On the Convergence of Continuous-Time Waveform Relaxation Methods for Singular Perturbation Initial Value Problems

Abstract

This paper extends the continuous-time waveform relaxation method to singular perturbation initial value problems. The sufficient conditions for convergence of continuous-time waveform relaxation methods for singular perturbation initial value problems are given.

1. Introduction

Singular perturbation initial value problems 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. Since 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 cannot be directly applied to singular perturbation initial value problems.

Waveform relaxation methods were introduced by Lelarasmee et al. [3]. In recent years, these methods have been widely applied due to their flexibility, convenience, and efficiency. The problems of convergence of waveform relaxation methods for systems of ordinary differential equations, differential algebraic equations, integral equations, delay differential equations, and fractional differential equations were discussed by main authors (cf. [4–7], and references therein). In’t Hout [8] consider the convergence of waveform relaxation methods for stiff nonlinear ordinary differential equations. The monograph of Jiang [9] which entitled “Waveform Relaxation Methods” introduced these methods for various ODEs, differential algebraic equations, integral equations, delay differential equations and fractional differential equations, and PDEs. For more comprehensive survey on these methods and their applications, the reader is refered to the monograph in [9] and the references therein. Natesan et al., Vigo-Aguiar and Natesan [10–14] considers the general second order singular perturbation problems.

In this paper, we apply continuous waveform relaxation methods to single stiff singular perturbation initial value problems and obtain the corresponding convergence results.

In the rest parts of the text, we define the maximum norm as follows: ‖𝑥‖_𝑇=max_{0≤𝑡≤𝑇}‖𝑥‖, (1.1) and the 𝜆 norm of exponential type: ‖𝑥‖_𝜆,𝑇=max_{0≤𝑡≤𝑇}𝑒^−𝜆𝑡‖𝑥‖, (1.2) where 𝜆 is any given positive number and ‖⋅‖ denotes an any given norm in 𝐶^𝑛.

2. Convergence Analysis of the First Type

2.1. Linear SPPs

Consider the following linear singular perturbation initial value problem 𝜖𝑑𝑦(𝑡)(𝑡)(𝑡)[]𝑦(0)𝑑𝑡+𝐴𝑦=𝑏,𝑡∈0,𝑇,0<𝜖≪1,=𝑦₀, (2.1) where the constant matrix 𝐴∈𝑅^𝑛×𝑛 and the input function 𝑏(𝑡)∈𝑅^𝑛, 𝑦₀ is the given initial value, and 𝜖 is the singular perturbation parameter. The constant matrix 𝐴 is split by 𝐴=𝐴₁−𝐴₂, then the system (2.1) can be written as 𝜖𝑑𝑦(𝑡)𝑑𝑡+𝐴₁𝑦(𝑡)=𝐴₂𝑦(𝑡)(𝑡)[]𝑦(0)+𝑏,𝑡∈0,𝑇,0<𝜖≪1,=𝑦₀, (2.2) then we can obtain the following iterate scheme: 𝑑𝑦^(𝑘+1)(𝑡)+1𝑑𝑡𝜖𝐴₁𝑦^(𝑘+1)(𝑡)=1𝜖𝐴₂𝑦^(𝑘)(𝑡)+𝑏(𝑡)𝜖[]𝑦,𝑡∈0,𝑇,0<𝜖≪1,_(𝑘+1)(0)=𝑦₀,𝑘=1,2,…, (2.3) here we can choose the initial iterative function 𝑦⁽⁰⁾(𝑡)=𝑦₀, the above iteration is called a continuous-time waveform relaxation process.

