Research Journal of Applied Sciences, Engineering and Technology 5(4): 1208-1212, 2013
ISSN: 2040-7459; e-ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: June 23, 2012
Accepted: July 28, 2012
Published: February 01, 2013
The Numerical Solution of a Class of Delay Differential Equations Boundary
Liu Shuiqiang, Zhu Hongpeng and Zeng Yuanhu
Network Information Center of Shaoyang, University Shaoyang, 422000, Hunan, China
Abstract: In this study, we propose a new method for the Cauchy solution of the Common Delay Differential
Equations 𝑥𝑥̇ (t) = ƒ(t, x(t), x(t - τ 1 (t)),…, x(t - τ m (t))). Moreover, they exist only and approximation of equations
boundary were especially proposed. From our analyze, we could get the results that Cauchy solution of the Common
Delay Differential Equations play an important role in the network information processing.
Keywords: Cauchy, common delay differential equations, equations boundary, network information processing
x (t ) = f (t , x(t ), x(t − τ 1 (t )), , x(t − τ m (t )))
INTRODUCTION
It is well known that a dynamical system is usually
expressed as the form of a calculus equation:
ẋ = ƒ(t, x), x ∈ Rn
(1)
the delay of dynamical systems is inevitable, even
through information systems based on the speed of light
is no exception. In this sense, the above equation is an
approximate description of dynamical systems, which
delay factor, is omitted. But he necessary precision will
not be reached even leads to the wrong system or do not
establish the mathematical model system, so as to use
the theories and methods of delay Differential
Equations to solve a variety of forms. Henry and
Penney (2004) study the differential equations and
boundary value problems computing and modeling.
(Guo, 2001) make a detailed research of the nonlinear
functional analysis. Shi et al. (2005) analyze the
differential equation theory and its application. Kuang
(1999) analyze the differential equation theory and its
application. Sun (1997) have a research of the
measurement of the motor winding temperature rise.
Dong (2011) analyzes the measurement and calculation
of the motor winding temperature rise.
In this study, we have a research of the numerical
solution of a class of delay differential equations
boundary. We propose a new method for the Cauchy
solution of the Common Delay Differential Equations.
Moreover, they exist only and approximation of
equations boundary were especially proposed. From our
analyze, we could get the results that Cauchy solution
of the Common Delay Differential Equations play an
important role in the network information processing.
PROBLEM FORMULATION
in this equation, τ i (t) ≥0, i = 1, 2,…, m which reflects
the lagged effect of the limited time at the state of the
current system.
Make τ(t) = max1≤i≤m τi (t), when τ (t) ≤r<∞,
Eq. (2) known as functional differential equations with
finite delay. Otherwise, known as functional differential
equations with infinite delay. The 2nd form is as
follows:
σ (t )
x (t ) =
∫
g (t , x(t ), x(t − τ ))dτ ,
σ (t ) ≥ 0.
0
(3)
We call Eq. (3) functional differential equations with
distributed delay.
When |σ (t)| ≤r<∞, functional differential Eq. (3)
known as functional differential equations with finite
delay. Otherwise, known as functional differential
equations with infinite delay.
Equation (3) reflects the state of a system affect the
current state of the system. The above two methods can
be integrated to a very general form within the
framework.
As r ∈ (0, ∞), C = C([-r, 0], Rn), ∀ φ ∈ C, If the
norm is defined as:
||φ|| =
Sup
|φ(θ) |, φ ∈ C
θ ∈ [−r, 0]
the C was given uniformly convergent topology. The
Hutchison initial time is σ ∈ R. As A ∈ (0, r) or ∞, for x
∈ C([σ - r, σ + A], Rn), and ∀ t ∈[σ, σ + A], If def.
x t = x t (θ) = x(t + θ), θ ∈ [-r, 0], then x t ∈ C. Assume
that Ω ⊂ R×C is an open set, ƒ: Ω → Rn. Then equation:
x (t ) = f (t , xt )
Common Delay Differential Equations can be
summarized in two forms; the 1st form is as follows:
(2)
(4)
called Delay Differential Equations on Ω.
Corresponding Author: Liu Shuiqiang, Network Information Center of Shaoyang, University Shaoyang, 422000, Hunan, China
1208
Res. J. Appl. Sci. Eng. Technol., 5(4): 1208-1212, 2013
−a1 x1 (t ) + b1 f1 ( x1 (t − τ 11 (t )), x2 (t − τ 12 (t ))) + I1
 x1 (t ) =

