Modified Laguerre pseudospectral method refined by multidomain Legendre pseudospectral approximation

Journal of Computational and Applied Mathematics 190 (2006) 304 – 324 www.elsevier.com/locate/cam

Modified Laguerre pseudospectral method refined by multidomain Legendre pseudospectral approximation夡 Ben-Yu Guoa, b,∗ , Li-Lian Wanga a Department of Mathematics, Shanghai Normal University, Guilin Road, 100, Shanghai 200234, China b Division of Computational Science of E-institute of Shanghai Universities, Shanghai, China

Received 8 June 2004; received in revised form 5 November 2004 Dedicated to Professor Roderick S.C. Wong on the occasion of his 60th birthday

Abstract A modified Laguerre pseudospectral method is proposed for differential equations on the half-line. The numerical solutions are refined by multidomain Legendre pseudospectral approximation. Numerical results show the spectral accuracy of this approach. Some approximation results on the modified Laguerre and Legendre interpolations are established. The convergence of proposed method is proved. © 2005 Elsevier B.V. All rights reserved. Keywords: Modified Laguerre pseudospectral method; Multidomain Legendre approximation; Differential equations on the half-line

1. Introduction Spectral methods have high accuracy. The usual spectral methods are only available for bounded domains. However it is also interesting to consider spectral methods for differential equations on unbounded domains, such as the KdV equation and the Klein–Gordon equation on the half-line and exterior problems. Some authors developed the Laguerre spectral method for the half-line, see [3,4,6,9,11]. In actual 夡

This work is supported in part by SF of Shanghai. The Fund of Education Ministry of China, N. 20040270002, N. 10471095, NSF of Shanghai, N. 04JC14062, The E-institutes of Shanghai Municipal Education Commission, N. E03004, The Special Founds for Major Specialities and The Found N. 04DB15 of Shanghai Education Commission. ∗ Corresponding author. Tel.: 86 21 64322851; fax: 86 21 64323364. E-mail address: [email protected] (B.-Y. Guo). 0377-0427/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2004.11.053

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305

computation, it is more preferable to use the Laguerre pseudospectral method, since it only needs to evaluate numerical solutions at the Laguerre interpolation nodes, say xj , 0
j
N. It is also much easier to deal with nonlinear terms. However, the distances between the adjacent interpolation nodes increase very fast as the mode N increases, especially for large xj . Therefore, although numerical solutions fit exact solutions well at the nodes, but as global approximate solutions, they only can simulate exact solutions roughly between the large interpolation nodes. In this paper, we propose a combined Laguerre and multidomain Legendre pseudospectral method for the half-line. We first use the modified Laguerre pseudospectral method to obtain numerical solutions on the half-line, which fit exact solutions well at the interpolation nodes. Then we refine them locally by using multidomain Legendre pseudospectral approximation. This technique can be also regarded as recovering the accuracy by reconstructing numerical solutions. The paper is organized as follows. In the next section, we describe the combined Laguerre and multidomain Legendre pseudospectral method. In Section 3, we discuss its implementation and present some numerical results demonstrating its efficiency. In Section 4, we establish some results on the modified Laguerre and Legendre interpolations, which play important role in the analysis of spectral methods. In Section 5, we prove the convergence of proposed method. The final section is for concluding discussions. 2. Combined spectral method Let I =(a, b), 0
a < b
∞, and
(x) be a certain weight function. For real r  0, we define the weighted Sobolev space H
r (I ) as usual, with the semi-norm |v|m,
,I and the norm vm,
,I . In particular, we denote r (I ) stands for the closure by (u, v)
,I and v
,I the inner product and norm of L2
(I ). The space H0,
of set D(I ) consisting of all infinitely differential functions with compact support in I. For
(x) ≡ 1, we drop the subscript
in notations. For any integer M  0, PM (I ) denotes the set of all algebraic polynomials of degree at most M. Moreover 0 PM = {v ∈ PM |v(a) = 0} and P0M = {v ∈ 0 PM |v(b) = 0}. Denote by c a generic positive constant independent of any function and M. Now let  =(0, ∞) and (x)=e−x . Denote by Ll (x) the Laguerre polynomials of degree l, l =0, 1, . . . . They satisfy the recurrence relation Ll (x) = jx Ll (x) − jx Ll+1 (x),