For any fixed 𝑘≥0, from (2.3), we have, upon premultiplying by 𝑒^{(𝑠−𝑡)𝐴1} and integrating from 0 to 𝑡, as following: 𝑦^(𝑘+1)(𝑡)=𝑒^{−(𝐴1𝑡)/𝜖}𝑦₀+^𝑡₀𝑒^{(𝑠−𝑡)𝐴1/𝜖}𝐴₂𝜖𝑦^(𝑘)(𝑠)+𝑏(𝑠)𝜖𝑑𝑠. (2.4) Let ℛ𝑦(𝑡)=^𝑡₀𝑒^{(𝑠−𝑡)𝐴1/𝜖}𝐴₂𝜖𝑦(𝑠)𝑑𝑠, (2.5) then (2.5) can be written as 𝑦^(𝑘+1)(𝑡)=ℛ𝑦^𝑘(𝑡)+𝜑(𝑡), (2.6) where 𝜑(𝑡)=𝑒^{−(𝐴1𝑡)/𝜖}𝑦₀+∫^𝑡₀𝑒^{(𝑠−𝑡)𝐴1/𝜖}(𝑏(𝑠)/𝜖)𝑑𝑠. It is easy to see that ℛ is a Volterra convolution operator with the kernel function 𝜅(𝑡)=𝑒^{−(𝐴1𝑡)/𝜖}(𝐴₂/𝜖): ℛ𝑦(𝑡)=𝜅(𝑡)∗𝑦(𝑡)=^𝑡₀𝜅(𝑠−𝑡)𝑦(𝑠)𝑑𝑠, (2.7) and ℛ is the waveform relaxation operator.

Theorem 2.1. Let the waveform relaxation operator ℛ be defined in 𝐶([0,𝑇],𝑅^𝑛). If the kernel function 𝜅(𝑡) is continuous in [0,𝑇] and satisfies ‖𝜅‖_𝑇≤𝐶, where 𝐶 is a constant, then the sequence of functions 𝑦^(𝑘)(𝑡) defined by (2.3) satisfy 𝑦^(𝑘)(𝑡)−𝑦^∗(𝑡)_𝑇≤(𝐶𝑇)^𝑘𝑦𝑘!⁽⁰⁾(𝑡)−𝑦^∗(𝑡)_𝑇, (2.8) where 𝑦^∗(𝑡) is the exact solution of system (2.1).

Proof. By the norm ‖⋅‖_𝑇 in 𝐶([0,𝑇]), we can obtain ℛ^𝑘_𝑇≤^𝑇₀𝜅^𝑘(𝑡)𝑑𝑠,𝑘=1,2,…, (2.9) where 𝜅¹(𝑡)=𝜅(𝑡), 𝜅^𝑘(𝑡)=𝜅(𝑡)∗𝜅^𝑘−1(𝑡).
In fact, from the given condition ‖𝜅‖_𝑇≤𝐶, for any 𝑡≤𝑇, we have 𝜅^𝑘(𝑡)≤𝐶^𝑡₀𝜅^𝑘−1(𝑡)𝑑𝑠, (2.10) it can be obtained, by induction, that 𝜅^𝑘(𝑡())≤𝐶𝐶𝑡^𝑘−1()!,𝑘−1 (2.11) so ℛ^𝑘_𝑇≤(𝐶𝑇)^𝑘,𝑘! (2.12) which complete the proof.

Finally, we mention that the estimate (2.8) is superlinear convergence estimate, which reveals a rapid convergence behavior when 𝑘→∞.

2.2. Nonlinear SPPs

Consider the following nonlinear singular perturbation initial value problem: 𝜖𝑑𝑦(𝑡)𝑦(𝑡)[]𝑦(0)𝑑𝑡=𝑓,𝑡,𝑡∈0,𝑇,0<𝜖≪1,=𝑦₀, (2.13) where 𝑦₀ is the given initial value, 𝜖 is the singular perturbation parameter. 𝑓∶𝑅^𝑛×[0,𝑇]→𝑅^𝑛 is given continuous function mapping, and 𝑦(𝑡) is unknown.

The continuous-time Waveform Relaxation algorithm for (2.13) is 𝑑𝑦^(𝑘+1)(𝑡)=1𝑑𝑡𝜖𝐹𝑦^(𝑘+1)(𝑡),𝑦^(𝑘)(𝑡)[]𝑦,𝑡,𝑡∈0,𝑇,0<𝜖≪1,_(𝑘+1)(0)=𝑦₀,𝑘=1,2,…, (2.14) where the splitting function 𝐹(𝑢,𝑣,𝑡) determines the type of the Waveform Relaxation algorithm, and we assume that 𝐹(𝑢,𝑣,𝑡) satisfy the following Lipschitz condition 𝐹𝑢₁,𝑣₁,𝑡−𝐹𝑢₂,𝑣₂,𝑡≤𝐿₁𝑢₁−𝑢₂+𝐿₂𝑣₁−𝑣₂. (2.15)

By integrating the inequality (2.15) of both side from 0 to 𝑡, we have 𝑦^(𝑘+1)(𝑡)=𝑦₀+^𝑡₀1𝜖𝐹𝑦^(𝑘+1)(𝑠),𝑦^(𝑘)(𝑠),𝑠𝑑𝑠. (2.16)

Let 𝑦(𝑡) denote the function that iterated by 𝑦(𝑡) from one iteration step, like (2.16), denote 𝑦(𝑡)=(ℛ𝑦)(𝑡).

