International Mathematical Forum, Vol. 7, 2012, no. 46, 2279 - 2292
Solution of Delay Differential Equations Using Nonic Spline Collocation Methods Shahid S. Siddiqi, Ghazala Akram and Huzaima Baig Department of Mathematics University of the Punjab, Lahore, Pakistan
[email protected] [email protected] [email protected] Abstract This paper provides a collocation scheme based on nonic C 4 -splines with five collocation points for the numerical solution of delay differential equation. It is also shown that the method is convergent and order of convergence is at least nine. For the numerical illustration three examples are presented. The method is also compared with the method given by Hwary and Mahmoud [6]. This comparison shows that the presented method is more accurate than the previous one.
Mathematics Subject Classification: 65L10 Keywords: Spline Collocation Methods; Delay Differential Equations; Stiff problems; Approximate solution
1
Introduction
The basic motivation of this paper is to solve delay differential equations using spline collocation method. The delay differential equations, which have some arguments with time-lags, are now widely in use in many areas of engineering, economics and biology. It is to be mentioned that a single first order delay differential equation may be defined as y (1)(x) = f (x, y(x), y(α(x))), x ∈ [a, b]
(1)
where f ∈ C 9 ([a, b] × R × R), is Lipschitiz continuous with respect to y. The function α(x) ≤ x, x ∈ [a, b] is the delay function, while the initial condition is given by y(x) = g(x), x ∈ [˜ a, a]
(2)
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Shahid S. Siddiqi, Gh. Akram and H. Baig
for a given function g(x), where a˜ = inf [(α(x))]. Delay differential equation arises in the situations when some hereditary function appears in the ordinary differential equation. In many phenomena of real world, the initial conditions or boundary conditions are not enough to predict the future behavior of the function, to deal with such complexities, it is necessary to have some knowledge of the previous behavior of the function. Banks and Kappel [3] developed a new framework to approximate the solution of delay systems. They presented third and fifth order spline based scheme as special cases of these frameworks. Oberle and Pesch [18] constructed Runge-Kutta-Fehlberg based methods for the numerical treatment of delay differential equation. In the methods, multipoint Hermite interpolation was used to approximate the retarded arguments. In the same year, Kappel and Kunisch [9] approximated the delay differential equation by a sequence of higher order ordinary differential equation. This scheme allows the use of spline functions to approximate the solution of delay differential equation. Kappel and Ito [16] used spline based scheme within the framework of semigroups on certain Hilbert space formula to solve differential equations with delays. The convergence of the scheme was presented in state spaces. The approximating scheme is uniformly differentiable. Akca et al. [1] used a collocation procedure, with polynomial spline of degree m ≥ 3 and continuity class C m−2 , to find the numerical solution of second order neutral delay differential equation. In the same year Micula and Acka [2] developed some numerical algorithms, based on natural spline functions of even degree, to solve delay differential equations. These spline functions interpolate the given derivatives at simple knots. It was shown that these methods are very suitable for numerical treatment of DDEs with initial conditions. Engelborghs et al. [7] presented three collocation methods based on piecewise polynomials. They found periodic solution of periodic two point boundary value problem of autonomous delay differential equations. In the same year, Huang [15] presented Runge-Kutta methods for the solution of system of differential equations with delays. Hawary and Mahmoud [5] developed four point spline collocation methods for solving delay differential equations, including stiff differential equations. They also analyzed the stability regions of these methods. It was also shown that the methods are consistent of order seven. They further showed that these methods could be applied successfully to ordinary initial value problem and to higher index differential algebraic equations. Calio and Marchetti [8], applied a particular collocation method, which approximates the unknown function by deficient spline function, to differential and integral equations with delays. They also studied existence and unique-
Solution of DDEs using nonic spline method
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ness of the numerical solution, convergence properties of the method and some features related to the estimation of error. Berezansky [4] discussed some properties of the solution of scalar nonlinear delay differential equations. These properties are related to sufficient conditions for oscillation of solutions and for asymptotical stability of positive equilibrium. Hawary and Mahmoud [6] developed three point spline collocation methods for solving delay differential equations, including stiff differential equations. Stability regions of these methods are also analyzed. The proposed methods were convergent of order five. However, for some spacial cases the methods are convergent of order six. Stability, of multistep formulas, based on linear delay differential equations y (x) = λy(x) + qy(x − τ ), x ≥ 0 y(x) = g(x), x ≤ 0
(3)
where λ, q ∈ C, τ ≥ 0, and g(x) is initial function, are studied in [11, 12, 13, 14, 19]. The paper is organized in four sections. Introduction of the delay differential equations is presented in Section 1. Section 2, gives a precise description of the method for the solution of delay differential equation. Error analysis and order of convergence is discussed in Section 3. In this section it is also shown that the order of convergence is O(h9). Numerical examples are provided in Section 4 along with their errors and graphical illustration.
