A LEGENDRE-GAUSS COLLOCATION METHOD FOR NONLINEAR DELAY DIFFERENTIAL EQUATIONS. Zhong-Qing Wang. Li-Lian Wang. (Communicated by Zhimin Zhang)

Transcription

1 DISCRETE AND CONTINUOUS doi: /dcdsb DYNAMICAL SYSTEMS SERIES B Volue 13, Nuber 3, May 2010 pp A LEGENDRE-GAUSS COLLOCATION METHOD FOR NONLINEAR DELAY DIFFERENTIAL EQUATIONS Zhong-Qing Wang Departent of Matheatics, Shanghai Noral University, Shanghai, , China Scientific Coputing Key Laboratory of Shanghai Universities Division of Coputational Science of E-institute of Shanghai Universities Li-Lian Wang Division of Matheatical Sciences, School of Physical and Matheatical Sciences Nanyang Technological University, , Singapore (Counicated by Zhiin Zhang Abstract. In this paper, we introduce an efficient Legendre-Gauss collocation ethod for solving nonlinear delay differential equations with variable delay. We analyze the convergence of the single-step and ulti-doain versions of the proposed ethod, and show that the schee enjoys high order accuracy and can be ipleented in a stable and efficient anner. We also ake nuerical coparison with other ethods. 1. Introduction. Delay differential equations (DDEs naturally arise in diverse science and engineering applications [25], and in recent years, a large body of literatures has been devoted to nuerical solutions of DDEs (see, e.g., [3]-[6] and [9, 10, 14, 26]. Aong the existing ethods, nuerical schees based on Taylor s expansions or quadrature forulas have been frequently used (cf. [11, 12, 25, 26, 28] and the references therein, e.g., the iplicit Runge-Kutta ethods, which can be systeatically designed and often provide accurate approxiations. Over the years, spectral ethod has becoe increasingly popular and been widely used in spatial discretizations of PDEs owing to its high order of accuracy (cf. [7, 8, 13, 16, 17, 18]. The solution of a DDE globally depends on its history due to the delay variable, so a global spectral ethod could be a good candidate for nuerical DDEs. Soe work has been done along this line, and we particularly point out that Ito, Tran and Manitius [27] proposed and analyzed a Legendre-tau ethod for linear DDEs 2000 Matheatics Subject Classification. Priary: 5L60, 65L05, 65Q05, 41A10. Key words and phrases. Legendre-Gauss collocation ethods, nonlinear delay differential equations, spectral accuracy. The first author is supported in part by National Basic Research Project of China N.2005CB321701, NSF of China, N , Shuguang Project of Shanghai Education Coission, N.08SG45, Science and Technology Coission of Shanghai Municipality Grant, N , Shanghai Leading Acadeic Discipline Project N.S30405 and The Fund for E- institute of Shanghai Universities N.E The second author is supported in part by Singapore AcRF Tier 1 Grant RG58/08, Singapore MOE Grant # T207B2202, Singapore # NRF2007IDM- IDM , and Leading Acadeic Discipline Project of Shanghai Municipal Education Coission Grant J

2 686 ZHONG-QING WANG AND LI-LIAN WANG with one constant delay, where the solution was approxiated by a truncated Legendre series, and the unknown expansion coefficients was solved fro a Lanczostau forulation. Moreover, Guo and Wang [21], and Guo and Yan [23] developed the Legendre-Gauss collocation ethods for ODEs based on Legendre polynoial expansions. Meanwhile, Guo and Wang [20], and Guo et al. [22] discussed the Laguerre-Gauss-type collocation ethods for ODEs. However, it is ore interesting but challenging to develop and analyze such type of high-order ethods for nonlinear DDEs with variable delay of the for { U (t = f(u(t, W(t, t, 0 < t T, (1 U(t = V (t, t 0, where W(t = U(t θ(t, f, θ and V are given functions and the delay variable: θ(t 0. We start with a single-step schee otivated by [21]. Basically, we approxiate the solution by a finite Legendre series, and collocate the nuerical schee at Legendre-Gauss points to deterine the coefficients. Due to the delay variable, it s inevitable to solve nonlinear systes likewise for the iplicit Runge- Kutta ethods, but the iterative solvers can be ipleented efficiently thanks to the fast transfor [2]. The schee has an infinite order of accuracy both in tie and the delay variable, and is particularly attractive for DDEs with highly oscillatory solutions and/or stiff behavior. Therefore, it enjoys soe rearkable advantages over the Runge-Kutta-type ethods. For a ore effective ipleentation, we suggest a ulti-doain schee to advance in tie subinterval by subinterval due to the following twofold considerations. Firstly, the resultant syste for the expansion coefficients can be solved ore stably and efficiently for a sall or odest nuber of unknowns. In particular, for large T, it is desirable to partition the solution interval (0, T and solve the subsystes successively. We also want to ake a point that we erely need to store the Legendre coefficients of the solution in relevant delay subintervals to recover the solution in the subinterval that needs to resolve. Hence, the schee can be ipleented efficiently and econoically. On the other hand, the ulti-doain schee provides us a flexibility to handle DDEs involving non-sooth initial data and/or solutions. In such cases, the partition of the interval can be adapted to the evolution process as the adaptive Runge-Kutta ethods, and we are able to place ore points in the subintervals that are needed. The rest of the paper is organized as follows. In the next section, we present and analyze the single-step Legendre-Gauss collocation ethod, and provide soe nuerical results to justify our theoretical analysis. The ulti-doain version is described, and the convergence result is also derived and nuerically illustrated in Section 3. The final section is for soe concluding discussions. 2. Single-step Legendre-Gauss collocation ethod. In this section, we describe and analyze a single-step nuerical integration process for the DDE (1 using the Legendre-Gauss interpolation, which serves as a base for the ulti-doain schee to be presented in the forthcoing section Preliinaries. Let L l ( be the Legendre polynoial of degree l defined on [ 1, 1]. The shifted Legendre polynoials L T,l (t, t [0, T], are defined by (cf. [21] ( 2t L T,l (t = L l T 1 = ( 1l l! d l {t l( dt l 1 t l }, l = 0, 1, 2,. T

3 LEGENDRE-GAUSS COLLOCATION METHOD FOR DDE 687 According to the properties of the standard Legendre polynoials, we have ( 2t (l + 1L T,l+1 (t (2l + 1 T 1 L T,l (t + ll T,l 1 (t = 0, l 1, L T,0 (t = 1, L T,1 (t = 2t T 1, L T,2(t = 6t2 T 2 6t T + 1. Moreover, there holds the recursive relation (2 L T,l+1 (t L T,l 1 (t = 2 T (2l + 1L T,l(t, l 1. (3 The set of L T,l (t fors a coplete L 2 (0, T orthogonal syste, naely, T 0 L T,l (tl T, (tdt = T 2l + 1 δ l,, (4 where δ l, is the Kronecker sybol. Thus for any v L 2 (0, T, we write v(t = v T,l L T,l (t, v T,l = 2l + 1 T v(tl T,l (tdt. (5 T l=0 We now turn to the Legendre-Gauss interpolation. Let {t N j, ωn j }N j=0 be the Legendre-Gauss quadrature nodes (in ( 1, 1 and arranged in ascending order and weights. Accordingly, we define the shifted Legendre-Gauss interpolation nodes and weights as t N T,j = T 2 (tn j + 1, ωn T,j = T 2 ωn j 0, j = 0, 1,, N. Let P N (0, T be the set of all real polynoials of degree at ost N. We recall that for any φ P 2N+1 (0, T, T 0 φ(tdt = T ( T φ 2 (t + 1 dt = T 2 N ( T N ωj N φ 2 (tn j + 1 = ωt,j N φ(tn T,j. (6 Denote by (u, v T and v T the inner product and the nor of the space L 2 (0, T, respectively. Define the following discrete inner product and nor associated with the Legendre-Gauss quadrature: N u, v T,N = u(t N T,j v(tn T,j ωn T,j, v T,N = v, v 1 2 T,N. j=0 j=0 Thanks to (6, for any φψ P 2N+1 (0, T and ϕ P N (0, T, j=0 (φ, ψ T = φ, ψ T,N, ϕ T = ϕ T,N. (7 For any v C(0, T, the shifted Legendre-Gauss interpolant I T,N v(t P N (0, T is deterined by I T,N v(t N T,j = v(tn T,j, 0 j N. In view of (7, we have that for any φ P N+1 (0, T, We can expand I T,N v(t as (I T,N v, φ T = I T,N v, φ T,N = v, φ T,N. (8 I T,N v(t = N ṽt,ll N T,l (t, (9 l=0