l = 0, 1, . . . ,

(1)

and form the L2 ()-orthogonal system, i.e., (Ll , Lm ), = l,m . In this paper, we introduce the modified Laguerre interpolation. Firstly let 0 H 1 ()={v ∈ H 1 ()|v(0)= 0} and take the approximation spaces as SM () = {(x)|(x) = 1/2 (x)(x), ∀(x) ∈ PM ()},

0 SM () = SM () ∩ 0 H

1

().

M Next, let {xj }M j =0 and {j }j =0 be the nodes and weights of the standard Laguerre–Gauss–Radau interpolation IL,M . Indeed, xj are the zeros of x jx LM+1 (x), and j = 1/(M + 1)L2M (xj ). Further, let
j = j −1 (xj ). Then by the property of Laguerre–Gauss–Radau interpolation (see [12]),

(, ) = (, )M, :=

M  j =0

(xj )(xj )
j ,

∀ ·  ∈ S2M ().

(2)

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L v(x) ∈ SM (), satisfying For any v ∈ C(¯ ), the modified Laguerre–Gauss–Radau interpolant I
L,M v)(xj ) = v(xj ), (I

0
j
M.

(3)

We now turn to the Legendre–Gauss–Lobatto interpolation. Let the reference interval Iˆ = (−1, 1), and Ll (x) be the Legendre polynomials of degree l, l = 0, 1, . . . . They satisfy the recurrence relation (2l + 1)Ll (x) = jx Ll+1 (x) − jx Ll−1 (x),

l = 1, 2, . . . ,

(4)

and form the L2 (Iˆ)-orthogonal system, i.e., (Ll , Lm )Iˆ = (2/(2l + 1))l,m . Let {x} ˆ N ˆ }N j =0 and { j =0 be the nodes and weights of the Legendre–Gauss–Lobatto interpolation. Indeed, xˆj are the zeros of (1 − x 2 )jx LN (x), and ˆ j = 2/N(N + 1)L2N (xˆj ). By Szegö [12] (, )Iˆ = (, )N,Iˆ :=

M 

(xˆj )(xˆj ) ˆj,

∀ ·  ∈ P2N −1 (Iˆ).

(5)

j =0

ˆ L,N v(x) ∈ PN (Iˆ), satisfying For any v ∈ C(I¯ˆ), the Legendre–Gauss–Lobatto interpolant I ˆ L,N v)(xˆj ) = v(xˆj ), (I

0
j
N.

(6)

Now, we describe the combined pseudospectral method. For clearness, we focus on the simple model problem −j2x U (x) + U (x) = f (x),

 > 0,

x ∈ ,

lim U (x) = U (0) = 0.

(7)

x→∞

We may derive a weighted variational formulation of (7), and then solve it by the standard Laguerre spectral method as in [6]. But in this paper, we prefer a natural (not weighted) variational formulation, and solve it by using the Laguerre functions as in [11], which is easier to match the multidomain Legendre pseudospectral approximation for refining numerical results. To do this, let a (u, v) = (jx u, jx v) + (u, v) . A weak formulation of (7) is to find U (x) ∈ 0 H 1 () such that a (U, v) = (f, v) ,

∀v ∈ 0 H 1 ().

(8)

If f (x) ∈ H −1 (), then (8) has a unique solution U (x) ∈ 0 H 1 (). Next, Let aM, (u, v) = (jx u, jx v)M, + (u, v)M, . The numerical solution u0M (x) ∈ 0 SM () is uniquely determined by aM, (u0M , ) = (f, )M, ,

 ∈ 0 SM ().