2
Description of the Method
Following is the single first order delay differential equation as considered in (1) y (1) (x) = f (x, y(x), y(α(x))), x ∈ [a, b] y(x) = g(x) for a ˜ ≤ x < a.
(4) (5)
The presented method uses five collocation points xi+cj = xi + cj h, j = 1, 2, . . . , 5 such that 0 < ci < cj ≤ 1 ∀i < j; i, j = 1, 2, . . . , 5, c5 = 1.
(6)
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In each subinterval Ii = [xi , xi+1 ], i = 0, 1, . . . , n − 1 of [a, b], the step size for each interval is taken as h = (b−a) . Moreover x0 = a, xn = b and xi = n a + ih, i = 0, 1, . . . , n are the grid points of the uniform partition of [a, b]. The nonic C 4 −spline function can be represented by (0)
S(x) = ξ 5 [(70ξ 4 + 35ξ 3 + 15ξ 2 + 5ξ + 1)Si 15 5 1 (1) (2) +(35ξ 4 + 15ξ 3 + 5ξ 2 + ξ)Si + ( ξ 4 + ξ 3 + ξ 2 )Si 2 2 2 5 4 1 3 (3) 1 4 (4) +( ξ + ξ )Si + ( ξ )Si ] 6 6 24 (0) 5 4 3 +ξ [(70ξ + 35ξ + 15ξ 2 + 5ξ + 1)Si+1 15 5 1 (1) (2) −(35ξ 4 + 15ξ 3 + 5ξ 2 + ξ )Si+1 + ( ξ 4 + ξ 3 + ξ 2 )Si+1 2 2 2 5 4 1 3 (3) 1 4 (4) +( ξ + ξ )Si+1 + ( ξ )Si+1 ] (7) 6 6 24 where ξ=
x − xi ∈ [0, 1], ξ = 1 − ξ, for x ∈ [xi , xi+1 ], h
and (0)
(1)
(2)
= hS (xi ), Si
Si
= S(xi ), Si
Si
= h4 S iv (xi ), i = 0, 1, . . . n.
(4)
(3)
= h2 S (xi ), Si
= h3 S (xi ), (8)
From equation (7), it follows that (0)
(1)
hS (x) = ξ 4 [(−630ξ 4 )Si + (−315ξ 4 + 20ξ 3 + 10ξ 2 + 4ξ + 1)Si −135 4 −15 4 1 (2) (3) ξ + 10ξ 3 + 4ξ 2 + ξ)Si + ( ξ + 2ξ 3 + ξ 2 )Si +( 2 2 2 −3 1 (4) +( ξ 4 + ξ 3 )Si ] 8 6 (0) (1) +ξ 4 [(630ξ 4)Si+1 + (−315ξ 4 + 20ξ 3 + 10ξ 2 + 4ξ + 1)Si+1 −135 4 −15 4 1 (2) (3) ξ + 10ξ 3 + 4ξ 2 + ξ )Si+1 + ( ξ + 2ξ 3 + ξ 2 )Si+1 −( 2 2 2 −3 4 1 3 (4) −( ξ + ξ )Si+1 ]. (9) 8 6 The nonic spline S is determined such that, in each interval Ii = [xi , xi+1 ], the following five collocation conditions S (xi+cj ) = f (xi+cj , S(xi+cj ), S(α(xi+cj ))), j = 1, 2, . . . , 5
(10)
hold. Denoting fi+φ = f (xi+φ , S(xi+φ ), S(α(xi+φ ))), 0 ≤ φ ≤ 1 and cj = 1 − cj , (9)
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Solution of DDEs using nonic spline method
can be rewritten as (0)
(1)
4 3 2 c4j [(630c4 j )Si+1 + (−315cj + 20cj + 10cj + 4cj + 1)Si+1 135 4 15 4 1 2 (3) (2) 2 cj + 10c3 cj + 2c3 −(− j + 4cj + cj )Si+1 + (− j + cj )Si+1 2 2 2 3 4 1 3 (4) −(− cj + cj )Si+1 ] 8 6 (0) (1) 4 4 = cj [(630cj )Si − (−315c4j + 20c3j + 10c2j + 4cj + 1)Si 135 4 15 1 (2) (3) −(− cj + 10c3j + 4c2j + cj )Si − (− c4j + 2c3j + c2j )Si 2 2 2 3 1 (4) −(− c4j + c3j )Si ] + hfi+cj . (11) 8 6 (1)
Substituting Si
(1)
= hfi and Si+1 = hfi+1 , it follows from equation (11) that S i+1 = AS i + hBf i , (1) Si = hfi , i = 0, 1, . . . , n − 1
where
(12)