4 688 ZHONG-QING WANG AND LI-LIAN WANG where by (4 and (8, ṽt,l N = 2l + 1 T (I T,Nv, L T,l T = 2l + 1 v, L T,l T,N, 0 l N. (10 T Furtherore, one verifies fro (4, (7 and (10 that for any ψ P N+1 (0, T, N+1 N ψ(t = ψ T,l N L T,l(t, I T,N ψ(t = ψ T,l N L T,l(t = ψ T,l N = ψ T,l N, 0 l N. l=0 which iplies that (cf. [21] l=0 ψ T,N ψ T, ψ P N+1 (0, T. (11 Let r be a nonnegative integer, H r (0, T be the usual Sobolev space as defined in [1], and denote by its nor and sei-nor by r,t and r,t, respectively. There holds the following estiates (see, e.g., Forulas ( and ( of [13]. Lea 2.1. For any u H r (0, T with integer 1 r N + 1, I T,N u u T ct r N r u r,t, (12 and (IT,N u u T ct r 1 N 3 2 r u r,t. (13 We notice that the Legendre-Gauss interpolation estiate in H 1 nor is optial (cf. [13]. The sei-nor in the upper bound can be iproved to the weighted sei-nor t r/2 (T t r/2 u (r T as in [19] The single-step schee. We next present the single-step schee for the delay differential equation (1. For this purpose, we denote the grid set by Λ N := { t N T,k : 0 k N } (0, T. The single-step Legendre-Gauss collocation approxiation to (1 is to find u N (t P N+1 (0, T, such that d dt un (t = f(u N (t, v N (t, t, t Λ N ; u N (0 = U(0 = V (0, (14 where the delay ter v N (t = { u N (t θ(t, V (t θ(t, t Λ N {t : t > θ(t}, t Λ N {t : t θ(t}. Here, we recall that θ(t 0 and V ( is a known function. Denote by Λ 0 N = { t Λ N : t θ(t }, Λ 1 N = { t Λ N : t > θ(t }, and set w N (t = u N (t θ(t. The collocation schee (14-(15 can be reforulated as: Find u N (t P N+1 (0, T such that { d f ( u N (t, w N (t, t, t Λ 1 dt un N (t =, f ( u N (t, V (t θ(t, t, t Λ 0 N, (16 suppleented with the initial condition u N (0 = V (0. We notice that it is an iplicit schee. An iportant proble is how to resolve (16. Indeed, we ay follow Labert [28] to design an algorith (ainly for ODEs to resolve the discrete syste (16 based on the Lagrange interpolation. However, it is known that Lagrange interpolation is not stable for large N. Hence, we propose a stable approach by expanding u N (t directly in ters of the shifted Legendre polynoials and deterining the unknown (15

5 LEGENDRE-GAUSS COLLOCATION METHOD FOR DDE 689 coefficients of the collocation schee (16. This approach is stable for large N, and uch easier to ipleent particularly for DDEs, since one only needs to store the coefficients of the nuerical solution at each step. One ay refer to Reark 2.1 of [21] for soe ore features of this ethod for ODEs. We next describe the nuerical ipleentation of the collocation schee (16. For this purpose, we expand the collocation solution as u N (t = N+1 l=0 û N T,l L T,l(t P N+1 (0, T, 0 < t T. (17 Furtherore, let [l] be the integer part of l. According to [21], we have d dt L T,l(t = 2 T [ l 1 2 ] (2l 4 1L T,l 2 1 (t. =0 On the other hand, L T,l (0 = ( 1 l. Hence, (16 is equivalent to where N+1 l=1 a N T,k,l = 2 T and f ft,k N = f = Further, let a N T,k,lû N T,l = f N T,k, 0 k N; [ l 1 2 ] =0 ( N+1 l=0 ( N+1 N+1 ( 1 l û N T,l = V (0, (18 (2l 4 1L T,l 2 1 (t N T,k, 0 k N, 1 l N + 1, N+1 û N T,l L T,l(t N T,k, l=0 l=0 û N T,l L T,l(t N T,k θ(tn T,k, tn T,k û N T,l L T,l(t N T,k, V (tn T,k θ(tn T,k, tn T,k l=0 ( f V (0 + N+1 + N+1 l=1 û N T,l (L T,l(t N T,k ( 1l, V (0 û N T,l (L T,l(t N T,k θ(tn T,k ( 1l, t N T,k l=1 ( f V (0 + N+1 l=1 V (t N T,k θ(tn T,k, tn T,k û N T,l (L T,l(t N T,k ( 1l,, t N T,k Λ1 N,, t N T,k Λ0 N,, t N T,k Λ1 N,, t N T,k Λ0 N. û N T = ( û N T,1, ûn T,2,, ûn T,N+1, F N T (û N T = ( f N T,0, fn T,1,, fn T,N, and A N T be the atrix with the entries an T,k,l, 0 k N, 1 l N + 1. Then we can rewrite (18 into the following atrix for: A N T ûn T = F N T (û N T ; N+1 û N T,0 = V (0 ( 1 l û N T,l. (19 In actual coputations, we solve { û N T,l } N+1 l=0 fro (19, and recover the collocation solution u N (t, 0 < t T fro (17. l=1

6 690 ZHONG-QING WANG AND LI-LIAN WANG We tabulate in Table 1 the condition nubers of A N T with T = 1 and various N, which indicates that the condition nubers grow like N 2 /4 as with the noral collocation schee using Lagrangian basis. Table 1. The condition nubers with T = 1. N Cond. Nu. N Cond. Nu Error analysis. In this subsection, we analyze the convergence of the schee (16. In particular, we shall prove the spectral accuracy of nuerical solution u N (t. Let I T,N be the Legendre-Gauss interpolation operator as defined before. Denote E N (t = u N (t I T,N U(t. Lea 2.1. Let U and u N be respectively solutions of (1 and (16. If U H r (0, T, with integer 2 r N + 1, then for any ε > 0, ( 1 2 ǫ t 1 (E N E N (0 2 T + T 1 E N (T E N (0 2 cǫ 1 T 2r 2 N 3 2r U 2 r,t + 2 GN T,1 2 T,N, (20 where f ( u N (t, w N (t, t f ( I T,N U(t, I T,N W(t, t, t Λ 1 G N T,1 (t = N, f ( u N (t, V (t θ(t, t f ( I T,N U(t, V (t θ(t, t, t Λ 0 N. (21 Proof. Let G N T,2 (t = I d T,N dt U(t d dt I T,NU(t. Then we have fro (1 that d dt I T,NU(t = f ( U(t, W(t, t G N T,2(t, t Λ 1 N, d dt I T,NU(t = f ( U(t, V (t θ(t, t (22 G N T,2(t, t Λ 0 N. Subtracting (22 fro (16 yields { d dt EN (t = G N T,1(t + G N T,2(t, t Λ j N, j = 0, 1, E N (0 = U(0 I T,N U(0. Clearly, t 1 (E N (t E N (0 P N (0, T. Thereby, by (7 and integration by parts, we deduce that (23

