)(),𝑦(𝑠)(𝐸𝑠−𝜏)−𝑓𝑠,𝑥,𝑥𝑠−𝜏,𝑦𝑠−𝜏)𝑑𝑠,_𝑛+1𝑦(𝑡)=𝐸_𝑛𝑦(𝑡)−^𝑡₀𝐸_𝑛𝑦^′(𝑠)−1𝜖×𝑔𝑠,𝑥_𝑛(𝑠),𝑥_𝑛(),𝑦𝑠−𝜏_𝑛(𝑠),𝑦_𝑛()(𝑠)()(𝑠),𝑦(𝑠−𝜏−𝑔𝑠,𝑥,𝑥𝑠−𝜏,𝑦𝑠−𝜏)𝑑𝑠. (9) Moreover, we can derive𝐸_𝑛+1𝑥(𝑡)=^𝑡₀𝑓𝑠,𝑥_𝑛(𝑠),𝑥_𝑛(𝑠−𝜏),𝑦_𝑛(𝑠),𝑦_𝑛()(𝑠)()(𝑠),𝑦()𝐸𝑠−𝜏−𝑓𝑠,𝑥,𝑥𝑠−𝜏,𝑦𝑠−𝜏𝑑𝑠,_𝑛+1𝑦(𝑡)=1𝜖^𝑡₀𝑔𝑠,𝑥_𝑛(𝑠),𝑥_𝑛(𝑠−𝜏),𝑦_𝑛(𝑠),𝑦_𝑛()(𝑠)()(𝑠),𝑦(𝑠−𝜏−𝑔𝑠,𝑥,𝑥𝑠−𝜏,𝑦𝑠−𝜏)𝑑𝑠. (10a)(10b)Now, the integration interval is split into two parts 𝐸_𝑛+1𝑥(𝑡)=^𝜏₀𝑓𝑠,𝑥_𝑛(𝑠),𝜑(𝑠−𝜏),𝑦_𝑛(𝑠)()(𝑠)(),𝑦(𝑠)(,𝜓𝑠−𝜏−𝑓𝑠,𝑥,𝜑𝑠−𝜏,𝜓𝑠−𝜏)𝑑𝑠+^𝑡_𝜏𝑓𝑠,𝑥_𝑛(𝑠),𝑥_𝑛(𝑠−𝜏),𝑦_𝑛(𝑠),𝑦_𝑛()(𝑠)(),𝑦(𝑠)(𝐸𝑠−𝜏−𝑓𝑠,𝑥,𝑥𝑠−𝜏,𝑦𝑠−𝜏)𝑑𝑠,_𝑛+1𝑦(𝑡)=1𝜖^𝜏₀𝑔𝑠,𝑥_𝑛(𝑠),𝜑(𝑠−𝜏),𝑦_𝑛(𝑠)()(𝑠)()(𝑠)()+,𝜓𝑠−𝜏−𝑔𝑠,𝑥,𝜑𝑠−𝜏,𝑦,𝜓𝑠−𝜏𝑑𝑠_𝑡𝜏𝑔𝑠,𝑥_𝑛(𝑠),𝑥_𝑛(𝑠−𝜏),𝑦_𝑛(𝑠),𝑦_𝑛()(𝑠)()(𝑠)().