(9)

We shall use the multidomain Legendre pseudospectral method to refine u0M (x). Let k = (ak , bk ) ⊂ (0, ∞), 1
k
K, such that k ∩l =] for k = l. The endpoints ak and bk are some Laguerre–Gauss–Radau nodes usually. But in this case, the lengths of (ak , bk ) might be very different, since the distribution of the interpolation nodes is not uniform, see [12]. In particular, for large M, the length of (ak , bk ) increases very fast as k increases. Thus, for refining numerical results on k , we should split k into Jk subintervals Ijk = (djk−1 , djk ) where ak = d0k < d1k < · · · < dJkk −1 < dJkk = bk . Their lengths hkj = djk − djk−1 .

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307

k,j

In each Ijk , we use the Legendre–Gauss–Lobatto interpolation with the nodes xi =(hkj /2)xˆi +(djk−1 + k,j djk )/2 and the weights i =(2/ hkj )ˆ i , 0
i
Njk . For any u, v ∈ C(¯ k ), we define the following discrete

inner product and norm on each Ijk , Nk

(u, v)N k ,I k = j

j

j 

k,j

k,j

k,j

uk,j (xi )v k,j (xi )i ,

1/2 . Njk ,Ijk

vN k ,I k = (v, v) j

i=0

j

Further, let Nk = (N1k , N2k , . . . , NJkk ), and define the discrete inner product and norm on k by (u, v)Nk ,k =

Jk  j =1

(u, v)N k ,I k , j

j

1/2

vNk ,k = (v, v)Nk ,k .

For any fixed k , we take the approximation spaces as VNk (k ) = {v ∈ H 1 (k )|v k,j ∈ PNj (Ijk ), 1
j
Jk },

V0Nk (k ) = VNk (k ) ∩ H01 (k ).

Let v k,j := v|I k . For any v ∈ C(¯ k ), the multidomain Legendre–Gauss–Lobatto interpolant IL,Nk v(x) ∈ j VNk (k ) is defined by k,j

k,j

(IL,Nk v)|I k (xi ) = v k,j (xi ), j

0
i
Njk .

(10)

If  · |I k ∈ PNj −1 (Ijk ), 1
j
Jk , then by (5), j

(, )Nk ,k = (, )k ,

∀ ·  ∈ V2Nk −1 (k ).

(11)

We now derive the algorithm for refining numerical results. Let U k := U |k , f k := f |k and W k (x) =

bk − x k x − ak k U (ak ) + U (bk ), bk − ak bk − ak

x ∈ k .

Due to (7), U k (x) ∈ H 1 (k ) satisfies ak (U k , v) = (f k , v)k ,

∀v ∈ H01 (k ).

(12)

Equivalently, it is to seek U∗k (x) = U k (x) − W k (x) ∈ H01 (k ) such that ak (U∗k , v) = (f k , v)k − ak (W k , v),

∀v ∈ H01 (k ).

Next, let ukNk be the approximation of U k on k , and k (x) = wM

bk − x 0 x − ak 0 uM (ak ) + u (bk ), bk − ak bk − ak M

x ∈ k .

(13)

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Let aNk ,k (u, v) = (jx u, jx v)Nk ,k + (u, v)Nk ,k . Then the multidomain Legendre pseudospectral approximation of (12) is to find ukNk (x) ∈ VNk (k ) such that aNk ,k (ukNk , ) = (f k , )Nk ,k ,

∀ ∈ V0Nk (k ).

(14)

k (x) ∈ V0 ( ) such that Equivalently, it is to seek ukNk ,∗ (x) = ukNk (x) − wM k Nk

aNk ,k (ukNk ,∗ , ) = BNk ,k (f, ),

∀ ∈ V0Nk (k ),

(15)

