Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 241984, 11 pages
doi:10.1155/2012/241984
Research Article
On the Convergence of Continuous-Time
Waveform Relaxation Methods for Singular
Perturbation Initial Value Problems
Yongxiang Zhao1, 2 and Li Li1
1
School of Mathematics and Statistics, Chongqing Three Gorges University,
Wanzhou 404000, China
2
Hunan Key Laboratory for Computation and Simulation in Science and Engineering,
School of Mathematics and Computational Science, Xiangtan University, Hunan 411105,
Xiangtan, China
Correspondence should be addressed to Yongxiang Zhao, zyxlily80@126.com
Received 26 April 2012; Revised 24 June 2012; Accepted 12 July 2012
Academic Editor: Jesus Vigo-Aguiar
Copyright q 2012 Y. Zhao and L. Li. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
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 multiplying the highest derivatives. Since the classical Lipschitz constant and onesided Lipschitz constant are generally of size O−1 0 < 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
2
Journal of Applied Mathematics
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:
xT maxx,
1.1
xλ,T max e−λt x ,
1.2
0≤t≤T
and the λ norm of exponential type:
0≤t≤T
where λ is any given positive number and · denotes an any given norm in Cn .
2. Convergence Analysis of the First Type
2.1. Linear SPPs
Consider the following linear singular perturbation initial value problem
dyt
dt
Ayt bt,
t ∈ 0, T , 0 < 1,
y0 y0 ,
2.1
where the constant matrix A ∈ Rn×n and the input function bt ∈ Rn , y0 is the given initial
value, and is the singular perturbation parameter. The constant matrix A is split by A A1 − A2 , then the system 2.1 can be written as
dyt
dt
A1 yt A2 yt
bt,
t ∈ 0, T , 0 < 1,
y0 y0 ,
2.2
then we can obtain the following iterate scheme:
dyk 1 t
dt
1
A1 yk
1
1
bt
A2 yk t
, t ∈ 0, T , 0 < 1,
1
0 y0 , k 1, 2, . . . ,
t yk
2.3
here we can choose the initial iterative function y0 t y0 , the above iteration is called a
continuous-time waveform relaxation process.
Journal of Applied Mathematics
3
For any fixed k ≥ 0, from 2.3, we have, upon premultiplying by es−tA1 and integrating from 0 to t, as following:
yk
1
t
t e−A1 t/ y0
es−tA1 /
0
A2 k
y s
bs
ds.
2.4
Let
Ry t t
es−tA1 /
0
A2
ysds,
2.5
then 2.5 can be written as
yk
1
t Ryk t
2.6
ϕt,
t
es−tA1 / bs/ds. It is easy to see that R is a Volterra conwhere ϕt e−A1 t/ y0
0
volution operator with the kernel function κt e−A1 t/ A2 /:
Ry t κt ∗ yt t
2.7
κs − tysds,
0
and R is the waveform relaxation operator.
Theorem 2.1. Let the waveform relaxation operator R be defined in C0, T , Rn . If the kernel
function κt is continuous in 0, T and satisfies κT ≤ C, where C is a constant, then the sequence
of functions yk t defined by 2.3 satisfy
yk t − y∗ t
T
≤
CT k 0
y t − y∗ t
k!
T
,
2.8
where y∗ t is the exact solution of system 2.1.
Proof. By the norm · T in C0, T , we can obtain
R
k
T
≤
T
κk t ds,
k 1, 2, . . . ,
2.9
0
where κ1 t κt, κk t κt ∗ κk−1 t.
In fact, from the given condition κT ≤ C, for any t ≤ T , we have
κ t ≤ C
k
t
0
κk−1 t ds,
2.10
4
Journal of Applied Mathematics
it can be obtained, by induction, that
κk t ≤ C
Ctk−1
,
k − 1!
2.11
so
Rk
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:
dyt
f yt, t , t ∈ 0, T , 0 < 1,
dt
y0 y0 ,
2.13
where y0 is the given initial value, is the singular perturbation parameter. f : Rn × 0, T →
Rn is given continuous function mapping, and yt is unknown.
The continuous-time Waveform Relaxation algorithm for 2.13 is
dyk 1 t 1 k 1
F y
t, yk t, t , t ∈ 0, T , 0 < 1,
dt
yk 1 0 y0 , k 1, 2, . . . ,
2.14
where the splitting function Fu, v, t determines the type of the Waveform Relaxation algorithm, and we assume that Fu, v, t satisfy the following Lipschitz condition
Fu1 , v1 , t − Fu2 , v2 , t ≤ L1 u1 − u2 L2 v1 − v2 .
2.15
By integrating the inequality 2.15 of both side from 0 to t, we have
yk
1
t y0
t
0
1
F yk
1
s, yk s, s ds.