𝑠−𝜏−𝑔𝑠,𝑥,𝑥𝑠−𝜏,𝑦,𝑦𝑠−𝜏𝑑𝑠 (11) From the Lipschitz conditions (2a) and (2b), we have 𝐸_𝑛+1𝑥(𝑡𝐸_𝑛+1)𝑦(𝑡≤𝑙)¹𝑙₃𝑘₁𝜖𝑘₃𝜖∫^𝑡₀𝐸_𝑛𝑥(𝑠∫)𝑑𝑠_𝑡0𝐸_𝑛𝑦(𝑠+𝑙)𝑑𝑠₂𝑙₄𝑘₂𝜖𝑘₄𝜖∫^𝑡₀𝐸_𝑛𝑥(∫𝑠−𝜏)𝑑𝑠_𝑡0𝐸_𝑛𝑦(,𝑠−𝜏)𝑑𝑠 (12) where 𝑙_𝑖=max_{0≤𝑠≤𝑇}𝑙_𝑖(𝑠),  𝑘_𝑖=max_{0≤𝑠≤𝑇}𝑘_𝑖(𝑠),  𝑖=1,…,4. Therefore, 𝐸₁𝑥(𝑡𝐸₁)𝑦(𝑡≤𝑙)¹𝑙₃𝑘₁𝜖𝑘₃𝜖∫^𝑡₀𝐸₀𝑥(𝑠∫)𝑑𝑠_𝑡0𝐸₀𝑦(𝑠+𝑙)𝑑𝑠₂𝑙₄𝑘₂𝜖𝑘₄𝜖∫^𝑡₀𝐸₀𝑥(∫𝑠−𝜏)𝑑𝑠_𝑡0𝐸₀𝑦(≤𝑙𝑠−𝜏)𝑑𝑠₁+𝑙₂𝑙₃+𝑙₄𝑘₁+𝑘₂𝜖𝑘₃+𝑘₄𝜖max_{−𝜏≤𝑠≤𝑇}𝐸₀𝑥(𝑠)𝑡max_{−𝜏≤𝑠≤𝑇}𝐸₀𝑦(𝑠𝑡.) (13) Moreover, we can derive 𝐸_𝑛𝑥(𝑡𝐸_𝑛)𝑦(𝑡≤𝑇)^𝑛𝑙𝑛!¹+𝑙₂𝑙₃+𝑙₄𝑘₁+𝑘₂𝜖𝑘₃+𝑘₄𝜖^𝑛×max_{−𝜏≤𝑠≤𝑇}𝐸₀𝑥(𝑠)max_{−𝜏≤𝑠≤𝑇}𝐸₀𝑦(𝑠≤())𝑇/𝜖^𝑛𝒪(𝜖)𝒪(𝜖)𝒪(1)𝒪(1)𝑛!max_{−𝜏≤𝑠≤𝑇}𝐸₀𝑥(𝑠)max_{−𝜏≤𝑠≤𝑇}𝐸₀𝑦(𝑠.) (14) Noting that 𝜖,𝑇, max_{−𝜏≤𝑠≤𝑇}‖𝐸₀𝑥(𝑠)‖, max_{−𝜏≤𝑠≤𝑇}‖𝐸₀𝑦(𝑠)‖, 𝑙_𝑖, 𝑘_𝑖, 𝑖=1,…,4 are constants. By using Stirling’s formula, we have 𝐸_𝑛𝑥(𝑡𝐸_𝑛)𝑦(𝑡≤)))((𝑇𝑒/𝜖/𝑛^𝑛√((×𝒪(𝜖)𝒪(𝜖)𝒪(1)𝒪(1)2𝜋𝑛1+𝒪1/𝑛))max_{−𝜏≤𝑠≤𝑇}𝐸₀𝑥(𝑠)max_{−𝜏≤𝑠≤𝑇}𝐸₀𝑦(𝑠;) (15) thus, (‖𝐸_𝑛𝑥(𝑡)‖, ‖𝐸_𝑛𝑦(𝑡)‖)^𝑇→𝟎 as 𝑛→∞.