k , ). where BNk ,k (f, ) := (f k , )Nk ,k − aNk ,k (wM

3. Implementation and numerical results ˆ l (x) = 1/2 (x)Ll (x) and ˆ l (x) = L ˆ l (x) − L ˆ l+1 (x). We first consider the implementation of (9). Let L ˆ l (0) = 1 for all l, we have that 0 SM () = span{ˆ 0 , ˆ 1 , . . . , ˆ M−1 }. Since L  Now, let u0M (x) = M−1 ˆ 0l ˆ l (x) and u = (uˆ 00 , uˆ 01 , . . . , uˆ 0M−1 )T . Also set A = (ail )i,l=0,1,...,M−1 , B = l=0 u (bil )i,l=0,1,...,M−1 and r = (r0 , r1 , . . . , rM−1 )T , with the elements ail = (jx ˆ l , jx ˆ i ) , bil = (ˆ l , ˆ i ) ˆ l (x) + L ˆ l+1 (x)). This with the orthogonality of and ri = (f, ˆ i )M, . Due to (1), we have jx ˆ l (x) = 21 (L 1 1 Laguerre polynomials leads to ail = ali = 2 i,l + 4 i−1,l . It is easy to show that bil = bli = 2i,l − i−1,l . Finally, we obtain the matrix form of scheme (9) as (A + B)u = r. This system is symmetric and tridiagonal. We now turn to the implementation of (15). As in [8], we let P0 k (Ijk ) := PN k (Ijk ) ∩ H01 (Ijk ), and
k,j (x) = Ll (x) ˆ L l that for

where xˆ = (2/ hkj )x − (djk−1 + djk )/ hkj , 1
k
K, 1
j
Jk and 0
l
Nj − 2, 

k,j
l (x) =


(x) − L
(x)), cl (L l l+2 k,j

k,j

0, k k,j N −2

j Clearly, the set {
l }l=0

k,j

x ∈ I¯jk ,

x∈

k,j

cl

x ∈ k \I¯jk .

Nj k Ij . We define

=

1 2

j

the local polynomials such

 hkj /(2l + 3),

(16)

forms a basis of P0 k (Ijk ). Next, we define the subinterval matching functions Nj

k,0 k,J  (x) =
 k (x) ≡ 0, and for 1
j
Jk − 1, as


⎧ k,j
k,j (x))/2,
(x) + L x ∈ I¯jk , (L ⎪ 1 ⎪ ⎨ 0 k,j

k,j +1 (x))/2, x ∈ I¯k ,
k,j +1 (x) − L  (x) = (L 0 1 j +1 ⎪ ⎪ ⎩ 0, x ∈ k \I¯jk ∪ I¯jk+1 .  Obviously


k,j

k,j  (djk ) = 1 and supp
⊂ [djk−1 , djk+1 ].

(17)

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309

Now let kj = ukNk ,∗ (djk ). We expand the numerical solution ukNk ,∗ (x) as ukNk ,∗ (x) =

Jk 

v k,j (x) +

j =1

J k −1 j =1

kj


k,j

Njk −2

(x),

v k,j (x) =



k,j k,j vˆl
l (x).

(18)

l=0

i , 0
i
Njk − 2 in (15) and using (16)–(18), we obtain that Taking  =
k,j

k,j −1

i ) = BNk ,k (f,
i ) − kj −1 aNk ,k (
 aNk ,k (v k,j ,
k,j

k,j

,
i ) − kj aNk ,k (
 k,j

k,j

,
i ). k,j

(19)

System (19) is the basic algorithm for refining numerical results in which all unknown variables are coupled. So it is not convenient for actual computation. To remedy this deficiency, we split this system into Njk −2 k,j k,j k,j vˆm,l
l (x), m = 1, 2, 3, four subsystems. Firstly, we try to find the three functions, vm (x) = l=0 such that i ) = BNk ,k (f,
i ), aNk ,k (v1 ,
k,j

k,j

k,j

k,j −1

i ) = −aNk ,k (
 aNk ,k (v2 ,
k,j

k,j

i ) = −aNk ,k (
 aNk ,k (v3 ,
k,j

k,j

k,j

0
i
Njk − 2,

,
i ), k,j

,
i ), k,j

(20)

0
i
Njk − 2,

(21)

0
i
Njk − 2.

(22)

Obviously Eqs. (20)–(22) can be solved separately in each subinterval Ijk . In other words, all the computations can be carried out in parallel. Moreover by (19)–(22), k,j

k,j

k,j

v k,j (x) = v1 (x) + kj −1 v2 (x) + kj v3 (x),

1
j
Jk .