−a2 x2 (t ) + b2 f 2 ( x1 (t − τ 21 (t )), x2 (t − τ 22 (t ))) + I 2
 x2 (t ) =
(6)
Here, α i >0, b i , I i (i = 1, 2) are constant, τ ij (t) ∈
C(R, [0, ∞)) are bounded, ƒ i ∈ C(R2, R) (i, j = 1, 2),
x i (t) ∈ C1([0, ∞), R) (i = 1, 2).
THE SOLUTION EXISTENCE AND
UNIQUENESS THEOREM
Assume that 𝜎𝜎 ∈ R, 𝜑𝜑 ∈ C, Ω ⊂ R×C is an open
set, operator ƒ ∈ C(Ω, Rn), hen following conclusions
are about Cauchy problems:
Fig. 1: The solution for the delayed state Eq. (5)
Definition 1: High-level overview of a variety of
Retarded Functional Differential Equations, just select
the appropriate operator ƒ(t, φ), Eq. (1.3) can represent
a variety of Retarded Functional Differential Equations.
For example, if we can take ƒ(t, φ) = ƒ(t, φ (0), φ(τ 1 (t)), … φ(-τ m (t))). Then（4）become（2). If we
want to determine the solution of Delay Differential
Equations, only give the state of the system at a given
moment is not enough, we need to give the state of the
system at some point before this moment for some time
that is, given an initial function. So, Delayed Functional
Differential Equations Cauchy problem in the form as
follows (Fig. 1):
 x (t ) = f (t , xt )

 xσ = ϕ
(5)
where, σ ∈ R is the initial moment and φ ∈ C is the
initial function.
Here is the definition of Delayed Functional
Differential Equation:
Definition 2: Assume that function x ∈ C([σ – r, σ +
A), Rn, when t ∈ [σ, σ + A), (t, x t ) ∈ Ω ⊏ R×C.
•
•
If x(t) satisfies the Eq. (3), then X(t) is called a
solution when the Eq. (4) in the interval [σ - r, σ +
A)
If X (t) is called a solution when the Eq. (4) in the
interval [σ - r, σ + A) and x σ = φ, then x (t) is
called a solution through (σ, φ), as know as the
solution of Delayed Functional Differential
Equations Cauchy problem (5), (2) form of Cauchy
problem numerical solution method for solving,
especially working on the equation has an
important role in the network information
processing:
Lemma 1: Assume that Ω⊂R×C is an open set,
operator ƒ0∈ C (Ω, Rn), The Cauchy
Problem（5） for any (𝜎𝜎, 𝜑𝜑) ∈Ω, the
solution exists.
Lemma 2: Assume that Ω⊂R×C is an open set, W⊂Ω
is a compact set, ƒ0∈ C (Ω, Rn) given. Then
there exists a neighborhood of W V∈Ω,
Make ƒ0∈ C0 (V, Rn). Move forward a
single step, there exists a neighborhood of ƒ0
U⊂ C0 (V, Rn), for ∀ ƒ∈U, (𝜎𝜎, 𝜑𝜑) ∈Ω,
problem (5) existence of the solution x(𝜎𝜎, 𝜑𝜑,
ƒ) (t).
Lemma 3: Assume that Ω⊂R×C is an open set, ƒ∈C
(Ω, Rn) and ƒ(t, 𝜑𝜑) each compact subset in Ω
about 𝜑𝜑 Satisfies the Lipschitz condition:
P
P
P
P
f (t , ϕ ) − f (t ,ψ ) ≤ L ϕ −ψ
(7)
Here, L∈R+ is Lipschitz constant. Then problem (5)
exists a unique solution.
The following discussion based on (7) is satisfied
under the premise that the existence and uniqueness is
guaranteed. Numerical method our idea is into long
differential
equations
appropriate
qualitative
considerations. So that you can facilitate the use of
classical numerical methods have been established to
solve problems, for example, Runge-Kutta method. The
following 2 cases to consider Introduction 1:
The 1st situation:
=
f (t , x (t − τ (t )))
 x (t )

=
t ∈ [ − r , 0]
 x (t ) ϕ (t ),
(8)
Here x(t) = (x 1 (t), x 2 (t),…, x n (t))T is the ndimensional vector function:
1209
𝜑𝜑 (t) = (𝜑𝜑 1 (t), 𝜑𝜑 2 (t),…, 𝜑𝜑 n (t))T
R
R
R
Res. J. Appl. Sci. Eng. Technol., 5(4): 1208-1212, 2013
here, x 1 (t), x 2 (t) ∈ C1 ([0, ∞) , R), ƒ i ∈ C(R2, R)(i, j = 1,
2).
Set x(t) = (x 1 (t), x 2 (t))T, in the transformation
x(t) = A(t) 𝑥𝑥� (t) 𝑥𝑥� (t) = 𝑥𝑥� 1 (t), 𝑥𝑥� 2 (t))T, A(t) as function
matrix:
is the Known n-dimensional vector function:
ƒ∈C(Ω, Rn), 𝜏𝜏(t) ∈C1(R, [0, ∞))
R
monotonically decreasing and:
lim
t→∞ τ(t)
r > r0 =
 e − a1t