2.2. Case 2

Consider the special case of (1): 𝑥^′(𝑡)=𝐴𝑥(𝑡)+𝐹𝑡,𝑥(𝑡),𝑥(𝑡−𝜏),𝑦(𝑡),𝑦(𝑡−𝜏),0≤𝑡≤𝑇,𝜖𝑦^′(𝑡)(𝑡)(𝑡)()(𝑡)(),𝑥(𝑡)(𝑡)(𝑡)(𝑡)=𝐵𝑦+𝐺𝑡,𝑥,𝑥𝑡−𝜏,𝑦,𝑦𝑡−𝜏0<𝜖≪1,=𝜑,𝑦=𝜓,𝑡≤0, (16) where 𝐹∶[−𝜏,𝑇]×𝑅^𝑛1×𝑅^𝑛1×𝑅^𝑛2×𝑅^𝑛2→𝑅^𝑛1,𝐺∶[−𝜏,𝑇]×𝑅^𝑛1×𝑅^𝑛1×𝑅^𝑛2×𝑅^𝑛2→𝑅^𝑛2 are given continuous mappings which satisfy the Lipschitz conditions (2a) and (2b); the matrices 𝐴=(𝑎_𝑖𝑗)∈𝑅^𝑛1×𝑛1,  𝐵=(𝑏_𝑖𝑗)∈𝑅^𝑛2×𝑛2 can be decomposed into 𝐴=𝐴₀+𝐴₁,  𝐵=𝐵₀+𝐵₁, respectively, where 𝐴₀=diag(𝑎₁₁,𝑎₂₂,…,𝑎_𝑛1𝑛1) and 𝐵₀=diag(𝑏₁₁,𝑏₂₂,…,𝑏_𝑛2𝑛2):𝐹𝑡,𝑥₁,𝑢₁,𝑦₁,𝑣₁−𝐹𝑡,𝑥₂,𝑢₂,𝑦₂,𝑣₂≤𝑝₁(𝑡)𝑥₁−𝑥₂+𝑝₂(𝑡)𝑢₁−𝑢₂+𝑝₃(𝑡)𝑦₁−𝑦₂+𝑝₄(𝑡)𝑣₁−𝑣₂,𝐺𝑡,𝑥₁,𝑢₁,𝑦₁,𝑣₁−𝐺𝑡,𝑥₂,𝑢₂,𝑦₂,𝑣₂≤𝑞₁(𝑡)𝑥₁−𝑥₂+𝑞₂(𝑡)𝑢₁−𝑢₂+𝑞₃(𝑡)𝑦₁−𝑦₂+𝑞₄(𝑡)𝑣₁−𝑣₂, (17a)(17b)where 𝑝_𝑖(𝑡),   𝑞_𝑖(𝑡)  (𝑖=1,…,4) are continuous bounded functions.