(23)

Inserting (23) into (18), we deduce that ukNk ,∗ (x) =

J k −1 j =1

k,j k,j +1 k,j (v2 (x) + v3 (x) +
 (x))kj

+

Jk 

k,j

(24)

v1 (x).

j =1

k,i −1 . For this purpose, we take  =
 , 1
i
Jk − 1 Thus it remains to evaluate the unknown values {kj }Jj k=1 in (15), and obtain that J k −1 j =1

k,j +1 aNk ,k (v2

k,j + v3

k,j k,i k,l +
 ,
 )kj = BNk ,k (f,
 )−

Jk 

aNk ,k (v1 ,
 ). k,j

k,i

(25)

j =1

Finally we resolve (25) and obtain the refined solution ukNk ,∗ (x) by (24). We can also derive the simple matrix forms for subsystems (20)–(22). 2 We now present some numerical results. We take the exact solution U (x) = e−(x− 0 ) / h0 with h0 > 0. As shown √ in Fig. 1, the√interested information of U (x) is mainly contained in the subinterval ( 0 , h0 ) = [ 0 − 3 h0 /2, 0 + 3 h0 /2].

310

B.-Y. Guo, L.-L. Wang / Journal of Computational and Applied Mathematics 190 (2006) 304 – 324 1.2 Exact solution Numer. solution at nodes Numer. solution

1

0.8

0.6

0.4

0.2

0

-0.2 0

20

40

60

80

100

120

x

Fig. 1. U (x) vs. u032 (x).

We take  = 1, 0 = 95 and h0 = 20, and use (9) with M = 32 to solve (7) numerically. In Fig. 1, we plot the exact solution U (x) by solid line, the numerical solution u0M (x) by dash dotted line, and the numerical solution at the interpolation nodes by ‘o’. Clearly u0M (x) fits U (x) very well at the nodes. But the errors are very big for large x which are not nodes. So the pure modified Laguerre pseudospectral method can not identify the interested structure of exact solution in detail. We next refine the numerical solution u0M (x). Since the numerical gradients between x25 and x32 are very big, we take K = 1, Jk = 1, a1 = x25 and b1 = x32 . We use (15) with N11 = 32 to solve (13) in the subinterval [a1 , b1 ], with the approximate boundary values u0M (a1 ) and u0M (b1 ). In Fig. 2, we plot the exact solution U (x) by solid line, the numerical solution u0M (x) by dash dotted line, and the refined numerical solution u132 (x) by dotted line. Clearly, the refined numerical result fits the exact solution very well, even for large x which are not the interpolation nodes. In fact we cannot distinguish U (x) and u132 (x) in Fig. 2. The proposed method is also suitable for solutions with several peaks. For instance, we consider the  2 exact solution as U (x) = 3j =0 e−(x− j ) / hj , hj > 0. We take 0 = 20, 1 = 100, 2 = 140, 3 = 180 and hj = 10, j = 0, 1, 2, 3, and use (9) with M = 64 to solve (7) numerically. In Fig. 3, we plot the exact solution U (x) by solid line, the numerical solution u0M (x) by dash dotted line, and the numerical solution at interpolation nodes by ‘o’. Fig. 3 indicates again that for large x, the numerical solution only fits the exact solution well at the nodes. We now refine the numerical solution u064 (x). Since the numerical gradients and residuals are very big between x44 and x64 , we take K = 1, Jk = 2, a1 = x44 , b1 = x64 , d01 = a1 , d11 = (b1 − a1 )/2 and d21 = b1 . We use (15) with N11 = N21 = 45 to resolve (13) in the subinterval [a1 , b1 ], with the approximate

B.-Y. Guo, L.-L. Wang / Journal of Computational and Applied Mathematics 190 (2006) 304 – 324

1.2

1

Exact solution Numer. solution Refined numer. solution

0.8

0.6

0.4

0.2

0

-0.2 50

60

70

80

90

100

110

120

x

Fig. 2. U (x), u032 (x) vs. u132 (x).