7 LEGENDRE-GAUSS COLLOCATION METHOD FOR DDE E N E N (0, d dt (t 1 (E N E N (0 T,N = 2 ( E N E N (0, d dt (t 1 (E N E N (0 T = 2 ( E N E N (0, t 2 (E N E N (0 T (24 +2 ( E N E N (0, t 1 d dt (EN E N (0 T = t 1 (E N E N (0 2 T + T 1 E N (T E N (0 2. On the other hand, we have fro (23 that d ( t 1 (E N (t E N (0 = t 2 (E N (t E N (0 + t 1 d dt dt EN (t = t 2( E N (t E N (0 + t 1( G N T,1 (t + GN T,2 (t, t Λ j N, j = 0, 1. The above fact with (7 leads to that 2 E N E N (0, d dt (t 1 (E N E N (0 T,N = 2 E N E N (0, t 2 (E N E N (0 T,N +2 E N E N (0, t 1 (G N T,1 + GN T,2 T,N = 2 t 1 (E N E N (0 2 T + AN T,1 + AN T,2, (25 where A N T,1 = 2 t 1 (E N E N (0, G N T,1 T,N, AN T,2 = 2 t 1 (E N E N (0, G N T,2 T,N. Since t 1 (E N (t E N (0 P N (0, T and G N T,2 (t P N(0, T, we use (7 to obtain that for any ǫ > 0, A N T,2 = 2 (t 1 (E N E N (0, G N T,2 T ǫ t 1 (E N E N (0 2 T + ǫ 1 G N T,2 2 T. Inserting the above estiate and (24 into (25 gives that (1 ǫ t 1 (E N E N (0 2 T + T 1 E N (T E N (0 2 ǫ 1 G N T,2 2 T + AN T,1. (26 We next estiate G N T,2 T. Clearly, by (12 with du dt and r 1 instead of u and r, we have that for integer r 2, Moreover, by (13, Therefore I T,N d dt U d dt U T ct r 1 N 1 r U r,t. (27 d dt (I T,NU U T ct r 1 N 3 2 r U r,t. (28 G N T,2 T d dt (I T,NU U T + I T,N d dt U d dt U T ct r 1 N 3 2 r U r,t. (29 On the other hand, by (7, A N T,1 1 2 t 1 (E N E N (0 2 T + 2 GN T,1 2 T,N. Substituting the above and (29 into (26, we obtain (20.

8 692 ZHONG-QING WANG AND LI-LIAN WANG We now consider several typical f and analyze the nuerical errors. Hereafter, β denotes any positive nuber less than 1 4. Case I. Consider (16 with the linear delay: θ(t = λt, 0 λ < 1. (30 Assue that f(x, y, t satisfies the following Lipschitz conditions in x and y. That is, there exists real nubers γ 1 0 and γ 2 0 such that f(x 1, y, t f(x 2, y, t γ 1 x 1 x 2, (31 and f(x, y 1, t f(x, y 2, t γ 2 y 1 y 2. (32 Theore 2.2. If the conditions (30-(32 hold, U H r (0, T, with integer 2 r N + 1, and for certain δ > 0, (1 + δt 2 γ (1 + δ 1 (1 λ 1 T 2 γ 2 2 β < 1 4, (33 then U u N 2 T c βt 2r N 3 2r U 2 r,t, (34 In particular U(T u N (T 2 c β T 2r 1 N 3 2r U 2 r,t. (35 ax t [0,T] U(t un (t 2 c β T 2r 1 N 3 2r U 2 r,t, (36 where c β is a positive constant depending only on β. Proof. Obviously, in this case, Λ 0 N =. Moreover, wn (t P N+1 (0, T. Therefore, by virtue of (11, (31 and (32, for any δ > 0, G N T,1 2 T,N (1 + δ f(un, w N, f(i T,N U, w N, 2 T,N + (1 + δ 1 f(i T,N U, w N, f(i T,N U, I T,N W, 2 T,N (1 + δγ 2 1 E N 2 T + (1 + δ 1 γ 2 2 w N I T,N W 2 T. (37 On the other hand, by virtue of (12 and a direct calculation, we deduce that for any ǫ > 0, w N I T,N W 2 T (1 + ǫ wn W 2 T + (1 + ǫ 1 W I T,N W 2 T (1 + ǫ(1 λ 1 U u N 2 T + (1 + ǫ 1 W I T,N W 2 T (1 + ǫ(1 λ 1 U u N 2 T + cǫ 1 T 2r N 2r W 2 r,t (1 + ǫ(1 λ 1 U u N 2 T + cǫ 1 T 2r N 2r U 2 r,t. (38 Due to (33, we have that (1 + δ 1 T 2 γ2 2 < 1 4. Hence, inserting the above two inequalities into (20, we obtain fro (33 that ( 1 2 ǫ t 1 (E N E N (0 2 T + T 1 E N (T E N (0 2 2(1 + δγ 2 1 E N 2 T + 2(1 + δ 1 (1 + ǫ(1 λ 1 γ 2 2 U u N 2 T + cǫ 1 (1 + δ 1 γ 2 2T 2r N 2r U 2 r,t + cǫ 1 T 2r 2 N 3 2r U 2 r,t (39 2(1 + δγ 2 1 EN 2 T + 2(1 + δ 1 (1 + ǫ(1 λ 1 γ 2 2 U un 2 T + cǫ 1 T 2r 2 N 3 2r U 2 r,t.