It is easy to show that the right hand sides of (16) also satisfy the Lipschitz conditions. If the right hand sides of (16) are considered as nonlinear terms, then we can also use the correction functionals constructed in Case 1 and get similar results to Theorem 1. Now, we construct the following correction functionals: 𝑥_𝑛+1(𝑡)=𝑥_𝑛(𝑡)+^𝑡₀Λ₁(𝑠,𝑡)𝑥^′_𝑛(𝑠)−𝐴₀𝑥_𝑛(𝑠)−𝐴₁𝑥_𝑛(𝑠)−𝐹𝑠,𝑥_𝑛(𝑠),𝑥_𝑛(),𝑦𝑠−𝜏_𝑛(𝑠),𝑦_𝑛()𝑦𝑠−𝜏𝑑𝑠,_𝑛+1(𝑡)=𝑦_𝑛(𝑡)+^𝑡₀Λ₂()×𝑦𝑠,𝑡_′𝑛(𝑠)−1𝜖𝐵₀𝑦_𝑛(𝑠)+𝐵₁𝑦_𝑛(𝑠)+𝐺𝑠,𝑥_𝑛(𝑠),𝑥_𝑛(),𝑦𝑠−𝜏_𝑛(𝑠),𝑦_𝑛(𝑠−𝜏)𝑑𝑠, (18) where Λ₁(𝑠,𝑡)=diag(𝜆₁₁(𝑠,𝑡),𝜆₁₂(𝑠,𝑡),…,𝜆_1𝑛1(𝑠,𝑡)), Λ₂(𝑠,𝑡)=diag(𝜆₂₁(𝑠,𝑡),𝜆₂₂(𝑠,𝑡),…,𝜆_2𝑛2(𝑠,𝑡)), in which 𝜆_1𝑖(𝑠,𝑡),  𝜆_2𝑗(𝑠,𝑡),  𝑖=1,2,…,𝑛₁, 𝑗=1,2,…,𝑛₂ are general Lagrange multipliers and 𝑥_𝑛,𝑦_𝑛,𝐹,𝐺 denote the restrictive variations; that is, 𝛿𝑥_𝑛=𝛿𝑦_𝑛=𝛿𝐹=𝛿𝐺=𝟎. Thus, we have 𝛿𝑥_𝑛+1(𝑡)=𝛿𝑥_𝑛(𝑡)+^𝑡₀Λ₁(𝑠,𝑡)𝛿𝑥^′_𝑛(𝑠)−𝐴₀𝛿𝑥_𝑛(𝑠)𝑑𝑠,𝛿𝑦_𝑛+1(𝑡)=𝛿𝑦_𝑛(𝑡)+^𝑡₀Λ₂(𝑠,𝑡)𝛿𝑦^′_𝑛(𝑠)−𝐵₀𝜖𝛿𝑦_𝑛(𝑠)𝑑𝑠, (19) and the stationary conditions are obtained as 1+Λ₁(𝑠,𝑡)_𝑠=𝑡=0,𝜕Λ₁(𝑠,𝑡)𝜕𝑠+𝐴₀Λ₁(𝑠,𝑡)=0,1+Λ₂(𝑠,𝑡)_𝑠=𝑡=0,𝜕Λ₁(𝑠,𝑡)+𝐵𝜕𝑠𝜖⁰Λ₂(𝑠,𝑡)=0. (20) Moreover, the general Lagrange multipliers can be readily identified by Λ₁(𝑠,𝑡)=−exp−𝐴₀(),Λ𝑠−𝑡₂()−𝐵𝑠,𝑡=−exp⁰𝜖(𝑠−𝑡). (21) Therefore, the variational iteration formula can be written as𝑥_𝑛+1(𝑡)=𝑥_𝑛(𝑡)−^𝑡₀𝑒^{−𝐴0(𝑠−𝑡)}𝑥^′_𝑛(𝑠)−𝐴𝑥_𝑛(𝑠)−𝐹𝑠,𝑥_𝑛(𝑠),𝑥_𝑛(),𝑦𝑠−𝜏_𝑛(𝑠),𝑦_𝑛()𝑦𝑠−𝜏𝑑𝑠,_𝑛+1(𝑡)=𝑦_𝑛(𝑡)−^𝑡₀𝑒^{((−𝐵0/𝜖)(𝑠−𝑡))}×𝑦^′_𝑛(𝑠)−1𝜖×𝐵𝑦_𝑛(𝑠)+𝐺𝑠,𝑥_𝑛(𝑠),𝑥_𝑛(𝑠−𝜏),𝑦_𝑛(𝑠),𝑦_𝑛(𝑠−𝜏)𝑑𝑠. (22a)(22b)

The following theorem shows that the sequences {𝑥_𝑛(𝑡)}^∞_𝑛=1,  {𝑦_𝑛(𝑡)}^∞_𝑛=1 defined by (22a) and (22b) with 𝑥₀(𝑡)=𝜑(𝑡),  𝑦₀(𝑡)=𝜓(𝑡) converge to the solution of (16).

Theorem 2. Let 𝑥(𝑡),  𝑥_𝑖(𝑡)∈(𝐶¹[0,𝑇])^𝑛1,𝑦(𝑡),𝑦_𝑖(𝑡)∈(𝐶¹[0,𝑇])^𝑛2,  𝑖=0,1,…. The sequences defined by (22a) and (22b) with 𝑥₀(𝑡)=𝜑(𝑡),𝑦₀(𝑡)=𝜓(𝑡) converge to the solutions of (16).