2.16
Let yt denote the function that iterated by yt from one iteration step, like 2.16,
denote yt Ryt.
Journal of Applied Mathematics
5
Theorem 2.2. Assume that the splitting function Fu, 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 zt, and denote zt Rzt, then
t
1
yt − zt 1
≤
F ys, ys, s − Fzs, zs, s ds
0
t
2.17
F ys, ys, s − Fzs, zs, s ds.
0
Equations 2.15–2.17 yield
yt − zt ≤
L1
t
yt − zs ds
0
L2
t
ys − zs ds.
2.18
0
From 2.18, we have, upon premultiplying by e−λt , the following:
e
−λt
L1 −λt
yt − zt ≤
e
t
ys − zs ds
0
L2 −λt
e
t
ys − zs ds
0
L2 −λt t λs −λs
L1 −λt t λs −λs
ys − zs ds
ys − zs ds
e
e
e e
e e
0
0
t
L
L1
2
−λs
−λs
−λt
e
≤
ys − zs
ys − zs
max e
max e
eλs ds,
0≤s≤t
0≤s≤t
0
2.19
≤
because of e−λt
have
t
0
eλs ds ≤ 1/λ, and from the definition of λ norm of the exponential type, we
yt − zt
λ,T
≤
L1
ys − zs
λ
λ,T
L2
ys − zs
λ
λ,T
.
2.20
It is easy to obtain
yt − zt
λ,T
≤
L2
ys − zs
λ − L1
λ,T
.
2.21
We can choose large enough λ such that γ L2 /λ − 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.
6
Journal of Applied Mathematics
3. Convergence Analysis of the Second Type
3.1. Linear SPPs
Consider the following linear singular perturbation initial value problem
x t
Axt
dyt
dt
Cxt
Byt rt,
t ∈ 0, T ,
Dyt st,
x0 x0 ,
3.1
0 < 1,
y0 y0 ,
where x0 and y0 are the given initial value, is the singular perturbation parameter, rt ∈ Rn1
and st ∈ Rn2 are given functions. The constant matrices A ∈ Rn1 ×n1 , B ∈ Rn1 ×n2 , C ∈ Rn2 ×n1 ,
D ∈ Rn2 ×n2 are split by A A1 − A2 , B B1 − B2 , C C1 − C2 , D D1 − D2 respectively,
xt ∈ Rn1 and yt ∈ Rn2 are unknowns. Then the system 3.1 can be written as
x t
A1 xt
B1 yt A2 xt
dyt
dt
C1 xt
D1 yt C2 xt
x0 x0 ,
B2 yt
D2 yt
rt,
st,
t ∈ 0, T ,
0 < 1,
3.2
y0 y0 .
The continuous-time Waveform Relaxation algorithm for 3.1 is as follows:
dxk 1 t
A1 xk 1 t B1 yk 1 t A2 xk t B2 yk t rt, t ∈ 0, T ,
dt
dyk 1 t C1 k 1
D1 k 1
C2 k
D2 k
1
x
y
x t
x t
st, 0 < 1,
t
t dt
xk 1 0 x0 , yk 1 0 y0 , k 1, 2, . . . ,
3.3
The matrix form of 3.3 reads
d
dt
xk
1
t
yk
1
t
⎞
A1 B1
xk
⎝ C1 D1 ⎠
yk
⎛
1
t
1
t
⎞
A2 B2
xk t
⎠
⎝
C2 D2
yk t
⎛
⎛
⎞
rt
⎝ st ⎠.
3.4
Solve the equations 3.4, we can derive
xk
1
t
yk
1
t
⎞ ⎞
A1 B1
xk 1 0
⎠
⎠
⎝
⎝
exp − C1 D1 t
yk 1 0
⎞⎛⎛
⎞
⎞
⎛⎛
t
A2 B 2
A1 B1
xk s
⎠
⎝
⎝
⎠
⎠
⎝
⎝
exp
C1 D1 s − t
C2 D2
yk s
0
⎛ ⎛
⎛
⎞⎞
rs
⎝ ss ⎠⎠ds.
3.5
Journal of Applied Mathematics
7
Denote εxk t xk t − xt, εyk t yk t − yt, where xt and yt are the exact solutions of 3.1. From 3.2 and 3.5, we can obtain
εxk 1 t
εyk 1 t
⎞⎛
⎞
⎞
A2 B 2
A1 B1
εxk s
⎠
⎝
⎠
⎠
⎝
⎝
exp
ds,
C1 D1 s − t
C2 D2
εyk s
0
⎛⎛
t
3.6
then 3.6 can be written as
εxk 1 t
εyk 1 t
R
εxk s
εyk s
t,
3.7
clearly, R is a Volterra convolution operator with the kernel function
⎞ ⎞⎛
⎞
A2 B2
A1 B1
κt exp⎝−⎝ C1 D1 ⎠t⎠⎝ C2 D2 ⎠,
t
Ry t κt ∗ yt κs − tysds,
⎛ ⎛
3.8
0
R is the Waveform Relaxation operator.