1.2 Exact solu. Numer. solu. at nodes Numer. solu.

1

0.8

0.6

0.4

0.2

0

-0.2 0

50

100

150 x

Fig. 3. U (x) vs. u064 (x).

200

250

311

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B.-Y. Guo, L.-L. Wang / Journal of Computational and Applied Mathematics 190 (2006) 304 – 324 1.2 Exact solution Refined numer. solution

1

0.8

0.6

0.4

0.2

0

-0.2 80

100

120

140

160

180

200

220

240

x

Fig. 4. U (x) vs. u145,45 (x).

boundary values u064 (a1 ) and u064 (b1 ). In Fig. 4, we plot the exact solution U (x) by solid line, and the refined numerical solution u145,45 (x) by dash dotted line. Obviously, the refined numerical solution fits the exact solution very well. 4. Some approximation results 4.1. Modified Laguerre interpolation We first consider the orthogonal projection PM : L2 () → PM (), defined by (PM v − v, ), = 0,

∀ ∈ PM ().

To describe approximation result, we introduce the weighted Sobolev space Ar (). Let r (x) = x r e−x . For the semi-norm and norm of Ar () are given by |v|Ar , = jrx vr , and vAr , = rinteger2 r  0, ( k=0 |v|Ak , )1/2 , respectively. For real r > 0, we define the space Ar () and its norm by space interpolation. We can prove the following result in the same manner as in [13]. Theorem 1. If v ∈ Ar () and integer r   0, then PM v − vA ,
cM (−r)/2 |v|Ar , . Theorem 2. If v ∈ H1 (), jx v ∈ Ar−1 () and integer r  1, then |PM v − v|1,,
cM 1−r/2 |jx v|Ar−1 , .

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313

Proof. We have |PM v − v|1,,
PM jx v − jx v, + PM jx v − jx PM v, . By Theorem 1, PM jx v − jx v,
cM 1/2−r/2 |jx v|Ar−1 , . Thus it suffices to estimate PM jx v − ∞ jx PM v, . Let jx v(x) = l=0 vˆˆ l Ll (x). By (1), jx v(x) =

∞ 

vˆˆ l (jx Ll (x) − jx Ll+1 (x)) =

l=0

∞ 

(vˆˆ l − vˆˆ l−1 )jx Ll (x).

l=1

 ∞ ˆ ˆ ˆ On the other hand, jx v(x) = ∞ l=1 vˆ l jx Ll (x). Therefore vˆ l = vˆ l − vˆ l−1 , and so vˆ l = − p=l+1 vˆ p .  Moreover, (1) implies that jx Ll (x) = − l−1 p=0 Lp (x). The above statements lead to that ⎛ ⎛ ⎞ ⎞ M ∞ M−1 M     Ll (x) ⎝ vˆp ⎠ , jx PM v(x) = − Ll (x) ⎝ vˆp ⎠ . PM jx v(x) = − l=0

p=l+1

Thus PM jx v(x) − jx PM v(x) = −

M 

l=0

⎛ ∞  Ll (x) ⎝

l=0

⎞ vˆp ⎠ = vˆˆ M

p=M+1

p=l+1

M 

Ll (x).

l=0

Accordingly, we use Theorem 1 to obtain that PM jx v

− jx PM v2,

2 = vˆˆ M

M 

2 Ll 2,
cM vˆˆ M
cMPM−1 v − v2,
cM 2−r |jx v|2Ar−1 , .

l=0

This completes the proof.  1 : H 1 () → P (), defined by We next consider the orthogonal projection 0 PM 0  0 M 1 (jx (0 PM v − v), jx ), = 0,

∀∈0 PM ().

Lemma 1. If v ∈ 0 H 1 (), jx v ∈ Ar−1 () and integer r  1, then 1 0 PM v − v1,,
cM 1/2−r/2 |jx v|Ar−1 , .

Proof. Let (x) =

x 0

PM−1 jy v(y) dy. By projection theorem and Theorem 1,

1 |0 PM v − v|1,,
| − v|1,, = PM−1 jx v − jx v,
cM 1/2−r/2 |jx v|Ar−1 , .