9 LEGENDRE-GAUSS COLLOCATION METHOD FOR DDE 693 Moreover ( 1 ( 1 ( E 2 ǫ N 2 T 2 ǫ (1 + ǫ E N E N (0 2 T + (1 + ǫ 1 E N (0 2 T ( 1 ( 2 ǫ (1 + ǫt 2 t 1 (E N E N (0 2 T + (1 + ǫ 1 T(E N (0 2. Therefore, we use (39 and (40 to derive that ( 1 2 ǫ E N 2 T + (1 + ǫt E N (T E N (0 2 (1 + ǫt 2( ( 1 2 ǫ t 1 (E N E N (0 2 T + T 1 E N (T E N (0 2 + cǫ 1 T(E N (0 2 (1 + ǫt 2( 2(1 + δγ 2 1 EN 2 T + 2(1 + δ 1 (1 + ǫ(1 λ 1 γ 2 2 U un 2 T (40 (41 + cǫ 1 T(E N (0 2 + cǫ 1 T 2r N 3 2r U 2 r,t, or equivalently, ( 1 2 ǫ 2(1 + ǫ(1 + δt 2 γ 2 1 E N 2 T + (1 + ǫt E N (T E N (0 2 2(1 + δ 1 (1 + ǫ 2 (1 λ 1 T 2 γ 2 2 U u N 2 T + cǫ 1 T(E N (0 2 + cǫ 1 T 2r N 3 2r U 2 r,t. Thanks to (12, we have that (42 U u N 2 T (1 + ǫ E N 2 T + (1 + ǫ 1 U I T,N U 2 T (1 + ǫ E N 2 T + cǫ 1 T 2r N 2r U 2 r,t. (43 The above with (42 yields ( 1 2 ǫ 2(1 + ǫ(1 + δt 2 γ1 2 ( (1 + ǫ 1 2 ǫ 2(1 + ǫ(1 + δt 2 γ 2 1 U u N 2 T + (1 + ǫ2 T E N (T E N (0 2 E N 2 T +(1 + ǫ 2 T E N (T E N (0 2 + cǫ 1 T 2r N 2r U 2 r,t 2(1 + δ 1 (1 + ǫ 3 (1 λ 1 T 2 γ2 2 U un 2 T +cǫ 1 T 2r N 3 2r U 2 r,t + cǫ 1 T(E N (0 2, (44 or equivalently, ( 1 2 ǫ 2(1 + ǫ(1 + δt 2 γ1 2 2(1 + δ 1 (1 + ǫ 3 (1 λ 1 T 2 γ2 2 U u N 2 T +(1 + ǫ 2 T E N (T E N (0 2 cǫ 1 T 2r N 3 2r U 2 r,t + cǫ 1 T(E N (0 2. (45 On the other hand, for any v H 1 (0, T (see (3.9 of [21], This, along with (12 and (13, leads to that ax t [0,T] v(t 2 2 T v 2 T + 2T dv dt 2 T. (46 (E N (0 2 = I T,N U(0 U(0 2 2 T I T,NU U 2 T + 2T d dt (I T,NU U 2 T ct 2r 1 N 3 2r U 2 r,t. (47

10 694 ZHONG-QING WANG AND LI-LIAN WANG 3 Next let ǫ = ( 4β > 0. Then (1 + ǫ 3 (1 + 2β = 3 2. Hence, by (33, ǫ + 2(1 + ǫ(1 + δt 2 γ (1 + δ 1 (1 + ǫ 3 (1 λ 1 T 2 γ 2 2 < (1 + ǫ 3 (1 + 2(1 + δt 2 γ (1 + δ 1 (1 λ 1 T 2 γ (1 + ǫ 3 (1 + 2β 1 = 1 2. A cobination of the above estiate, (45 and (47 leads to (34. Further, by (45 and (47, we deduce that (E N (T 2 2 E N (T E N ( E N (0 2 c β T 2r 1 N 3 2r U 2 r,t. Moreover, an arguent siilar to (47 yields Consequently, I T,N U(T U(T 2 ct 2r 1 N 3 2r U 2 r,t. (48 U(T u N (T 2 2 I T,N U(T U(T E N (T 2 c β T 2r 1 N 3 2r U 2 r,t. This leads to (35. Next, by using (37, (38, (34 and (12, we deduce that G N T,1 2 T,N (1 + δγ2 1 EN 2 T + (1 + δ 1 (1 + ǫ(1 λ 1 γ 2 2 U un 2 T + cǫ 1 (1 + δ 1 γ 2 2 T 2r N 2r U 2 r,t 2(1 + δγ 2 1 ( U I T,NU 2 T + U un 2 T + c βγ 2 2 T 2r N 3 2r U 2 r,t c β (γ γ2 2 T 2r N 3 2r U 2 r,t. (49 Hence, by (7, (23, (28, (29, (49 and (33, we deduce that d dt (U un 2 T 2 d dt EN 2 T + 2 d dt (U I T,NU 2 T = 2 d dt EN 2 T,N + 2 d dt (U I T,NU 2 T (50 4 G N T,1 2 T,N + 4 GN T,2 2 T + ct 2r 2 N 3 2r U 2 r,t c β T 2r 2 N 3 2r U 2 r,t. Finally, by virtue of (46, (34 and (50, we obtain (36. Reark 2.1. The condition (33 is necessary for the proof, but it should not be essential. In fact, soe nuerical exaples do not eet this condition, but the nuerical schee still converges. This reark also applies to ultiple-doain cases. Case II. Assue that the delay function satisfies: t θ(t 0, t [0, T]. (51 Moveover, f(x, y, t satisfies the Lipschitz condition (31. Theore 2.3. If the conditions (51 and (31 hold, U H r (0, T, with integer 2 r N + 1, and then In particular T 2 γ 2 1 β < 1 4, (52 U u N 2 T c β T 2r N 3 2r U 2 r,t, (53 U(T u N (T 2 c β T 2r 1 N 3 2r U 2 r,t. (54 ax U(t t [0,T] un (t 2 c β T 2r 1 N 3 2r U 2 r,t. (55

11 LEGENDRE-GAUSS COLLOCATION METHOD FOR DDE 695 Proof. Obviously, in this case, Λ 1 N =. Hence, by virtue of (11 and (31, Inserting the above inequality into (20, we obtain G N T,1 2 T,N γ2 1 EN 2 T. (56 ( 1 2 ǫ t 1 (E N E N (0 2 T + T 1 E N (T E N (0 2 2γ 2 1 E N 2 T + cǫ 1 T 2r 2 N 3 2r U 2 r,t. (57 Therefore, we use (40, (57 and (47 to derive that ( 1 2 ǫ EN 2 T + (1 + ǫt E N (T E N (0 2 (1 + ǫt 2 ( ( 1 2 ǫ t 1 (E N E N (0 2 T + T 1 E N (T E N (0 2 +cǫ 1 T(E N (0 2 2(1 + ǫt 2 γ 2 1 EN 2 T + cǫ 1 T 2r N 3 2r U 2 r,t, (58 or equivalently, ( 1 2 ǫ 2(1+ǫT 2 γ 2 1 E N 2 T +(1+ǫT E N (T E N (0 2 cǫ 1 T 2r N 3 2r U 2 r,t. Let ǫ = 3 8β > 0. Then by (52, ǫ + 2(1 + ǫt 2 γ 2 1 = (1 + ǫ(1 + 2T 2 γ (1 + ǫ(1 + 2β 1 = β < 1 2. A cobination of the above estiate, (43 and (59 leads to (53. Further, by (59 and (47, we deduce that (E N (T 2 2 E N (T E N ( E N (0 2 c β T 2r 1 N 3 2r U 2 r,t. This, along with (48 yields (54. We can derive the result (55 easily. Reark 2.2. We see fro (34, (35, (53 and (54 that the errors U(T u N (T and U u N T decay rapidly as N and r increase. The convergence rate is O(N 3 2 r. Thus, the soother the exact solution, the saller the nuerical errors. In other words, the schee (16 possesses the spectral accuracy. Reark 2.3. If dr U dt r L (0, T, N r 1 and T 1, then we use Theores 2.1 and 2.2 to deduce that U u N T c 1 2 β T r N 2 r d r U L dt r, (60 (0,T U(T u N (T c 1 2 β T r N 3 2 r d r U L dt r. (61 (0,T In particular, if 1 N T 1, then we ay take r = N + 1 in (60 and (61, to reach that U u N T = O(T 2N+1, U(T u N (T = O(T 2N+1 2. ( Nuerical results. In this subsection, we present soe nuerical results to illustrate the efficiency of our single-step algorith. ❶ Linear variable delay (Case I: { d dt u(t = 1 ( 2 e t t 2 u + 1 u(t, for 0 t T, 2 2 (63 u(0 = 1. (59 As pointed out in [15], the exact solution is u(t = e t. Obviously, the conditions (31 and (32 hold with γ 1 = 1 2 and γ 2 = 1 2 e T 2. Moreover, the inequality (33 is