Theorem 3.1. Let the waveform relaxation operator R be defined in C0, T , Rn . If the kernel
function κt is continuous in 0, T and satisfies κT ≤ M, where M is a constant, then the sequence
of functions εxk 1 t, εyk 1 tT defined by 3.6 satisfy
⎞
⎛
max εx0 s
k
T
⎠.
⎠≤
⎝
Mk ⎝0<s<T 0
k
k!
max εy s
εy t
0<s<T
⎛
εxk t
⎞
3.9
Proof. Taking the norm in both side of 3.6 which reads
⎛
⎞
⎛ k
⎞
εx s
A2 B2
A1 B 1
⎝
⎠ ≤ exp⎝ C1 D1 s − t⎠ C2 D2 ⎝
⎠ds.
k
0
εyk 1 t
ε
s
y
εxk 1 t
⎞
⎛
t
3.10
From the induction of 3.10 and the condition κT ≤ M, we can derive
⎛
⎝
εx1 t
εy1 t
⎞
⎛
max εx0 s
⎞
⎠ ≤ M⎝0<s<T
⎠t.
max εy0 s
0<s<T
3.11
8
Journal of Applied Mathematics
Moreover, we can derive
⎞
⎞
⎛
max εx0 s
k
T
0<s<T
k
⎠≤
⎠,
⎝
M ⎝
k!
max εy0 s
εyk t
0<s<T
⎛
εxk t
3.12
thus, εxk t, εyk tT → 0 as k → ∞ which complete the proof.
3.2. Nonlinear SPPs
Consider the following nonlinear singular perturbation initial value problem:
x t f xt, yt, t ,
t ∈ 0, T ,
dyt
g xt, yt, t , 0 < 1,
dt
x0 x0 ,
y0 y0 ,
3.13
where x0 and y0 are given initial values, is the singular perturbation parameter, f : Rn1 ×
Rn2 × 0, T → Rn1 and g : Rn1 × Rn2 × 0, T → Rn2 are given continuous function mappings.
The continuous-time waveform relaxation algorithm for 3.13 is
xk
1
t F xk
1
t, xk t, yk
1
t, yk t, t ,
t ∈ 0, T ,
dyk 1 t 1 k 1
G x
t, xk t, yk 1 t, yk t, t , 0 < 1,
dt
k 1
x
yk 1 0 y0 , k 1, 2, . . . ,
0 x0 ,
3.14
where the splitting functions Fu1 , u2 , u3 , u4 , t and Gu1 , u2 , u3 , u4 , t determine the type of
the waveform relaxation algorithm. we can derive from 3.14 that
xk t F xk t, xk−1 t, yk t, yk−1 t, t ,
t ∈ 0, T ,
dyk t 1 k
G x t, xk−1 t, yk t, yk−1 t, t , 0 < 1,
dt
xk 0 x0 , yk 0 y0 , k 1, 2, . . . .
3.15
Denote
εxk 1 t xk 1 t − xk t,
εyk 1 t yk 1 t − yk t.
3.16
Theorem 3.2. Assume that the matrices ∂F/∂ui and ∂G/∂ui i 1, 2, 3, 4 of the splitting functions Fu1 , u2 , u3 , u4 , t and Gu1 , u2 , u3 , u4 , t are continuous, then the continuous-time waveform
relaxation algorithm 3.14 is convergent.
Journal of Applied Mathematics
9
Proof. Subtracting 3.14 from 3.15, we have
dεxk 1 t
∂F k 1
∂F k
∂F k 1
∂F k
εx t
εx t
εy t
ε t,
dt
∂u1
∂u2
∂u3
∂u4 y
dεyk 1 t 1 ∂G k 1
∂G k
∂G k 1
∂G k
εx t
εx t
εy t
εy t ,
dt
∂u1
∂u2
∂u3
∂u4
3.17
the matrix form of 3.17 reads
d
dt
εxk 1 t
εyk 1 t
⎛ ∂F
∂F ⎞
k 1 ⎜ ∂u1
∂u3 ⎟
⎟ εx t
⎜
⎜
⎟
⎝ 1 ∂G 1 ∂G ⎠ εyk 1 t
∂u1 ∂u3
⎛ ∂F
∂F ⎞
k ⎜ ∂u2
∂u4 ⎟
⎟ εx t
⎜
.