Theorem 2.2. Assume that the splitting function 𝐹(𝑢,𝑣,𝑡) in WR iteration process (2.14) is Lipschitz continuous with respect to 𝑢 and 𝑣, then the continuous-time Waveform Relaxation algorithm (2.14) is convergent.

Proof. We introduce another continuous function 𝑧(𝑡), and denote 𝑧(𝑡)=(ℛ𝑧)(𝑡), then (𝑡)−𝑧(𝑡=1𝑦)𝜖^𝑡₀𝐹(𝑠)(𝑠)(𝑧(𝑠)(𝑠))≤1𝑦,𝑦,𝑠−𝐹,𝑧,𝑠𝑑𝑠𝜖^𝑡₀𝐹𝑦(𝑠),𝑦(𝑠),𝑠−𝐹(𝑧(𝑠),𝑧(𝑠),𝑠)𝑑𝑠. (2.17) Equations (2.15)–(2.17) yield (𝑡)−𝑧(𝑡≤𝐿𝑦)¹𝜖^𝑡₀(𝑡)−𝑧(𝑠𝐿𝑦)𝑑𝑠+²𝜖^𝑡₀𝑦(𝑠)−𝑧(𝑠)𝑑𝑠. (2.18) From (2.18), we have, upon premultiplying by 𝑒^−𝜆𝑡, the following: 𝑒^−𝜆𝑡(𝑡)−𝑧(𝑡≤𝐿𝑦)¹𝜖𝑒^{−𝜆𝑡𝑡}₀(𝑠)−𝑧(𝑠𝐿𝑦)𝑑𝑠+²𝜖𝑒^{−𝜆𝑡𝑡}₀𝑦(𝑠)(𝑠≤𝐿−𝑧)𝑑𝑠₁𝜖𝑒^{−𝜆𝑡𝑡}₀𝑒^𝜆𝑠𝑒^−𝜆𝑠(𝑠)−𝑧(𝑠𝐿𝑦)𝑑𝑠+²𝜖𝑒^{−𝜆𝑡𝑡}₀𝑒^𝜆𝑠𝑒^−𝜆𝑠𝑦(𝑠)(𝑠≤𝐿−𝑧)𝑑𝑠₁𝜖max_{0≤𝑠≤𝑡}𝑒^−𝜆𝑠(𝑠)−𝑧(𝑠+𝐿𝑦)²𝜖max_{0≤𝑠≤𝑡}𝑒^−𝜆𝑠𝑦(𝑠)−𝑧(𝑠)𝑒^{−𝜆𝑡𝑡}₀𝑒^𝜆𝑠𝑑𝑠, (2.19) because of 𝑒^−𝜆𝑡∫^𝑡₀𝑒^𝜆𝑠𝑑𝑠≤1/𝜆, and from the definition of 𝜆 norm of the exponential type, we have 𝑦(𝑡)−𝑧(𝑡)_𝜆,𝑇≤𝐿₁(𝑠)−𝑧(𝑠𝜆𝜖𝑦)_𝜆,𝑇+𝐿₂𝑦(𝑠)(𝑠𝜆𝜖−𝑧)_𝜆,𝑇. (2.20) It is easy to obtain 𝑦(𝑡)−𝑧(𝑡)_𝜆,𝑇≤𝐿₂𝜆𝜖−𝐿₁𝑦(𝑠)−𝑧(𝑠)_𝜆,𝑇. (2.21) We can choose large enough 𝜆 such that 𝛾=𝐿₂/(𝜆𝜖−𝐿₁)<1. Thus, the Waveform Relaxation operator ℛ is a contractive operator under this norm. From the contractive mapping principle, we can derive that the continuous-time Waveform Relaxation algorithm (2.14) is convergent.