 0
= 0 (usually there is 𝜏𝜏(t) ~ λe-λt)
max
τ(t)
t≥0
=
x (t ) f ( x (t − τ (t )))
In this equation,
 x ( s ) = f ( s, x( s ))

 x( s ) = ϕ ( s ), s ∈ A = {s | s = t − τ (t ), t ∈ [−r , 0]}  [−r , 0]
f = ( f1 , f2 )T
f1 =−
b1 f1 ( x1 (t τ (t ))e − a1 (t −τ (t )) , x2 (t − τ (t ))e − a2 (t −τ (t )) )
𝑑𝑑𝑑𝑑
where, 𝑥𝑥̇ (S) = 𝑥𝑥̇ (t). , ̃ƒ (S, x(S)) = ƒ (t(S)). . Because
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
max
r>r 0 =
τ(t), so A ≠ ∅, as a matter of fact,
t≥0
S 0 = S(0) = -𝜏𝜏(0)∈ [-r, 0], the, (8) conversion for the
following appropriate qualitative issues:
 x ( s ) = f ( s, x ( s ))



 x ( s0 ) = ϕ ( s0 ) s0 = −τ (0)
0 

e

− a2 t
Then (11) becomes:
Let S(t) = t - 𝜏𝜏 (t), and 𝜏𝜏 (t) monotonically
decreasing makes S(t) monotonically increasing,
therefore, there is a continuous inverse function t = t(S).
Substituted into the (8) then we got:
𝑑𝑑𝑑𝑑
R
f2 =
b2 f 2 ( x1 (t − τ (t ))e − a1 (t −τ (t )) , x2 (t − τ (t ))e − a2 (t −τ (t )) )
At this point, the method described above can be
solved. The 2nd situation:
) f (t , x(t ), x(t − τ 1 (t )), x(t − τ 2 (t )), , x(t − τ m (t )))
 x (t=

=
x
(
t
 ) ϕ (t ), t ∈ [−r , 0]
(9)
at this point, we can choose the following Runge-Kutta
method of Simpson formula to solve the problem.
Here x(t) = (x 1 (t), x 2 (t),…, x n (t))T is the ndimensional vector function, 𝜑𝜑(t) = (𝜑𝜑 1 (t), 𝜑𝜑 2 (t),…,
𝜑𝜑 n (t))T is the Known n-dimensional vector function:
R
 xm +1 =xm + 6h (k1 + 2k2 + 2k3 + k4 )


k1 = f (tm , xm )

h
h
k2 = f (tm + 2 , xm + 2 k1 )

h
h
k3 = f (tm + 2 , xm + 2 k2 )

h

k4 = f (tm + h, xm + 2 k3 )

 x0 = x( s0 )
(12)
R
R
ƒ∈C(Ω, Rn), 𝜏𝜏 i (t) ∈C (R, [0, ∞))
R
(i = 1, 2, …, m) monotonically decreasing also:
lim
t→∞ τ i (t)
R
= 0, (usually there is 𝜏𝜏(t) ~ λe-λt)
max
r>r 0 = t ≥ 0, 0 ≤ i ≤ m τ i (t)
(10)
R
of course we can also use other formulas of the RungeKutta method.
Now we study (5) form of the Cauchy problem to
the following special case of solving:
 x (t ) =
− a1 x1 (t ) + b1 f1 ( x1 (t − τ (t )), x2 (t − τ (t )))
1



x
(
t
)
=
− a2 x2 (t ) + b2 f 2 ( x1 (t − τ (t )), x2 (t − τ (t )))
 2

x1 (t ) ϕ1 (t ), =
x2 (t ) ϕ 2 (t ), t ∈ [− r , 0],
r > r0 max τ (t )
=
=

t ≥0

First, we consider the case when m = 1, In this
case, (13) becomes:
=
 x (t ) f (t , x(t ), x(t − τ (t )), )