A = A˜−1 D,
and
B = A˜−1 H,
with ⎡ ⎢ ⎢
⎢ A˜ = ⎢ ⎢ ⎣
⎡
D=
⎡
H=
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
3 28 3 28 3 28 3 28
1 1 1 1
⎢ ⎢ ⎢ ⎢ ⎢ ⎣
1 1 1 1
− − − −
3 28 3 28 3 28 3 28
1 63c1 1 63c2 1 63c3 1 63c4
− − − −
− − − −
1 63c1 1 63c2 1 63c3 1 63c4
2 315c2 1 2 315c2 2 2 315c2 3 2 315c2 4
− − − −
− − − −
2 315c21 2 315c22 2 315c23 2 315c24
315c41 −20c31 −10c21 −4c1 −1 630c41 315c42 −20c32 −10c22 −4c2 −1 630c42 315c43 −20c33 −10c23 −4c3 −1 630c43 315c44 −20c34 −10c24 −4c4 −1 630c44
1 630c3 1 1 630c3 2 1 630c3 3 1 630c3 4
− − − −
1 630c31 1 630c32 1 630c33 1 630c34
1 630c41 c4 1
1 − 84 1 − 84 1 − 84 1 − 84
1 84 1 84 1 84 1 84
+ + + +
− − − −
1 315c1 1 315c2 1 315c3 1 315c4
1 315c1 1 315c2 1 315c3 1 315c4
+ + + +
− − − −
1 1260c2 1 1 1260c2 2 1 1260c2 3 1 1260c2 4
1 1260c21 1 1260c22 1 1260c23 1 1260c24
1 1680 1 1680 1 1680 1 1680
1 1680 1 1680 1 1680 1 1680
0
0
0
0
1 630c42 c4 2
0
0
0
0
1 630c43 c4 3
0
0
0
0
1 630c44 c4 4
− − − −
− − − −
1 3780c1 1 3780c2 1 3780c3 1 3780c4
1 3780c1 1 3780c2 1 3780c3 1 3780c4
⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎦
⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎦
3 2 315c4 1 −20c1 −10c1 −4c1 −1 4 630c1 3 2 315c4 2 −20c2 −10c2 −4c2 −1 630c4 2 3 2 315c4 3 −20c3 −10c3 −4c3 −1 630c4 3 3 2 315c4 4 −20c4 −10c4 −4c4 −1 630c4 4
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
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(2)
(3)
and S i = (Si , Si , Si )T , f i = (fi , fi+c1 , fi+c2 , fi+c3 , fi+c4 , fi+1 )T . It can be shown that, when α(xi+cj ) ≤ a, the initial condition (2.2) becomes S(α(xi+cj )) = g(α(xi+cj )), while for α(xi+cj ) ∈ [xk , xk+1 ], k ≤ i, k = 0, 1, . . . , n−1, S(α(xi+cj )) can be determined as (0)
4 3 2 S(α(xi+cj )) = c5 j [(70cj + 35cj + 15cj + 5cj + 1)Sk 15 5 1 (1) (2) +(35c4j + 15c3j + 5c2j + cj )Sk + ( c4j + c3j + c2j )Sk 2 2 2 5 4 1 3 (3) 1 4 (4) +( cj + cj )Sk + ( cj )Sk ] 6 6 24 (0) 5 4 3 +cj [(70cj + 35cj + 15c2 j + 5cj + 1)Sk+1 15 4 5 3 1 2 (2) (1) 3 2 c + cj + cj )Sk+1 −(35c4 j + 15cj + 5cj + cj )Sk+1 + ( 2 j 2 2 5 4 1 3 (3) 1 4 (4) +( cj + cj )Sk+1 + ( cj )Sk+1 ], j = 1, 2, . . . , 5. (13) 6 6 24
For 0 < c1 < c2 < c3 < c4 < 1, A˜−1 is nonsingular as ˜ =− |A|
3
(c1 − c2 )(c1 − c3 )(c2 − c3 )(c1 − c4 )(c2 − c4 )(c3 − c4 ) = 0. 3000564000(c1 − 1)3 (c2 − 1)3 (c3 − 1)3 (c4 − 1)3
Error Analysis and Order of Convergence
The nonic spline S(x) is defined s.t S(x) = g(x), ∀x ≤ a and S(xi ) approximates y(xi ) for i = 0, 1, . . . , n. Moreover to determine S(x), y(α(x)) is replaced by S(α(x)). By numerical experiments it is concluded that the method is stable in the following subintervals of [0, 1].