3. Convergence Analysis of The Second Type

3.1. Linear SPPs

Consider the following linear singular perturbation initial value problem 𝑥^′(𝑡)(𝑡)(𝑡)(𝑡)[],𝜖(𝑡)+𝐴𝑥+𝐵𝑦=𝑟,𝑡∈0,𝑇𝑑𝑦(𝑡)(𝑡)(𝑡)𝑥(0)𝑑𝑡+𝐶𝑥+𝐷𝑦=𝑠,0<𝜖≪1,=𝑥₀,𝑦(0)=𝑦₀, (3.1) where 𝑥₀ and 𝑦₀ are the given initial value, 𝜖 is the singular perturbation parameter, 𝑟(𝑡)∈𝑅^𝑛1 and 𝑠(𝑡)∈𝑅^𝑛2 are given functions. The constant matrices 𝐴∈𝑅^𝑛1×𝑛1, 𝐵∈𝑅^𝑛1×𝑛2, 𝐶∈𝑅^𝑛2×𝑛1, 𝐷∈𝑅^𝑛2×𝑛2 are split by 𝐴=𝐴₁−𝐴₂, 𝐵=𝐵₁−𝐵₂, 𝐶=𝐶₁−𝐶₂, 𝐷=𝐷₁−𝐷₂ respectively, 𝑥(𝑡)∈𝑅^𝑛1 and 𝑦(𝑡)∈𝑅^𝑛2 are unknowns. Then the system (3.1) can be written as 𝑥^′(𝑡)+𝐴₁𝑥(𝑡)+𝐵₁𝑦(𝑡)=𝐴₂𝑥(𝑡)+𝐵₂𝑦(𝑡)(𝑡)[],𝜖(𝑡)+𝑟,𝑡∈0,𝑇𝑑𝑦𝑑𝑡+𝐶₁𝑥(𝑡)+𝐷₁𝑦(𝑡)=𝐶₂𝑥(𝑡)+𝐷₂𝑦(𝑡)(𝑡)𝑥(0)+𝑠,0<𝜖≪1,=𝑥₀,𝑦(0)=𝑦₀. (3.2) The continuous-time Waveform Relaxation algorithm for (3.1) is as follows: 𝑑𝑥^(𝑘+1)(𝑡)𝑑𝑡+𝐴₁𝑥^(𝑘+1)(𝑡)+𝐵₁𝑦^(𝑘+1)(𝑡)=𝐴₂𝑥^(𝑘)(𝑡)+𝐵₂𝑦^(𝑘)(𝑡)+𝑟(𝑡),𝑡∈[0,𝑇],𝑑𝑦^(𝑘+1)(𝑡)+𝐶𝑑𝑡𝜖¹𝑥^(𝑘+1)(𝑡)+𝐷₁𝜖𝑦^(𝑘+1)(𝑡)=𝐶₂𝜖𝑥^(𝑘)(𝑡)+𝐷₂𝜖𝑥^(𝑘)(𝑡)+1𝜖𝑠(𝑡)𝑥,0<𝜖≪1,_(𝑘+1)(0)=𝑥₀,𝑦^(𝑘+1)(0)=𝑦₀,𝑘=1,2,…, (3.3) The matrix form of (3.3) reads 𝑑𝑥𝑑𝑡^(𝑘+1)(𝑡)𝑦^(𝑘+1)(𝑡)+𝐴₁𝐵₁𝐶₁𝜖𝐷₁𝜖𝑥^(𝑘+1)(𝑡)𝑦^(𝑘+1)(𝑡)=𝐴₂𝐵₂𝐶₂𝜖𝐷₂𝜖𝑥^(𝑘)(𝑡)𝑦^(𝑘)(𝑡)+𝑟(𝑡)𝑠(𝑡)𝜖. (3.4) Solve the equations (3.4), we can derive 𝑥^(𝑘+1)(𝑡)𝑦^(𝑘+1)(𝑡)−𝐴=exp¹𝐵₁𝐶₁𝜖𝐷₁𝜖𝑡𝑥^(𝑘+1)(0)𝑦^(𝑘+1)(0)+^𝑡₀𝐴exp¹𝐵₁𝐶₁𝜖𝐷₁𝜖()𝐴𝑠−𝑡²𝐵₂𝐶₂𝜖𝐷₂𝜖𝑥^(𝑘)(𝑠)𝑦^(𝑘)(𝑠)+𝑟(𝑠)𝑠(𝑠)𝜖𝑑𝑠. (3.5)