=
 x(t ) ϕ (t ), t ∈ [−r , 0]
(13)
Here x(t) = (x 1 (t), x 2 (t),…, x n (t))T is the ndimensional vector function, 𝜑𝜑 (t) = (𝜑𝜑 1 (t), 𝜑𝜑 2 (t),…,
𝜑𝜑 n (t))T is the Known n-dimensional vector function:
R
(11)
1210
R
R
Res. J. Appl. Sci. Eng. Technol., 5(4): 1208-1212, 2013
ƒ∈C(Ω, Rn), r>r 0 =
max
τ(t)
t≥0
Here x(t) = (x 1 (t), x 2 (t),…, x n (t))T is ndimensional vector function:
τ(t) ∈C(R, [0, ∞)) monotonically decreasing also:
lim
t→∞ τ(t)
𝜑𝜑 (t) = (𝜑𝜑 1 (t), 𝜑𝜑 2 (t),…, 𝜑𝜑 n (t))T
R
= 0, (usually get 𝜏𝜏(t) ~ λe-λt)
Numerical solution in the form given below using
the Runge-Kutta method for solving the above
problems.
Make (a, b, c) to a given Runge-Kutta method,
which i.e., a = (a ij ) s×s , b = (b 1 , b 2 ,… b s )T, c = (c 1 , c 2 ,…
c s )T, ∑𝑠𝑠𝑖𝑖=1 𝑏𝑏𝑖𝑖 = 1, ci ∈ [0, 1](1≤i ≤s). The specific form
of the corresponding Runge-Kutta method as follows:
s

xn −1 + h∑ aij f (tn −1 + c j h, X j )
Xi =
j =1


s
x =
xn −1 + h∑ b j f (tn −1 + c j h, X j )
 i
j =1
R
𝜏𝜏 i (t) ∈ C(R, [0, ∞))
R
(i = 1, 2,…, m) monotonically decreasing and
lim
t→∞ τ i (t)
R
(usually get 𝜏𝜏 i (t) ~ λe-λt)
R
s

(n)
xn −1 + h∑ aij f (tn −1 + c j h, X j , Z1( nj ) , Z 2( nj ) , , Z mj
)
Xi =
j =1


s
(n)
x =
xn −1 + h∑ b j f (tn −1 + c j h, X j , Z1( nj ) , Z 2( nj ) , , Z mj
)
i

j =1

(14)
(19)
(i = 1, 2, , s)
x n ~ x(t n ), 𝑍𝑍𝑖𝑖𝑖𝑖𝑛𝑛
(i = 1, 2, , s )
(18)
̃ x(t n + c j h - 𝜏𝜏 i (t n + c j h))
R
when t n + c j h - 𝜏𝜏 i (t n + c j h) ≤0, we get:
R
(15)
Z ij( n ) = ϕ (tn + c j h − τ i (tn + c j h))(i = 1, 2, , m)
On the time delay options x(t - 𝜏𝜏(t)) processing is
the key to solving problems. If you use the symbol “~”
exact and numerical solutions of the approximate
relations, then x n~x(t n ), 𝑍𝑍𝑗𝑗𝑛𝑛 ̃ x(t n + c j h - 𝜏𝜏(t n + c j h)) It
said the n-th iteration approximation Delay Term which
t n = t 0 + n. h, t 0 = 0 especially, if 𝜏𝜏(t) is a constant,
when we take a step stance: h = 𝜏𝜏/k(where k is a
positive integer), get t n + c j h - 𝜏𝜏 = t n-k + c j h, then 𝑍𝑍𝑗𝑗𝑛𝑛 ̃
x(t n-k + c j h).
when t n + c j h - 𝜏𝜏(t n + c j h) ≤0, we get:
(16)
(20)
Parametric c i , b i , a ij can be expressed as the
Butcher array form, is:
c1
a11  a1s


cs
as1  ass
b1  bs

Determine the coefficients implicit Runge-Kutta
formula, follow these steps:
•
(13) and (14) as a whole constitute solving (11) RungeKutta method. Matlab or C++ programming for solving
the above numeric format can be.
For the case when m≥2. At this point, (10) becomes:
) f (t , x(t ), x(t − τ 1 (t )), x(t − τ 2 (t )), , x(t − τ m (t )))
 x (t=