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Solution of DDEs using nonic spline method
Table 1: Some cases that make the method stable 0.01 ≤ c1
0.999 ≤ c2
.9999 ≤ c3
.99999 ≤ c4
0.1 ≤ c1
0.9659 ≤ c2
.995 ≤ c3
.996 ≤ c4
0.2 ≤ c1
0.93 ≤ c2
.99 ≤ c3
.991 ≤ c4
0.3 ≤ c1
0.86 ≤ c2
.99 ≤ c3
.9989 ≤ c4
0.4 ≤ c1
0.799 ≤ c2
.9849 ≤ c3
.9989 ≤ c4
0.5 ≤ c1
0.74 ≤ c2
.984 ≤ c3
.9989 ≤ c4
0.6 ≤ c1
0.63 ≤ c2
.983 ≤ c3
.9969 ≤ c4
0.7 ≤ c1
0.76 ≤ c2
.94 ≤ c3
.95 ≤ c4
0.8 ≤ c1
0.85 ≤ c2
.869 ≤ c3
.889 ≤ c4
0.9 ≤ c1
0.90001 ≤ c2
0.90002 ≤ c3
0.90003 ≤ c4
The following theorem is proved to determine the error bound. Theorem 1 For f ∈ C 9 ([a, b] × R × R), the method is consistent and is of order nine. Proof For α(xi+cj ) ∈ [xk , xk+1 ], the discretizatioin error, can be written as ⎡ ⎢ ⎢ ⎣
di = ⎢
y(xi+1 ) 2 h y (xi+1 ) h3 y (xi+1 ) h4 y (iv) (xi+1 ) ⎡
⎢ ⎢ ⎢ ⎢ −hB ⎢ ⎢ ⎢ ⎢ ⎣
where
⎤
⎡
⎥ ⎢ ⎥ ⎢ ⎥ −A⎢ ⎦ ⎣
y(xi ) 2 h y (xi ) h3 y (xi ) h4 y (iv) (xi )
⎤ ⎥ ⎥ ⎥ ⎦
f (xi , pi (xi ), pk (xk )) f (xi+c1 , pi (xi+c1 ), pk (xk+c1 )) f (xi+c2 , pi (xi+c2 ), pk (xk+c2 )) f (xi+c3 , pi (xi+c3 ), pk (xk+c3 )) f (xi+c4 , pi (xi+c4 ), pk (xk+c4 )) f (xi+1 , pi (xi+1 ), pk (xk+1 ))
⎤ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦
i = 0, 1, . . . , n, k ≤ i
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Shahid S. Siddiqi, Gh. Akram and H. Baig
pi (x) = ξ 5 [(70ξ 4 + 35ξ 3 + 15ξ 2 + 5ξ + 1)y(xi) 15 5 1 +(35ξ 4 + 15ξ 3 + 5ξ 2 + ξ)y (xi ) + ( ξ 4 + ξ 3 + ξ 2 )y (xi ) 2 2 2 5 4 1 3 1 4 (iv) +( ξ + ξ )y (xi ) + ( ξ )y (xi )] 6 6 24 +ξ 5 [(70ξ 4 + 35ξ 3 + 15ξ 2 + 5ξ + 1)y(xi+1 ) 15 5 1 −(35ξ 4 + 15ξ 3 + 5ξ 2 + ξ )y (xi+1 ) + ( ξ 4 + ξ 3 + ξ 2 )y (xi+1 ) 2 2 2 5 4 1 3 1 4 (iv) +( ξ + ξ )y (xi+1 ) + ( ξ )y (xi+1 )] (14) 6 6 24 is the nonic Hermite interpolation polynomial which interpolates y, y , y , y , y (iv) at x = xi and x = xi+1 , i = 0, 1, . . . , n − 1 and pk (x) ≡ S(α(x)). Since |pi(x) − y(x)| ≤ Lh10 , x ∈ Ii , i = 1, 2, . . . , n, it follows that di = d˜i + O(h10 ), i = 1, 2, . . . , n, where ⎡ ⎡
d˜i =
⎢ ⎢ ⎢ ⎣
y(xi+1 ) h2 y (xi+1 ) h3 y (xi+1 ) h4 y (iv) (xi+1 )
⎤
⎡
⎥ ⎢ ⎥ ⎢ ⎥ −A⎢ ⎦ ⎣
y(xi ) h2 y (xi ) h3 y (xi ) h4 y (iv) (xi )
⎤
⎢ ⎢ ⎢ ⎥ ⎢ ⎥ ⎥ − hB ⎢ ⎢ ⎦ ⎢ ⎢ ⎣
y (xi ) y (xi+c1 ) y (xi+c2 ) y (xi+c3 ) y (xi+c4 ) y (xi+1 )
⎤ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎦
Using Taylor’s series expansion y(x) = q9 (x) + O(h10 ), x ∈ [xi , xi+1 ], y ∈ C 10 [a, b] where, q9 (x) =
9
hk (k) y (xi )ck . k! k=0
It can be observed that the error reduces to zero for a polynomial of degree ≤ 9 (that is for y = q9 , d˜i = di = 0). Thus according to lemma 8.11 in [10], the methods is consistent of order nine for all values listed in Table 1. To determine the convergence of the method the following theorem is proved. Theorem 2
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Solution of DDEs using nonic spline method