12 696 ZHONG-QING WANG AND LI-LIAN WANG satisfied for T = 0.3. But for T = 1, the inequality (33 is no longer valid. In Fig. 1, we plot the nuerical errors at t = T with T = 0.3, 1 and various values of N. They indicate that the nuerical errors decay exponentially as N increases. In particular, we can observe fro Fig. 1 that even if the condition (33 is not satisfied (for instance, T=1, our algorith is still valid. 2 4 T=1 T=0.3 6 log 10 error N Figure 1. The nuerical errors of Legendre-Gauss collocation ethod at t = T log 10 error N ❷ Figure 2. The nuerical errors of Legendre-Gauss collocation ethod at t = 1. Nonlinear variable delay (Case I: { d dt u(t = 1 2u2( t, for 0 t 1, 2 u(0 = 0. This is a nonlinear DDE with the exact solution u(t = sin(t, see [15]. In Fig. 2, we plot the nuerical errors at t = 1 using the single-step schee (16 and Newton-Raphson iteration ethod with various values of N. They indicate that the nuerical errors decay exponentially as N increases. We also note that the condition (32 is not satisfied for proble (64, but our algorith is still valid. (64

13 ❸ LEGENDRE-GAUSS COLLOCATION METHOD FOR DDE 697 Linear constant delay (Case II: { d ( dt u(t = u t π, 0 t π 2 2, (65 u(t = sin(t, π 2 t 0. The exact solution is u(t = sin(t. Obviously, the condition (31 hold with γ 1 = 0. Moreover, the inequality (51 is satisfied. In Fig. 3, we plot the nuerical errors at t = π 2 with various values of N. They indicate that the nuerical errors decay exponentially as N increases log 10 error N Figure 3. The nuerical errors of Legendre-Gauss collocation ethod at t = π Multiple-doain Legendre-Gauss collocation ethod. In the last section, we investigated the single-step Legendre-Gauss collocation ethod. The nuerical errors decay very rapidly as N and r increase. While the foregoing singlestep collocation ethod provide accurate results, in actual coputation, it is not convenient to resolve the discrete syste (19 with very large ode N. As coented in the introduction, it is advisable to partition the interval (0, T into a finite nuber of subintervals and solve the equations subsequently on each subinterval The ultiple-doain schee. We now describe the ultiple-doain schee. Let M and N, 1 M be any positive integers. We first decopose the interval (0, T] into M subintervals (T 1, T ], 1 M, such that the set of T includes all breaking points, where T 0 = 0 and T M = T. Denote by τ = T T 1, 1 M. We shall use u N (t P N +1(0, τ to approxiate the solution U in the subinterval (T 1, T ]. Firstly, replacing T and N by τ 1 and N 1 in (16 and all other forulas in Subsection 2.2, we can derive an alternative algorith, with which we obtain the nuerical solution u N1 1 (t P N 1+1(0, τ 1. Then we evaluate the nuerical solutions u N(t P N +1(0, τ, 2 M, step by step. Finally, the global nuerical solution of (1 is given by u N (T 1 + t = u N (t, 0 t τ, 1 M. (66

14 698 ZHONG-QING WANG AND LI-LIAN WANG We now present the nuerical schee for u N (t. Denote by t N τ,k and ωn τ,k, 0 k N the nodes and the corresponding Christoffel nubers of the shifted Legendre- Gauss interpolation on the interval (0, τ. Let and Λ 0 N, = {tn τ,k T 1 + t N τ,k θ(t 1 + t N τ,k 0, 0 k N }, Λ j N, ={t N τ,k T 1 + t N τ,k θ(t 1 + t N τ,k (T j 1, T j ], 0 k N }, 1 j. The ultiple-doain collocation ethod for (1 is to seek u N (t P N +1(0, τ, such that d dt un (t = f(u N (t, w(t, j T 1 + t, t Λ j N,, j > 0, d (t = f(u N (t, V (T 1 + t θ(t 1 + t, T 1 + t, t Λ 0 N,, where dt un u N (0 = un 1 1 (τ 1, 2 M, w j (t = un (T 1 + t θ(t 1 + t = u Nj j (T 1 T j 1 + t θ(t 1 + t. Denote by U (t = U(T 1 + t for 0 t τ. Then by (1, d dt U (t = f(u (t, W j (t, T 1 + t, t Λ j N,, j > 0, d dt U (t = f(u (t, V (T 1 + t θ(t 1 + t, T 1 + t, t Λ 0 N,, U (0 = U 1 (τ 1, 2 M, U 1 (0 = U(0 = V (0, where W j (t = U(T 1 + t θ(t 1 + t = U j (T 1 T j 1 + t θ(t 1 + t. We see fro (67 and (68 that the local nuerical solution u N (t is actually an approxiation to the local exact solution U (t, with the approxiate initial data u N (0 = un 1 1 (τ Error analysis. We next analyze the nuerical errors. Denote E N (t = u N (t I τ,n U (t. Lea 3.1. Let U and u N be respectively solutions of (68 and (67. If U H r (0, τ, with integer 2 r N + 1, then for any ε > 0, where G N τ,1 (t = ( 1 2 ǫ t 1 (E N cǫ 1 τ 2r 2 N 3 2r EN (0 2 τ + τ 1 E N U 2 r,τ + 2 G N τ,1 2 τ,n, (τ E N (0 2 f(u N (t, wj (t, T 1 + t f(i τ,n U (t, W j (t, T 1 + t, t Λ j N,, j > 0, (67 (68 (69 f(u N (t, V (T 1 + t θ(t 1 + t, T 1 + t f(i τ,n U (t, V (T 1 + t θ(t 1 + t, T 1 + t, t Λ 0 N,.

15 LEGENDRE-GAUSS COLLOCATION METHOD FOR DDE 699 Proof. Let G N τ,2 (t = I τ,n d dt U (t d dt I τ,n U (t. Then, following the sae line as in the derivation of (23, we obtain fro (67 and (68 that d dt EN E N E N1 Like (24, we have (t = G N τ,1 (t + GN τ (t,,2 t Λj N,, j 0, (0 I τ,n U (0, 2 M, (0 = un 1 (0 = U(0 I τ 1,N 1 U(0. 2 E N = t 1 (E N EN (0, d dt (t 1 (E N E N Next, we use (70 to derive that d dt (t 1 (E N (t E N = t 2 (E N (t EN (0 2 τ + τ 1 (0 = t 2 (E N EN E N (t E N (0 τ,n (τ E N (0 2. (0 + t 1 d dt EN (t (0 + t 1 (G N τ,1 (t + GN τ (t, t,2 Λj N,, j 0. Hence, by (7 with τ and N instead of T and N, we deduce fro the above inequality that 2 E N = 2(E N EN (0, d dt (t 1 (E N + 2 t 1 (E N EN (0, t 2 (E N EN (0, GN EN (0 τ,n EN τ,1 (0 τ τ,n + 2(t 1 (E N EN (0, GN τ,2 τ (70 (71 2 t 1 (E N E N (0 2 τ + ( ǫ t 1 (E N + 2 G N τ,1 2 τ,n + ǫ 1 G N τ,2 2 τ. According to (29 with the paraeters N and τ, we have E N (0 2 τ (72 G N τ,2 τ cτ r 1 A cobination of (71-(73 leads to (69. N 3 2 r U r,τ. (73 We now consider several typical f, and analyze the nuerical errors. Hereafter, let τ = ax τ j and N = in N j. Moreover, we always assue that for any 1 j M 1 j M 1 i j M, τi τ j is bounded above. Case I. Consider (67 with the linear delay: θ(t = λt, 0 < λ < 1. (74 Assue that f(x, y, t satisfies the Lipschitz conditions (31 and (32. Theore 3.2. If ❶ the conditions (74, (31 and (32 hold, ❷ (1 λt T 1 for all 2 M, ❸ U H r (0, T with integer 2 r N + 1,