Proof. By a similar process to the proof of Theorem 1, we can easily obtain Obviously from system (16) we have𝑥(𝑡)=𝑥(𝑡)−^𝑡₀𝑥^′(𝑠)(𝑡)(𝑠)()(𝑠),𝑦()𝑦(𝑡)(𝑡)−−𝐴𝑥−𝐹𝑠,𝑥,𝑥𝑠−𝜏,𝑦𝑠−𝜏𝑑𝑠,=𝑦_𝑡0𝑦^′(𝑠)−1𝜖(𝑡)(𝑠)()(𝑠),𝑦(𝐵𝑦+𝐺𝑠,𝑥,𝑥𝑠−𝜏,𝑦𝑠−𝜏)𝑑𝑠. (23a)(23b)Introduce 𝐸_𝑖𝑥(𝑡)=𝑥_𝑖(𝑡)−𝑥(𝑡),  𝐸_𝑖𝑦(𝑡)=𝑦_𝑖(𝑡)−𝑦(𝑡),  𝑖=0,1,…, where 𝐸_𝑗𝑥(0)=𝐸_𝑗𝑦(0)=0,  𝑗=0,1,…. Now from (22a), (22b)-(23a), and (23b) we obtain 𝐸_𝑛+1𝑥(𝑡)=^𝑡₀𝑒^{−𝐴0(𝑠−𝑡)}×𝐴₁𝐸_𝑛𝑥(𝑠)+𝐹𝑠,𝑥_𝑛(𝑠),𝑥_𝑛(𝑠−𝜏),𝑦_𝑛(𝑠),𝑦_𝑛()(𝑠)()(𝑠)()𝐸𝑠−𝜏−𝐹𝑠,𝑥,𝑥𝑠−𝜏,𝑦,𝑦𝑠−𝜏𝑑𝑠,_𝑛+1𝑦(𝑡)=^𝑡₀𝑒^{((−𝐵0/𝜖)(𝑠−𝑡))}×1𝜖𝐵₁𝐸_𝑛𝑦(𝑠)+𝐺𝑠,𝑥_𝑛(𝑠),𝑥_𝑛(𝑠−𝜏),𝑦_𝑛(𝑠),𝑦_𝑛()(𝑠)()(𝑠),𝑦()𝑠−𝜏−𝐺𝑠,𝑥,𝑥𝑠−𝜏,𝑦𝑠−𝜏𝑑𝑠. (24) From the Lipschitz conditions (17a) and (17b), we have 𝐸_𝑛+1𝑥(𝑡𝐸_𝑛+1)𝑦(𝑡≤𝑒)^{−‖𝐴0‖(𝑠−𝑡)}00𝑒^{−(‖𝐵0‖/𝜖)(𝑠−𝑡)}×𝐴₁+𝑝₁𝑝₃𝑞₁𝜖𝐵₁+𝑞₃𝜖×∫^𝑡₀𝐸_𝑛𝑥(𝑠∫)𝑑𝑠_𝑡0𝐸_𝑛𝑦(𝑠+𝑒)𝑑𝑠_{−‖𝐴0‖(𝑠−𝑡)}00𝑒^{−(‖𝐵0‖/𝜖)(𝑠−𝑡)}×𝑝₂𝑝₄𝑞₂𝜖𝑞₄𝜖×∫^𝑡₀𝐸_𝑛𝑥(∫𝑠−𝜏)𝑑𝑠_𝑡0𝐸_𝑛𝑦(;𝑠−𝜏)𝑑𝑠 (25) similarly, we can derive 𝐸_𝑛𝑥(𝑡𝐸_𝑛)𝑦(𝑡(𝜖)𝑇)≤𝑒^𝑛𝐴𝑛!¹+𝑝₁+𝑝₂𝑝₃+𝑝₄𝑞₁+𝑞₂𝜖𝐵₁+𝑞₃+𝑞₄𝜖^𝑛×max_{−𝜏≤𝑠≤𝑇}𝐸₀𝑥(𝑠)max_{−𝜏≤𝑠≤𝑇}𝐸₀𝑦(𝑠≤𝑒(𝜖))))((𝑇𝑒/𝜖/𝑛^𝑛√((×𝒪(𝜖)𝒪(𝜖)𝒪(1)𝒪(1)2𝜋𝑛1+𝒪1/𝑛))max_{−𝜏≤𝑠≤𝑇}𝐸₀𝑥(𝑠)max_{−𝜏≤𝑠≤𝑇}𝐸₀𝑦(𝑠,) (26) where 𝑒(𝜖)=max_{0≤𝑠≤𝑡,0≤𝑡≤𝑇}(𝑒^{−‖𝐴0‖(𝑠−𝑡)},𝑒^{−(‖𝐵0‖/𝜖)(𝑠−𝑡)}), 𝑝_𝑖=max_{0≤𝑠≤𝑇}𝑝_𝑖(𝑠), 𝑞_𝑖=max_{0≤𝑠≤𝑇}𝑞_𝑖(𝑠), 𝑖=1,…,4. Noting that 𝜖,  𝑇,  ‖𝐴₁‖,‖𝐵₁‖,  𝑒(𝜖),  𝑝_𝑖,  𝑞_𝑖,  𝑖=1,…,4, max_{−𝜏≤𝑠≤𝑇}‖𝐸₀𝑥(𝑠)‖ and max_{−𝜏≤𝑠≤𝑇}‖𝐸₀𝑦(𝑠)‖ are constants, we can derive from (26) that (‖𝐸_𝑛𝑥(𝑡)‖, ‖𝐸_𝑛𝑦(𝑡)‖)^𝑇→𝟎 as 𝑛→∞.