Denote 𝜀^𝑘_𝑥(𝑡)=𝑥^𝑘(𝑡)−𝑥(𝑡), 𝜀^𝑘_𝑦(𝑡)=𝑦^𝑘(𝑡)−𝑦(𝑡), where 𝑥(𝑡) and 𝑦(𝑡) are the exact solutions of (3.1). From (3.2) and (3.5), we can obtain 𝜀_𝑥^𝑘+1(𝑡)𝜀_𝑦^𝑘+1(𝑡)=^𝑡₀𝐴exp¹𝐵₁𝐶₁𝜖𝐷₁𝜖()𝐴𝑠−𝑡²𝐵₂𝐶₂𝜖𝐷₂𝜖𝜀^𝑘_𝑥(𝑠)𝜀^𝑘_𝑦(𝑠)𝑑𝑠, (3.6) then (3.6) can be written as 𝜀_𝑥^𝑘+1(𝑡)𝜀_𝑦^𝑘+1(𝑡)=ℛ𝜀^𝑘_𝑥(𝑠)𝜀^𝑘_𝑦(𝑠)(𝑡), (3.7) clearly, ℛ is a Volterra convolution operator with the kernel function 𝜅(𝑡)−𝐴=exp¹𝐵₁𝐶₁𝜖𝐷₁𝜖𝑡𝐴₂𝐵₂𝐶₂𝜖𝐷₂𝜖,ℛ𝑦(𝑡)=𝜅(𝑡)∗𝑦(𝑡)=^𝑡₀𝜅(𝑠−𝑡)𝑦(𝑠)𝑑𝑠, (3.8)ℛ is the Waveform Relaxation operator.

Theorem 3.1. Let the waveform relaxation operator ℛ be defined in 𝐶([0,𝑇],𝑅^𝑛). If the kernel function 𝜅(𝑡) is continuous in [0,𝑇] and satisfies ‖𝜅‖_𝑇≤𝑀, where 𝑀 is a constant, then the sequence of functions (𝜀_𝑥^𝑘+1(𝑡),𝜀_𝑦^𝑘+1(𝑡))^𝑇defined by (3.6) satisfy 𝜀^𝑘_𝑥(𝑡𝜀_𝑘𝑦)(𝑡≤𝑇)^𝑘𝑀𝑘!^𝑘max_0<𝑠<𝑇𝜀⁰_𝑥(𝑠)max_0<𝑠<𝑇𝜀⁰_𝑦(𝑠). (3.9)

Proof. Taking the norm in both side of (3.6) which reads 𝜀_𝑥^𝑘+1(𝑡𝜀_𝑦𝑘+1)(𝑡≤)^𝑡0𝐴exp¹𝐵₁𝐶₁𝜖𝐷₁𝜖()𝐴𝑠−𝑡²𝐵₂𝐶₂𝜖𝐷₂𝜖𝜀^𝑘_𝑥(𝑠𝜀_𝑘𝑦)(𝑠)𝑑𝑠. (3.10) From the induction of (3.10) and the condition ‖𝜅‖_𝑇≤𝑀, we can derive 𝜀¹_𝑥(𝑡𝜀_1𝑦)(𝑡)≤𝑀max_0<𝑠<𝑇𝜀⁰_𝑥(𝑠)max_0<𝑠<𝑇𝜀⁰_𝑦(𝑠)𝑡. (3.11) Moreover, we can derive 𝜀^𝑘_𝑥(𝑡𝜀_𝑘𝑦)(𝑡≤𝑇)^𝑘𝑀𝑘!^𝑘max_0<𝑠<𝑇𝜀⁰_𝑥(𝑠)max_0<𝑠<𝑇𝜀⁰_𝑦(𝑠), (3.12) thus, (‖𝜀^𝑘_𝑥(𝑡)‖,‖𝜀^𝑘_𝑦(𝑡)‖)^𝑇→0 as 𝑘→∞ which complete the proof.

3.2. Nonlinear SPPs

Consider the following nonlinear singular perturbation initial value problem: 𝑥^′(𝑡)𝑥(𝑡)(𝑡)[],𝜖(𝑡)=𝑓,𝑦,𝑡,𝑡∈0,𝑇𝑑𝑦𝑥(𝑡)(𝑡)𝑥(0)𝑑𝑡=𝑔,𝑦,𝑡,0<𝜖≪1,=𝑥₀,𝑦(0)=𝑦₀, (3.13) where 𝑥₀ and 𝑦₀ are given initial values, 𝜖 is the singular perturbation parameter, 𝑓∶𝑅^𝑛1×𝑅^𝑛2×[0,𝑇]→𝑅^𝑛1 and 𝑔∶𝑅^𝑛1×𝑅^𝑛2×[0,𝑇]→𝑅^𝑛2 are given continuous function mappings.