16 700 ZHONG-QING WANG AND LI-LIAN WANG ❹ for certain δ > 0 and c 0 > 0, (1 + δτ1 2 γ1 2 + (1 + δ 1 (1 λ 1 τ1 2 γ2 2 β < 1 4, then for any 1 M, In particular, τ γ 1 < 1 2, τ γ 2 c 0, > 1, (75 U u N 2 L 2 (T 1,T c βτ 2r N 3 2r U 2 r,t, (76 U(T u N (T 2 c β τ 2r 1 N 3 2r U 2 r,t. (77 ax t [T 1,T ] U(t un (t 2 c β τ 2r 1 N 3 2r U 2 r,t. (78 Proof. Clearly, in this case, Λ 0 N, = for 1 and Λ N, = for > 1. Therefore, N by (31, (32, (11, (46 and the fact ω N τ = τ,k, we deduce that for any ǫ > 0 and > 1, 1 G N τ,1 2 τ,n (1 + ǫ j=1 k=0 t N τ,k Λj N, 2ω f(i τ,n U (t N τ,,k wj (tn τ, T,k 1 + t N τ N,k τ,k 1 + (1 + ǫ 1 j=1 t N τ,k Λj N, ( f(u N (t N τ,k, wj (t N τ,k, T 1 + t N τ,k ( f(i τ,n U (t N τ,k, wj (tn τ,k, T 1 + t N τ,k f(i τ,n U (t N τ,k, W j (t N τ,k, T 1 + t N τ,k 2ω N τ,k (1 + ǫ 1 γ j=1 t N τ,k Λj N, + (1 + ǫγ 2 1 EN 2 τ,n (w j (tn τ W j,k (tn τ,k 2 ω N τ,k (1 + ǫ 1 τ γ2 2 ax U j u Nj j 2 L 1 j 1 (0,τ + (1 + j ǫγ2 1 EN 2 τ (1 + ǫ 1 τ γ2 2 ax ( 2 U j u Nj j 2 τ 1 j 1 τ j + 2τ j d j dt (U j u Nj j 2 τ j + (1 + ǫγ 2 1 E N 2 τ. Inserting the above inequality into (69, we obtain that for > 1, ( 1 2 ǫ t 1 (E N EN (0 2 τ + τ 1 E N (τ E N(0 2 2(1 + ǫγ1 E 2 N 2 τ + cǫ 1 τ 2r 2 N 3 2r U 2 r,τ +2(1 + ǫ 1 τ γ2 2 ax ( 2 U j u Nj j 2 τ 1 j 1 τ j + 2τ j d j dt (U j u Nj j 2 τ j. On the other hand, like (40, ( 1 2 ǫ EN 2 τ ( 1 2 ǫ ((1 + ǫτ 2 t 1 (E N EN (0 2 τ + (1 + ǫ 1 τ E. N(0 2 (79 (80 (81

17 LEGENDRE-GAUSS COLLOCATION METHOD FOR DDE 701 Therefore, we use (80 and (81 to derive that for > 1, ( 1 2 ǫ EN 2 τ + (1 + ǫτ E N (τ E N (0 2 cǫ 1 τ E N (0 2 ( + (1 + ǫτ 2 ( 1 2 ǫ t 1 (E N E N (0 2 τ + τ 1 E N (τ E N (0 2 ( (1 + ǫτ 2 2(1 + ǫγ1 E 2 N 2 τ + 2(1 + ǫ 1 τ γ2 2 ax ( 2 U j u Nj j 2 τ 1 j 1 τ j j + 2τ j d dt (U j u Nj j 2 τ j + cǫ 1 τ 2r N3 2r U 2 r,τ + cǫ 1 τ E N (0 2, (82 or equivalently, ( 1 2 ǫ 2(1 + ǫ2 τ 2 γ 2 1 E N 2(1 + ǫ 1 (1 + ǫτ 3 γ2 2 ax ( 2 U j u Nj j 1 j 1 τ j +cǫ 1 τ 2r U 2 r,τ + cǫ 1 τ E N (0 2. N 3 2r Furtherore, like (43, we have 2 τ + (1 + ǫτ E N (τ E N (0 2 2 τ j + 2τ j d dt (U j u Nj j 2 τ j (83 U u N 2 τ (1 + ǫ E N 2 τ + cǫ 1 τ 2rN 2r U 2 r,τ. (84 The above two inequalities with (75 lead to that for > 1, ( 1 2 ǫ 2(1 + ǫ2 τγ U u N 2 τ + (1 + ǫ 2 τ E N (1 + ǫ( 1 2 ǫ 2(1 + ǫ2 τ 2 γ2 1 EN +cǫ 1 τ 2rN 2r U 2 r,τ 2 τ + (1 + ǫ 2 τ E N (τ E N (0 2 (τ E N (0 2 2(1 + ǫ 1 (1 + ǫ 2 τγ ax ( 2 U j u Nj j 2 τ 1 j 1 τ j + 2τ j d j dt (U j u Nj j 2 τ j +cǫ 1 τ 2r N 3 2r U 2 r,τ + cǫ 1 τ E N (0 2. Thanks to (46, an arguent siilar to (47 yields E N (0 2 = u N (0 I τ,n U (0 2 2 N u 1 1 (τ 1 U 1 (τ U (0 I τ,n U (0 2 2 N u 1 1 (τ 1 U 1 (τ cτ 2r 1 N 3 2r U 2 r,τ. (85 Siilarly (U (τ u N (τ 2 2(E N (τ 2 + 2(U (τ I τ,n U (τ 2 2(E N (τ 2 + cτ 2r 1 N3 2r U 2 r,τ.

18 702 ZHONG-QING WANG AND LI-LIAN WANG A cobination of the above three inequalities gives that for > 1, ( 1 2 ǫ 2(1 + ǫ2 τ 2 γ 2 1 U u N 2 τ (1 + ǫ2 τ U (τ u N (τ 2 ( 1 2 ǫ 2(1 + ǫ2 τ 2 γ 2 1 U u N 2 τ (1 + ǫ2 τ E N (τ 2 + cτ 2r N3 2r + (1 + ǫ 2 τ ( E N U 2 r,τ ( 1 2 ǫ 2(1 + ǫ2 τ 2 γ2 1 U u N 2 τ (τ E N (0 2 + E N (0 2 + cτ 2r N3 2r U 2 r,τ 2(1 + ǫ 1 (1 + ǫ 2 τ 3 γ2 2 ax ( 2 U j u Nj j 2 τ 1 j 1 τ j + 2τ j d j dt (U j u Nj j 2 τ j + cǫ 1 τ u N 1 1 (τ 1 U 1 (τ cǫ 1 τ 2r N 3 2r U 2 r,τ. Thus by (50, (34, (35 and (75, we obtain fro (86 that for a suitable value of ǫ, Siilarly U u N 2 L 2 (T 1,T 2 = U 2 u N2 2 2 τ 2 c β τ 2r N 3 2r ( U 1 2 r,τ 1 + U 2 2 r,τ 2. U(T 2 u N (T 2 2 = U 2 (τ 2 u N2 2 (τ 2 2 c β τ 2r 1 N 3 2r ( U 1 2 r,τ 1 + U 2 2 r,τ 2. Since τi τ j is bounded above for 1 i j M, we use (86, (79, (75 and repeat the above process, to obtain (76 and (77. In particular, by virtue of (46, (76 and a result siilar to (50, we get (78. Reark 3.1. The estiates (76-(78 indicate the spectral accuracy of schee (67. Reark 3.2. If dr U dt r L (0, T, N r 1 and τ 1, then for 1 M, (86 U u N L 2 (0,T c 1 2 β τ r+ 1 2 N 3 2 r dr U dt r L (0,T, (87 U(T u N (T c 1 2 β 1 2 τ r N 3 2 r dr U dt r L (0,T, (88 ax U(t t [T un (t c 1 2 β 1 2 τ r N 3 2 r dr U 1,T ] dt r L (0,T. (89 1 If, in addition, N τ 1, then we ay take r = N + 1 in (87 and (88, to reach that for 1 M, U u N L 2 (0,T = O(τ 2N, U(T u N (T = O(τ 2N. (90 Case II. Assue that the delay function satisfies: T 1 + t θ(t 1 + t T 1, for 1 M and t (0, τ. (91 Moveover, f(x, y, t satisfies the Lipschitz condition (31 and (32. Theore 3.3. If ❶ the conditions (31, (32 and (91 hold, ❷ U H r (0, T with integer 2 r N + 1, ❸ τ γ 1 < 1 2, 1, ❹ for certain c 0 > 0 and > 1, τ γ 2 c 0, then we have the sae results (76-(78.