3. Numerical Examples

In this section, some numerical examples are given to show the efficiency of the VIM for solving SPIVPs.

Example 3. Consider SPIVPD (cf. [2]): 𝑥^′(𝑡)=𝑥(𝑡−1)𝑦(𝑡−1)−1000𝑥(𝑡)+2𝑦²(𝑡)+𝑅_𝑥(𝑡),𝑡>0,𝜖𝑦^′(𝑡)=𝑥(𝑡−1)−𝑦(𝑡−1)−(1+𝑥(𝑡))𝑦(𝑡)+𝑅_𝑦(𝑡),𝑥(𝑡)0<𝜖≪1,=𝑒^−0.5𝑡+𝑒^−0.2𝑡,𝑦(𝑡)=−𝑒^−0.5𝑡+𝑒^−0.2𝑡,𝑡≤0, (27) where 𝑅_𝑥(𝑡)=999.5𝑒^−0.5𝑡+999.8𝑒^−0.2𝑡+𝑒^{−(𝑡−1)}−𝑒^{−0.4(𝑡−1)}−2𝑒^−𝑡−2𝑒^−0.4𝑡+4𝑒^−0.7𝑡,𝑅_𝑦(𝑡)=(0.5𝜖−1)𝑒^−0.5𝑡+(1−0.2𝜖)𝑒^−0.2𝑡−𝑒^{−0.5(𝑡−1)}−𝑒^−𝑡+𝑒^−0.4𝑡. (28)
By using the VIM in Case 1, we construct the following iteration formula:𝑥_𝑛+1(𝑡)=𝑥_𝑛(𝑡)−^𝑡₀𝑥^′_𝑛(𝑠)−𝑥_𝑛(𝑠−1)𝑦_𝑛(𝑠−1)+1000𝑥_𝑛(𝑠)−2𝑦²_𝑛(𝑠)−𝑅_𝑥(𝑠)𝑦𝑑𝑠,_𝑛+1(𝑡)=𝑦_𝑛(𝑡)−^𝑡₀𝑦^′_𝑛(𝑠)−1𝜖×𝑥_𝑛(𝑠−1)−𝑦_𝑛(𝑠−1)−1+𝑥_𝑛(𝑠)×𝑦_𝑛(𝑠)+𝑅_𝑦(𝑠)𝑑𝑠. (29a)(29b)
To get iterate sequence, we start with an initial approximation 𝑥₀(𝑡)=𝑒^−0.5𝑡+𝑒^−0.2𝑡,  𝑦₀(𝑡)=−𝑒^−0.5𝑡+𝑒^−0.2𝑡 and let 𝜖=10⁻⁶. By means of formulas (29a) and (29b), we have 𝑥₁(𝑡)=𝑒^−0.5𝑡+𝑒^−0.2𝑡,𝑦₁(𝑡)=−𝑒^−0.5𝑡+𝑒^−0.2𝑡. (30) Figure 1 shows the efficiency of VIM for SPIVPDs.