The continuous-time waveform relaxation algorithm for (3.13) is 𝑥^(𝑘+1)(𝑡)=𝐹𝑥^(𝑘+1)(𝑡),𝑥^(𝑘)(𝑡),𝑦^(𝑘+1)(𝑡),𝑦^(𝑘)(𝑡),𝑡,𝑡∈[0,𝑇],𝑑𝑦^(𝑘+1)(𝑡)=1𝑑𝑡𝜖𝐺𝑥^(𝑘+1)(𝑡),𝑥^(𝑘)(𝑡),𝑦^(𝑘+1)(𝑡),𝑦^(𝑘)(𝑡)𝑥,𝑡,0<𝜖≪1,_(𝑘+1)(0)=𝑥₀,𝑦^(𝑘+1)(0)=𝑦₀,𝑘=1,2,…, (3.14) where the splitting functions 𝐹(𝑢₁,𝑢₂,𝑢₃,𝑢₄,𝑡) and 𝐺(𝑢₁,𝑢₂,𝑢₃,𝑢₄,𝑡) determine the type of the waveform relaxation algorithm. we can derive from (3.14) that 𝑥^(𝑘)(𝑡)=𝐹𝑥^(𝑘)(𝑡),𝑥^(𝑘−1)(𝑡),𝑦^(𝑘)(𝑡),𝑦^(𝑘−1)(𝑡),𝑡,𝑡∈[0,𝑇],𝑑𝑦^(𝑘)(𝑡)=1𝑑𝑡𝜖𝐺𝑥^(𝑘)(𝑡),𝑥^(𝑘−1)(𝑡),𝑦^(𝑘)(𝑡),𝑦^(𝑘−1)(𝑡)𝑥,𝑡,0<𝜖≪1,_(𝑘)(0)=𝑥₀,𝑦^(𝑘)(0)=𝑦₀,𝑘=1,2,…. (3.15) Denote 𝜀_𝑥^𝑘+1(𝑡)=𝑥^𝑘+1(𝑡)−𝑥^𝑘(𝑡),𝜀_𝑦^𝑘+1(𝑡)=𝑦^𝑘+1(𝑡)−𝑦^𝑘(𝑡). (3.16)

Theorem 3.2. Assume that the matrices (𝜕𝐹/𝜕𝑢_𝑖) and (𝜕𝐺/𝜕𝑢_𝑖)(𝑖=1,2,3,4) of the splitting functions 𝐹(𝑢₁,𝑢₂,𝑢₃,𝑢₄,𝑡) and 𝐺(𝑢₁,𝑢₂,𝑢₃,𝑢₄,𝑡) are continuous, then the continuous-time waveform relaxation algorithm (3.14) is convergent.