19 LEGENDRE-GAUSS COLLOCATION METHOD FOR DDE 703 Proof. Obviously, in this case, Λ N, =. Therefore, by (31, (32 and a siilar arguent as in the derivation of (79, we deduce that for any ǫ > 0, 1 G N τ,1 2 τ,n j=1 t N τ,k Λj N, ( f(u N (tn τ,k, wj (tn τ,k, T 1 + t N τ,k 2ω f(i τ,n U (t N τ, W j,k (t N τ, T,k 1 + t N τ N,k τ,k ( + f(u N (tn τ, V (T,k 1 + t N τ θ(t,k 1 + t N τ, T,k 1 + t N τ,k t N τ,k Λ0 N, f(i τ,n U (t N τ,k, V (T 1 + t N τ,k θ(t 1 + t N τ,k, T 1 + t N τ,k 2ω N τ,k (1 + ǫ 1 τ γ2 2 ax ( 2 U j u Nj j 2 τ 1 j 1 τ j + 2τ j d j dt (U j u Nj j 2 τ j + (1 + ǫγ 2 1 E N 2 τ. (92 Moreover, (80-(86 still hold. Furtherore, by (56, (53 and (12, we obtain G N1 τ 1,1 2 τ 1,N 1 γ 2 1 E N1 1 2 τ 1 2γ 2 1( U 1 I τ1,n 1 U 1 2 τ 1 + U 1 u N1 1 2 τ 1 c β γ 2 1 τ2r 1 N3 2r 1 U 1 2 r,τ 1. Therefore, we can derive the sae results (76-( Nuerical results. In this subsection, we present soe nuerical results to illustrate the efficiency of our ultiple-doain algorith. ❶ Linear variable delay (Case I. We consider the exaple (63 with T = 1 and unifor step-size grid. As pointed out before, the conditions (31 and (32 hold with γ 1 = 1 2 and γ 2 = 1 2 e 1 2. Furtherore, the inequality (75 is satisfied for τ 0.25, but for τ 0.5, the inequality (75 is no longer valid. Moreover, (1 λt T 1 with λ = 1 2. In Fig. 4, we plot the nuerical errors at t = 1 with unifor τ and N. They indicate that the nuerical errors decay exponentially as N increases and τ decreases. In particular, we can observe fro Fig. 4 that even if the condition (75 is not satisfied, our ultiple-doain algorith is still valid (see the case τ 0.5. ❷ Nonlinear variable delay (Case I. We consider the exaple (64. In Fig. 5, we plot the nuerical errors at t = 1 using the ultiple-doain schee (67 and Newton-Raphson iteration ethod with unifor τ and N. They indicate that the nuerical errors decay exponentially as N increases and τ decreases. Again, we observe fro Fig. 5 that our algorith is still valid even if the condition (32 is not satisfied. ❸ Linear variable delay (Case II. d 1 u(t = u(t 1, for t > 0, dt t + 1 u(t = 1, for 0.5 t 0, (93 u(t = 2 3 (t + 2, for 2 t < 0.5.

20 704 ZHONG-QING WANG AND LI-LIAN WANG 4 τ =0.5 τ = log 10 error N Figure 4. The nuerical errors of Legendre-Gauss collocation ethod at t = τ =1/2 τ =1/6 τ =1/10 6 log 10 error N Figure 5. The nuerical errors of Legendre-Gauss collocation ethod at t = n =2 0 n =4 0 n =6 0 n 0 =8 8 log 10 error N Figure 6. The nuerical errors of Legendre-Gauss collocation ethod at t = ξ 2.

21 LEGENDRE-GAUSS COLLOCATION METHOD FOR DDE log 10 error N Figure 7. The nuerical errors of Legendre-Gauss collocation ethod at t = 20. As pointed out in [24], the exact solution is { u(t = 3 t + t log(t + 1, on [0, 1], log 2 + t, on [1, 2]. The first derivative of the solution is not continuous at t = 0.5 and t = 0, therefore the second derivative also has a jup at the points where the tie lag is equal to 0.5 and 0, for instance, t = ξ 1 = 1 and t = ξ 2 = 2. Denote by n 0 = 2k the total nuber of steps. Obviously, the conditions (31 and (32 hold with γ 1 = 0 and γ 2 = 1. Furtherore, by choosing suitably T, we can ensure that the condition (91 holds. In Fig. 6, we plot the nuerical errors at t = ξ 2 with τ = ξ 1 k, 1 k, τ = ξ 2 ξ 1, k + 1 n and unifor N. k They indicate that the nuerical errors decay exponentially as N increases and τ decreases. As pointed out, in actual coputation, even if the conditions for Theores 3.1 and 3.2 are not satisfied, the nuerical solutions of our ethod atch the exact solutions very well. We next present soe other nuerical exaples to illustrate the efficiency of our ethod, which can be found in [5]. ❹ Nonlinear constant delay (Case II: { d u(t = 3u(t 1(1 + u(t, for t > 0, dt (94 u(t = t, for 1 t 0. This is a well-known equation fro biology, solution of which has the breaking points at the integers 0, 1,. In Table 1, we copare the errors of our ethod LGC3(5 (N = 3, 5 and the step-size is constant with the ethod UCIRK4(6 (the uniforly corrected iplicit Runge-Kutta ethod of order 4(6 presented in [5] and the ethod IRKSR4(6 (the iplicit Runge-Kutta superconvergence rate 4(6 presented in [4], at the point t = 20 with respect to the reference solution u(20 = Since the three ethods have the sae esh and the sae nuber of interpolation nodes, we can observe that our ethod provides ore accurate nuerical results.