⎟
⎜
⎝ 1 ∂G 1 ∂G ⎠ εyk t
∂u2 ∂u4
3.18
Denote
εk t εxk t, εyk t
T
,
⎛ ∂F
∂F ⎞
⎜ ∂u1
∂u3 ⎟
⎟
⎜
At ⎜
⎟,
⎝ 1 ∂G 1 ∂G ⎠
∂u1 ∂u3
⎛ ∂F
∂F ⎞
⎜ ∂u2
∂u4 ⎟
⎜
⎟
Bt ⎜
⎟.
⎝ 1 ∂G 1 ∂G ⎠
∂u2 ∂u4
3.19
Then, we can derive
dεk 1 t
Atεk 1 t
dt
Btεk t.
3.20
Assume that the basis matrix φt satisfies
dφt
Atφt,
dt
3.21
then the solution of 3.20 can be written as
ε
k 1
t φt
t
0
φ−1 sBsεk sds.
3.22
10
Journal of Applied Mathematics
From 3.22, we have, upon taking the norm in both side and premultiplying by e−λt , that
e
−λt
ε
k 1
t e
t
−λt
φtφ−1 sBsεk sds ,
3.23
0
furthermore,
max e
−λt
0≤t≤T
ε
k 1
t
max e
−λt
t
0≤t≤T
−1
φtφ sBsε sds
k
,
3.24
0
so
ε
k 1
t
λ,T
≤ max e
−λt
t
0≤t≤T
−1
λs −λs k
−1
λs −λs
φtφ sBse e
≤ max e
−λt
0≤t≤T
t
φtφ sBs e e
0
≤ M εk t
≤
ε sds
0
M k
ε t
λ
max e−λt
λ,T 0≤t≤T
λ,T
t
ε s ds
k
3.25
eλs ds
0
,
where M max0≤t≤T, 0≤s≤t {φtφ−1 sBs}, and we can choose large enough λ such that
M/λ < 1, then the iterative error sequences {εk 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.
References
1 E. M. Dejager and F. R. Jiang, The Theory of Singular Perturbation, Elservier Science B.V., Amsterdam,
The Netherlands, 1996.
2 R. E. O’Malley Jr., Singular Perturbation Methods for Ordinary Differential Equations, Springer, New York,
NY, USA, 1990.
3 E. Lelarasmee, A. E. Ruehli, and A. L. Sangiovanni-Vincentelli, “The waveform relaxation method for
time-domain analysis of large scale integrated circuits,” IEEE Transactions, vol. 1, pp. 131–145, 1982.
4 A. Bellen, Z. Jackiewicz, and M. Zennaro, “Time-point relaxation Runge-Kutta methods for ordinary
differential equations,” Journal of Computational and Applied Mathematics, vol. 45, no. 1-2, pp. 121–137,
1993.
5 M. R. Crisci, N. Ferraro, and E. Russo, “Convergence results for continuous-time waveform methods
for Volterra integral equations,” Journal of Computational and Applied Mathematics, vol. 71, no. 1, pp.
33–45, 1996.
Journal of Applied Mathematics
11
6 M. L. Crow and M. D. Ilić, “The waveform relaxation method for systems of differential/algebraic
equations,” Mathematical and Computer Modelling, vol. 19, no. 12, pp. 67–84, 1994.
7 U. Miekkala, “Remarks on waveform relaxation method with overlapping splittings,” Journal of
Computational and Applied Mathematics, vol. 88, no. 2, pp. 349–361, 1998.
8 K. J. In’t Hout, “On the convergence of waveform relaxation methods for stiff nonlinear ordinary
differential equations,” Applied Numerical Mathematics, vol. 18, no. 1–3, pp. 175–190, 1995.
9 Y. L. Jiang, Waveform Relaxation Methods, Science Press, Beijing, China, 2009.
10 S. Natesan, J. Jayakumar, and J. Vigo-Aguiar, “Parameter uniform numerical method for singularly
perturbed turning point problems exhibiting boundary layers,” Journal of Computational and Applied
Mathematics, vol. 158, no. 1, pp. 121–134, 2003.
11 S. Natesan, J. Vigo-Aguiar, and N. Ramanujam, “A numerical algorithm for singular perturbation
problems exhibiting weak boundary layers,” Computers & Mathematics with Applications, vol. 45, no.
1–3, pp. 469–479, 2003.
12 J. Vigo-Aguiar and S. Natesan, “A parallel boundary value technique for singularly perturbed twopoint boundary value problems,” The Journal of Supercomputing, vol. 27, pp. 195–206, 2004.
13 J. Vigo-Aguiar and S. Natesan, “An efficient numerical method for singular perturbation problems,”
Journal of Computational and Applied Mathematics, vol. 192, no. 1, pp. 132–141, 2006.
14 J. Vigo-Aguiar and H. Ramos, “Dissipative Chebyshev exponential-fitted methods for numerical
solution of second-order differential equations,” Journal of Computational and Applied Mathematics, vol.
158, no. 1, pp. 187–211, 2003.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014