(a)

(b)

Figure 1 

Results for problem 1.

Example 4. Consider SPIVPD (cf. [2]):𝑥^′(𝑡)=2𝑥(𝑡−1)+𝑦(𝑡−1)−1000𝑥(𝑡)+𝑦(𝑡)+𝑅_𝑥(𝑡),𝑡>0,𝜖𝑦^′(𝑡)=𝑥(𝑡−1)−𝑦(𝑡−1)+3𝑥(𝑡)−𝑦(𝑡)+𝑅_𝑦(𝑡),𝑥(𝑡)0<𝜖≪1,=1+10𝑒^{−(𝑡+1)/2}+5𝑒^{−(𝑡+1)/𝜖}𝑦(𝑡),𝑡≤0,=−1−9𝑒^{−(𝑡+1)/2}+4𝑒^{−(𝑡+1)/𝜖},𝑡≤0, (31) where 𝑅_𝑥(𝑡)=10004𝑒^{−(𝑡+1)/2}+54996−𝜖𝑒^{−(𝑡+1)/𝜖}−11𝑒^−𝑡/2−14𝑒^−𝑡/𝜖𝑅+1000,_𝑦(𝑡)=9𝜖2−39𝑒^{−(𝑡+1)/2}+15𝑒^{−(𝑡+1)/𝜖}−19𝑒^−𝑡/2−𝑒^−𝑡/𝜖−6. (32)

By using the VIM in Case 2, we construct the following iteration formula: 𝑥_𝑛+1(𝑡)=𝑥_𝑛(𝑡)−^𝑡₀𝑒^{1000(𝑠−𝑡)}𝑥^′_𝑛(𝑠)−2𝑥_𝑛(𝑠−1)−𝑦_𝑛(𝑠−1)+1000𝑥_𝑛(𝑠)−𝑦_𝑛(𝑠)−𝑅_𝑥(𝑠)𝑦𝑑𝑠,_𝑛+1(𝑡)=𝑦_𝑛(𝑡)−^𝑡₀𝑒^{(𝑠−𝑡)/𝜖}×𝑦^′_𝑛(𝑠)−1𝜖×𝑥_𝑛(𝑠−1)−𝑦_𝑛(𝑠−1)+3𝑥_𝑛(𝑠)−𝑦_𝑛(𝑠)+𝑅_𝑦(𝑠)𝑑𝑠. (33)(34)

To get iterate sequence, we start with an initial approximation 𝑥₀(𝑡)=1+10𝑒^{−(𝑡+1)/2}+5𝑒^{−(𝑡+1)/𝜖},  𝑦₀(𝑡)=−1−9𝑒^{−(𝑡+1)/2}+4𝑒^{−(𝑡+1)/𝜖} and let 𝜖=10⁻³. By means of formulas (33) and (34), we have 𝑥₁(𝑡)=1+10𝑒^{−(𝑡+1)/2}+5𝑒^{−(𝑡+1)/𝜖},𝑦₁(𝑡)=−1−9𝑒^{−(𝑡+1)/2}+4𝑒^{−(𝑡+1)/𝜖}. (35) Figure 2 shows the efficiency of VIM for SPIVPDs.

(a)

(b)

Figure 2 

Results for problem 2.

4. Conclusion

The VIM used in this paper is the variational iteration algorithm I; there are also variational iteration algorithms II and III [29]. In this paper, we apply the VIM to obtain the analytical or approximate analytical solutions of SPIVPDs. The convergence results of VIM for solving SPIVPDs are given. The illustrative examples show the efficiency of the method. When considering the system (16), the choice of correction functionals of Case 1 or Case 2 relies on the practical problems and this choice will result in the difference of the speed of convergence.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by projects NSF of China (11126329, 11271311, and 11201510), Projects Board of Education of Chongqing City (KJ121110), and Key Laboratory for Nonlinear Science and System Structure.