For f ∈ C 9 ([a, b]×R×R) be Lipschitz continuous, the spline approximation S(x) given by (9) and (10) converges to the solution y(x) of (4) as h → 0, for all values of collocation point listed in Table 1 and (j)
lim h−j S0 = y (j) (x0 ), j = 0, 1, . . . , 4.
h→0
(15)
Moreover, the convergence order is nine, i.e., 1 (k) Si | ≤ Lk h9 , k = 0, 1 k h 1 (k) (k) |y (xi ) − k Si | ≤ Lk h9−k , k = 2, 3, 4, i = 1, 2, . . . , n, h |y (k) (xi ) −
(16) (17)
if the initial values (8) satisfy (15) (with i = 0). It may be mentioned that, the global error estimates can be determined as (j)
|y (j)(x) − Si (x)| ≤ Lj h9−j , j = 0, 1, . . . , 8, x ∈ [a, b].
(18)
Proof The Lipschitz condition follows |y (xi ) − S (xi )| = |f (xi , y(xi), y(α(xi))) − f (xi , S(xi ), S(α(xi )))| ≤ L[|y(xi ) − S(xi )| + |y(α(xi)) − S(α(xi ))|] ≤ L[L0 h9 + L0 h9 ] = L1 h9 where L1 = 2LL0 .
4
Numerical Examples
In this section two numerical examples are given to demonstrate the reliability of the method. In the first example, the absolute error is compared with other methods. The comparison shows that the present method is more accurate than the previously developed method. The second example is chosen because this example exhibit difficult characteristics of the delay differential equation, a combination of stiffness and delay, and a non smooth character of the initial function. Example 1 The following single nonhomogeneous linear delay differential equation with stiffness parameter is considered [17], as 3π ) − Asin(t) 2 3π θ ∈ [− , 0], 2
y (t) = Ay(t) + y(t − y(θ) = epθ + sinθ,
(19)
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Shahid S. Siddiqi, Gh. Akram and H. Baig
where A = p − e−3πp/2 . It is to be noted that the exact solution for the equation is given as under y(t) = ept + sint. The comparison of the presented method with routine HRKF4 [18] and with quintic spline method [6] is shown in Table 2. Table 2: Test results for Example 1, with h = pi/40 p −0.1 −0.1 −0.1 −0.1 −0.1 −1 −1 −1 −1 −1 −2 −2 −2 −2 −2
Time Oberle and Pesch [18] Hawary and Mahmoud [6] 3π/4 2.1E − 4 6.0E − 12 3π/2 4.7E − 5 9.3E − 12 9π/4 1.7E − 4 1.8E − 11 3π 1.3E − 4 2.9E − 11 15π/4 4.5E − 5 4.4E − 11 3π/4 3.1E − 7 9.8E − 15 3π/2 2.1E − 7 2.2E − 14 9π/4 1.7E − 7 3.6E − 14 3π 9.8E − 8 7.1E − 14 15π/4 9.8E − 7 1.5E − 13 3π/4 1.5E − 7 1.5E − 15 3π/2 1.6E − 7 3.3E − 15 9π/4 1.6E − 7 5.2E − 15 3π 6.4E − 9 7.5E − 15 15π/4 Failed 9.7E − 15
Present Method 4.4E − 16 5.6E − 15 9.1E − 15 6.7E − 15 4.2E − 15 7.78E − 16 1.11E − 16 2.26E − 14 2.35E − 14 1.11E − 15 2.22E − 16 2.22E − 16 7.77E − 16 1.15E − 16 1.11E − 15
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Figure 1: Example 1 for p = −0.1
Example 2 The following delay differential equation is considered y (t) = costy(t − 1) − y(t), y(θ) = θ2 , θ ∈ [−1, 0].