=
 x(t ) ϕ (t ), t ∈ [−r , 0]
= 0 (i = 1, 2,…, m)
At this point, you can get the corresponding
Runge-Kutta iteration scheme as follows:
(i = 1, 2, , s )
Z (j n ) = ϕ (tn + c j h − τ (tn + c j h))
R
is the Known n-dimensional, ƒ∈C(Ω, Rn)
max
r>r 0 = t ≥ 0, 1 ≤ i ≤ m τ i (t)
Thus, the specific form of the corresponding
Runge-Kutta method in Delay Differential:
s

xn −1 + h∑ aij f (tn −1 + c j h, X j , Z (j n ) )
Xi =
j =1


s
x =
xn −1 + h∑ b j f (tn −1 + c j h, X j , Z (j n ) )
 i
j =1
R
•
•
(17)
1211
Find the S-order polynomial S-zeros c 1 , c 2 ,… c s of
P s (2c - 1) (This can look-up table), here P s (x) is
the s-order Legendre polynomial
For each i(1≤i≤s), Solution of equations
∑𝑠𝑠𝑗𝑗−1 𝑎𝑎𝑖𝑖𝑖𝑖 𝑐𝑐𝑗𝑗𝑘𝑘−1 = 1/k 𝑐𝑐𝑖𝑖𝑘𝑘 , k = 1, 2, …, s
From the current equations ∑𝑠𝑠𝑗𝑗−1 𝑏𝑏𝑗𝑗 𝑐𝑐𝑗𝑗𝑘𝑘−1 = 1/k,
k = 1, 2, …, s
Determine the elements of {b i } is here c 1 ≠ 0.
Res. J. Appl. Sci. Eng. Technol., 5(4): 1208-1212, 2013
Commonly used formula for the Butcher is:
5 − 15
10
5
36
10 + 3 15
72
25 + 6 15
180
1
2
5 + 15
10
5
18
10 − 3 15
45
2
9
f2 =
−4 x2 (t ) +
25 − 6 15
180
10 − 3 15
72
10 + 3 15
45
5
36
4
9
5
18
1
f 2 ( x1 (t − τ 21 (t )), x2 (t − τ 22 (t ))) + 1
5
Based on the above analysis, we can use the
following format to solve the problem:
s

xn −1 + h∑ aij f (tn −1 + c j h, X j , Z1( nj ) , Z 2( nj ) , , Z mj( n ) )
Xi =
j =1


s
x =
xn −1 + h∑ b j f (tn −1 + c j h, X j , Z1( nj ) , Z 2( nj ) , , Z mj( n ) )
 i
j =1
NUMERICAL EXAMPLE
(i = 1, 2, , s )
x n~x(t n ), 𝑍𝑍𝑖𝑖𝑖𝑖𝑛𝑛 ̃ x(t n + c j h - 𝜏𝜏 i (t n + c j h))
R
Now we work on the introduction of (5).
Specifically consider the problems:
x1 (t ) =
−4 x1 (t ) +
1
f1 ( x1 (t − τ 11 (t )), x2 (t − τ 12 (t ))) + 2
20
x2 (t ) =
−4 x2 (t ) +
1
f 2 ( x1 (t − τ 21 (t )), x2 (t − τ 22 (t ))) + 1
5
CONCLUSION
The Cauchy solution was given a new method to
the key, especially science and numeral of the exist and
only and approximation and equations boundary was
belong to itself, which has an important role in the
network information processing.
1
1
1
1
f1 ( x1 ,=
x2 ) cos( x1 ) + x1 + sin( x2 ) − x2 + 1
4
8
3
6
REFERENCES
1
1
1
1
x1 ) − x1 + cos( x2 ) + x2 + 2
2
6
4
8
(i, j = 1, 2) and λ 11 = 0.8, λ 12 = 1.3,
𝜏𝜏 ij (t) =
λ 21 = 4.7, λ 22 = 1.0.
R
λ ij e-λ ij t).
Can get r 0 =
max
τ (t) = 4.7
t ≥ 0 ij
R
Known initial conditions as follows:
T
( x=
(sin t , cos t )T , t ∈ [ −5, 0]
1 (t ), x2 (t ))
Make 𝑥𝑥̇ (t) = (x 1 (t), x 2 (t))T,
f (t , x(t ), x(t − τ 11 (t )), x(t − τ 12 (t )), x(t − τ 21 (t )), x(t − τ 22 (t ))) =
( f1 , f 2 )T
here,
f1 =
−4 x1 (t ) +
1
f1 ( x1 (t − τ 11 (t )), x2 (t − τ 12 (t ))) + 2
20
R
Z ij( n ) = ϕ (tn + c j h − τ i (tn + c j h))(i = 1, 2, , m)
here,
f 2 ( x1 , x=
sin(
2)
When t n + c j h - 𝜏𝜏 i (t n + c j h) ≤0, we get:
Dong, G., 2011. Measurement and calculation of the
motor winding temperature rise. Small Spec. Ele.
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Guo, D., 2001. Nonlinear Functional Analysis [M]. 2nd
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