22 706 ZHONG-QING WANG AND LI-LIAN WANG Table 2. Nuerical results for Eq. (94. Nuber of steps LGC3 UCIRK4 IRKSR4 LGC5 UCIRK6 IRKSR e e e e-9 2.0e e e e e e e e e e e e e e-06 In particular, our ethod is still valid even for large tie step-size. In Fig. 7, we plot the nuerical errors at t = 20 with τ = 1, 1 20 and unifor N. They indicate that the nuerical errors decay exponentially as N increases log 10 error N ❺ Figure 8. The nuerical errors of Legendre-Gauss collocation ethod at t = ξ 2. Nonlinear variable delay (Case II: d dt u(t = t u(t log(t + 1 1u(t, for t > 0, t + 1 u(t = 1, for 1 t 0. The solution has the breaking points: ξ 0 = 0, ξ 1 = , ξ 2 = , etc.. In Table 2, we copare the nuerical errors of various ethods at the point t = ξ 2 with respect to the reference solution: u(ξ 2 = Since the three ethods have the sae esh and the sae nuber of interpolation nodes, we can observe that our ethod provides again uch better nuerical results. In particular, our ethod is still valid even for large tie step-size. In Fig. 8, we plot the nuerical errors at t = ξ 2 with τ 1 = ξ 1, τ 2 = ξ 2 ξ 1 and unifor N. They indicate that the nuerical errors decay exponentially as N increases. 4. Concluding discussions. In this paper, we proposed the single-step and ultiple-doain Legendre-Gauss collocation integration processes for nonlinear DDEs. These approaches have several attractive features. (95

23 LEGENDRE-GAUSS COLLOCATION METHOD FOR DDE 707 Table 3. Nuerical results for Eq. (95. Nuber of steps LGC3 UCIRK4 IRKSR4 LGC5 UCIRK6 IRKSR e e e e e e e e e e e e-10 The single-step Legendre-Gauss collocation ethod is easy to be ipleented for nonlinear DDEs, and possesses the spectral accuracy. It enjoys coputational efficiency and accuracy over soe popular existing ethods. By using the ultiple-doain Legendre-Gauss collocation ethod, we can use oderate ode N to evaluate the nuerical solutions ore stably and effectively. Moreover, it can be adapted to solutions with jup-discontinuities.for any fixed ode N, the nuerical errors decay as τ 0. For any fixed step size in tie, the nuerical error decays very rapidly as N increases. The reason of this phenoena ight be that there exists the factor N r in the error estiations of our new ethod, whereas there is no such factor in the error estiations of the corresponding iplicit Runge-Kutta ethod. In fact, in the derivation of algorith of our ethod, we benefit fro the orthogonality of Legendre polynoials, while in the derivation of algorith of the iplicit Runge-Kutta ethod, one used the Lagrange interpolation on the Legendre- Gauss interpolation nodes, which is not stable for large N. In particular, our ethods are uch easier to be ipleented than the iplicit Runge-Kutta ethod for DDEs, since we only need to save the coefficients of nuerical solutions in each step. The nuerical errors of our ethods are characterized by the sei-nors of exact solutions in certain Sobolev spaces. These sharp nors are in particular necessary for probles with degenerated initial data. The nuerical results deonstrated the spectral accuracy of proposed algoriths and coincided with the theoretical analysis very well. REFERENCES [1] R. A. Adas, Sobolev Spaces, Acadic Press, New York-London, [2] B. K. Alpert and V. Rokhlin, A fast algorith for the evaluation of Legendre expansion, SIAM J. Sci. and Stat. Cop., 12 (1991, [3] C. T. H. Baker, C. A. H. Paul and D. R. Willé, Issues in the nuerical solution of evolutionary delay differential equations, Adv. Cop. Math., 3 (1995, [4] A. Bellen, One-step collocation for delay differential equations, J. Cop. Appl. Math., 10 (1984, [5] A. Bellen and M. Zennaro, Nuerical solution of delay differential equations by unifor corrections to an iplicit Runge-Kutta ethod, Nuer. Math., 47 (1985, [6] A. Bellen and S. Maset, Nuerical solution of constant coefficient linear delay differential equations as abstract Cauchy probles, Nuer. Math., 84 (2000, [7] C. Bernardi and Y. Maday, Spectral ethods, in Handbook of Nuerical Analysis, edited by P. G. Ciarlet and J. L. Lions, North-Holland, Asterda, [8] J. P. Boyd, Chebyshev and Fourier Spectral Methods, Second Edition, Dover Publications, Inc., Mineola, NY, [9] E. Bueler, Error bounds for approxiate eigenvalues of periodic-coefficient linear delay differential equations, SIAM J. Nuer. Anal., 45 (2007, [10] E. A. Butcher, H. Ma, E. Bueler, V. Averina and Z. Szabo, Stability of linear tie-periodic delay-differential equations via Chebyshev polynoials, Int. J. Nuer. Meth. Engrg., 59 (2004,

24 708 ZHONG-QING WANG AND LI-LIAN WANG [11] J. C. Butcher, Integration processes based on Radau quadrature forulas, Math. Coput., 18 (1964, [12] J. C. Butcher, The Nuerical Analysis of Ordinary Differential Equations, Runge-Kutta and General Linear Methods, John wiley & Sons, Chichester, [13] C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods: Fundaentals in Single Doains, Springer-Verlag, Berlin, [14] A. El-Safty, M. S. Sali and M. A. El-Khatib, Convergence of the spline functions for delay dynaic systes, Int. J. Coput. Math., 80 (2003, [15] D. J. Evans and K. R. Raslan, The adoian decoposition ethod for solving delay differential equation, Int. J. Coput. Math., 82 (2005, [16] D. Funaro, Polynoial Approxiations of Differential Equations, Springer-Verlag, Berlin, [17] D. Gottlieb and S. A. Orszag, Nuerical Analysis of Spectral Methods: Theory and Applications, SIAM-CBMS, Philadelphia, [18] B. Y. Guo, Spectral Methods and Their Applications, World Scietific, Singapore, [19] B. Y. Guo and L. L. Wang, Jacobi approxiations in non-uniforly Jacobi-weighted Sobolev spaces, J. Approx. Theory, 1 (2004, [20] B. Y. Guo and Z. Q. Wang, Nuerical integration based on Laguerre-Gauss interpolation, Coput. Meth. in Appl. Mech. and Engin., 196 (2007, [21] B. Y. Guo and Z. Q. Wang, Legendre-Gauss collocation ethods for ordinary differential equations, Adv. in Cop. Math., 30 (2009, [22] B. Y. Guo, Z. Q. Wang, H. J. Tian and L. L. Wang, Integration processes of ordinary differential equations based on Laguerre-Radau interpolations, Math. Cop., 77 (2008, [23] B. Y. Guo and J. P. Yan, Legendre-Gauss collocation ethods for initial value probles of second ordinary differential equations, Appl. Nuer. Math., 59 (2009, [24] I. Györi, F. Hartung and J. Turi, On nuerical solutions for a class of nonlinear delay equations with tie- and state-dependent delays, Proceedings of the World Congress of Nonlinear Analysts, Tapa, Florida, August 1992, Walter de Gruyter, Berlin, New York, 1996, [25] E. Hairer, S. P. Norsett and G. Wanner, Solving Ordinary Differential Equation I: Nonstiff Probles, Second Edition, Springer-Verlag, Berlin, [26] E. Hairer and G. Wanner, Solving Ordinary Differential Equation II: Stiff and Differential- Algebraic Probles, Second Edition, Springer-Verlag, Berlin, [27] K. Ito, H. T. Tran and A. Manitius, A fully-discrete spectral ethod for delay-differential equations, SIAM J. Nuer. Anal., 28 (1991, [28] J. D. Labert, Nuerical Methods for Ordinary Differential Systes: The Initial Value Proble, John Wiley and Sons, Chichester, Received Noveber 2008; revised October E-ail address: zqwang@shnu.edu.cn E-ail address: lilian@ntu.edu.sg