(20)
It is to be noted that the analytic solution of the problem is given as under 1 y(t) = (t − 2)[tcost + (t − 2)sint]. 2 In this example the initial function does not satisfy the exact solution. The exact solution of the problem and absolute error at various values are shown in Table 3.
5
Conclusion
The major goal of the paper is to develop a class of five point spline collocation method for the solution of delay differential equation. The collocation scheme is developed using nonic Hermite interpolation spline with four interior collocation points. The convergence of the developed method is of O(h9 ). It may be noted that the method has the same order of accuracy for delay differential equations as it has for ordinary differential equations because spline approximation is directly used to interpolate the delay function.
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Table 3: The absolute error of Example 2 Time 10 20 30 40 50 60 70 80 90 100
Exact Solution Absolute Error −50.972 3.69E − 008 22.135 8.81E − 013 −32.252 2.56E − 012 31.099 9.25E − 012 855.70 9.44E − 012 −2168.9 3.18E − 012 3296.5 4.18E − 011 −3367.8 5.18E − 011 1687.2 3.70E − 011 1793.8 4.32E − 012
Figure 2: Example 2
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, A class of three-point spline collocation methods for solving delay differential equations, Tishreen University, J.Studies and Scientific Research 28 (2006), 163–178.
[7] K. Engelborghs, K. J. Int. Hout D. Roose, and T. Luzyanina, Collocation methods for the computation of periodic solutions of delay differential equations, Siam J. Science and Computations 22 (2000), 1593–1609. [8] F.Calio, E. Marchetti, and R.Pavani, About the deficient spline collocation method for particular differential and integral equations with delays, Rend. Sem. Mat.Univ. Pol. Torino 63, 3 (2003), 287–300. [9] F.Kappel and K. Kunisch, Spline approximations for neutral functional differential equations, Siam J. Numerical Analysis 18, 6 (1981), 1058– 1080. [10] E. Hairer, S.p.Norsett, and G. Wanner, Solving Ordinary Differential Equations- Nonstiff Problems, second ed., Springer, New York, 1993. [11] H. Hayashi, Numerical Solution of Retarded and Neutral delay differential equations using continuos Runge-Kutta methods, Ph.D. thesis, University of Toronto, Toronto, Canada, 1996. [12] T. Hong-Jiong and K. Jiao-Xun, The numerical stability of linear multistep methods for delay differential equations with many delays, Siam J. Numerical Analysis 33 (1996), 883–889.
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[13] K. J. In’t Hout, The stability of a class of Runge-Kutta methods for delay differential equations, Applied Numerical Methods 9 (1992), 347–355. [14] HU and Guang-Da, Stability of runge-kutta methods for delay differential systems with multiple delays, IMA J. Numerical Analysis 19 (1999), 349– 359. [15] C. Huang, Dissipativity of Runge-Kutta methods for dynamical systems with delays, IMA J. Numerical Analysis 20 (2000), 153–166. [16] K. Ito and R. Kapple, A uniformly differentialble approximation scheme for delay systems using spline, Tech. Report, Technische Universitat Graz, Austeria, 1987. [17] K. Ito, H.T. Tran, and A. Manitius, A fully discrtet spectral methods for delay differential equations, Siam J. Numerical Analysis 28 (1991), 1121– 1140. [18] H. J. Oberle and H. J. Pesch, Numerical treatment of delay differential equations by hermitte interpolation, Numerical Math. 37 (1981), 235–255. [19] L. Torell, Stability of numerical methods for delay differential equations, J. Computation and Appl.Math. 25 (1989), 15–26. Received: April, 2012