Proof. Subtracting (3.14) from (3.15), we have 𝑑𝜀_𝑥^𝑘+1(𝑡)=𝑑𝑡𝜕𝐹𝜕𝑢₁𝜀_𝑥^𝑘+1(𝑡)+𝜕𝐹𝜕𝑢₂𝜀^𝑘_𝑥(𝑡)+𝜕𝐹𝜕𝑢₃𝜀_𝑦^𝑘+1(𝑡)+𝜕𝐹𝜕𝑢₄𝜀^𝑘_𝑦(𝑡),𝑑𝜀_𝑦^𝑘+1(𝑡)=1𝑑𝑡𝜖𝜕𝐺𝜕𝑢₁𝜀_𝑥^𝑘+1(𝑡)+𝜕𝐺𝜕𝑢₂𝜀^𝑘_𝑥(𝑡)+𝜕𝐺𝜕𝑢₃𝜀_𝑦^𝑘+1(𝑡)+𝜕𝐺𝜕𝑢₄𝜀^𝑘_𝑦(𝑡), (3.17) the matrix form of (3.17) reads 𝑑𝜀𝑑𝑡^𝑥𝑘+1(𝑡)𝜀_𝑦^𝑘+1(𝑡)=𝜕𝐹𝜕𝑢₁𝜕𝐹𝜕𝑢₃1𝜖𝜕𝐺𝜕𝑢₁1𝜖𝜕𝐺𝜕𝑢₃𝜀_𝑥^𝑘+1(𝑡)𝜀_𝑦^𝑘+1(𝑡)+𝜕𝐹𝜕𝑢₂𝜕𝐹𝜕𝑢₄1𝜖𝜕𝐺𝜕𝑢₂1𝜖𝜕𝐺𝜕𝑢₄𝜀^𝑘_𝑥(𝑡)𝜀^𝑘_𝑦(𝑡). (3.18) Denote 𝜀^𝑘(𝑡)=𝜀^𝑘_𝑥(𝑡),𝜀^𝑘_𝑦(𝑡)^𝑇,𝐴(𝑡)=𝜕𝐹𝜕𝑢₁𝜕𝐹𝜕𝑢₃1𝜖𝜕𝐺𝜕𝑢₁1𝜖𝜕𝐺𝜕𝑢₃,𝐵(𝑡)=𝜕𝐹𝜕𝑢₂𝜕𝐹𝜕𝑢₄1𝜖𝜕𝐺𝜕𝑢₂1𝜖𝜕𝐺𝜕𝑢₄. (3.19) Then, we can derive 𝑑𝜀^𝑘+1(𝑡)(𝑡)𝜀𝑑𝑡=𝐴^𝑘+1(𝑡)+𝐵(𝑡)𝜀^𝑘(𝑡). (3.20) Assume that the basis matrix 𝜙(𝑡) satisfies 𝑑𝜙(𝑡)(𝑡)𝜙(𝑡),𝑑𝑡=𝐴 (3.21) then the solution of (3.20) can be written as 𝜀^𝑘+1(𝑡)=𝜙(𝑡)^𝑡₀𝜙⁻¹(𝑠)𝐵(𝑠)𝜀^𝑘(𝑠)𝑑𝑠. (3.22) From (3.22), we have, upon taking the norm in both side and premultiplying by 𝑒^−𝜆𝑡, that 𝑒^−𝜆𝑡𝜀^𝑘+1(𝑡)=𝑒^{−𝜆𝑡𝑡}₀𝜙(𝑡)𝜙⁻¹(𝑠)𝐵(𝑠)𝜀^𝑘(𝑠)𝑑𝑠, (3.23) furthermore, max_{0≤𝑡≤𝑇}𝑒^−𝜆𝑡𝜀^𝑘+1(𝑡)=max_{0≤𝑡≤𝑇}𝑒^{−𝜆𝑡𝑡}₀𝜙(𝑡)𝜙⁻¹(𝑠)𝐵(𝑠)𝜀^𝑘(𝑠)𝑑𝑠, (3.24) so 𝜀^𝑘+1(𝑡)_𝜆,𝑇≤max_{0≤𝑡≤𝑇}𝑒^{−𝜆𝑡𝑡}₀𝜙(𝑡)𝜙⁻¹(𝑠)𝐵(𝑠)𝑒^𝜆𝑠𝑒^−𝜆𝑠𝜀^𝑘(𝑠)𝑑𝑠≤max_{0≤𝑡≤𝑇}𝑒^{−𝜆𝑡𝑡}₀𝜙(𝑡)𝜙⁻¹(𝑠)𝐵(𝑠)𝑒^𝜆𝑠𝑒^−𝜆𝑠𝜀^𝑘(𝑠𝜀)𝑑𝑠≤𝑀^𝑘(𝑡)_𝜆,𝑇max_{0≤𝑡≤𝑇}𝑒^{−𝜆𝑡𝑡}₀𝑒^𝜆𝑠≤𝑀𝑑𝑠𝜆𝜀^𝑘(𝑡)_𝜆,𝑇, (3.25) where 𝑀=max_{0≤𝑡≤𝑇,0≤𝑠≤𝑡}{‖𝜙(𝑡)𝜙⁻¹(𝑠)𝐵(𝑠)‖}, and we can choose large enough 𝜆 such that 𝑀/𝜆<1, then the iterative error sequences {𝜀^𝑘(𝑡)} are convergent.

Acknowledgments

This work is supported by Projects from NSF of China (11126329 and 10971175), Specialized Research Fund for the Doctoral Program of Higher Education of China (20094301110001), NSF of Hunan Province (09JJ3002), and Projects from the Board of Education of Chongqing City (KJ121110).