According to Lemma 2.2 of [6], v,
2|v|1,, for any v ∈ 0 H1 (). A combination of the above two estimates leads to the desired result.  We now derive an important result on the modified Laguerre approximation.

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1 v(x) = e−x/2 P 1 (ex/2 v(x)). Then the projector P¯ 1 : Theorem 3. For any v ∈ 0 H 1 (), let 0 P¯M 0 M 0 M 1 0 H () → 0 SM () satisfies 1 1 v − v), jx ) + 41 (0 P¯M v − v, ) = 0, (jx (0 P¯M

∀ ∈ 0 SM ().

(26)

Moreover, for integer r  1, 1 v − v1,
cM 1/2−r/2 |jx (ex/2 v)|Ar−1 , . 0 P¯M

(27)

Proof. Result (26) comes from Lemma 3.2 of [11]. Next, by Lemmas 1 and 2.2 of [6], 1 1 x/2 1 x/2 |0 P¯M v − v|1,
21 0 PM (e v) − ex/2 v, + |0 PM (e v) − ex/2 v|1,, 1 x/2
2|0 PM (e v) − ex/2 v|1,,
cM 1/2−r/2 |jx (ex/2 v)|Ar−1 , .

(28)

1, Furthermore, by the definition of 0 P¯M 1 1 x/2 1 x/2 v − v = 0 PM (e v) − ex/2 v,
2|0 PM (e v) − ex/2 v|1,, . 0 P¯M

(29)

A combination of (28) and (29) leads to the desired result. 
L,M , respectively. We first In the end of this subsection, we deal with the interpolations IL,M and I study the stability of the interpolation IL,M . Lemma 2. For any v ∈ H1 () ∩ A1 (), 1/2

1/2

IL,M v,
c(v, + v, |v|1,, + (ln M)1/2 vA1 , ).   Proof. We have IL,M v2, =AM +BM where AM = 0
xj
1 v 2 (xj )j and BM = xj >1 v 2 (xj )j . As shown in the proof of Lemma 2.1 of [14], for any v ∈ H1 (), e−x/2 vL∞ ()
v, + Since

M

j =0 j

1/2

1/2

2v, |v|1,, .

= 1, we deduce that

AM
v2L∞ (0,1) (1)

√

 0
xj <1

j
ce−x/2 v2L∞ ()
c(v2, + 2v, |v|1,, ).

(1)

Next, let Ll (x) be the generalized Laguerre polynomial of degree l, i.e., Ll (x) = 1/ l! ex /xdl / (1) dx l (e−x x l+1 ). Denote by j , 1
j
M the zeros of LM (x), arranged in increasing order. Since (1) jx LM+1 (x) = −LM (x)(see [12]), we have xj = j , 1
j
M. Moreover by the properties of the

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315

Laguerre–Gauss and Laguerre–Gauss–Radau quadratures (see [12]),  x jx LM+1 (x) 1 j = (x) dx jx (x jx LM+1 (x))|x=xj  x − xj  (1) x LM (x) 1 = (x) dx (1) jx (x L (x))|x=xj  x − xj M

=



1 (1) j jx LM ( j )

(1)

LM (x)



x − j

x (x) dx = −1 j j .

By (2.4) and (2.7) of [10], we know that j
c j e− j (4M − j )−1/2 and j +1 − j j (4M − j )−1/2 . Moreover, we have from Lemma 2.6 of [14] that x 1/2 e−x/2 v2L∞ (0,∞)
2v2A1 , . Thus 3/2

BM =



v 2 (xj ) −1 j j
c

xj >1


cv2A1 ,





v 2 (xj ) j e− j (4M − j )−1/2 1/2

xj >1 −1/2

j

(4M − j )−1/2 cv2A1 ,

xj >1


cv2A1 ,

M 1

1/2



−1 j ( j +1 − j )

xj >1

−1 d
c ln Mv2A1 , .

The proof is completed.  Theorem 4. If v ∈ Ar (), jx v ∈ Ar−1 () and integer r  1, then for 0

1, IL,M v − v,,
c(ln M)1/2 M +1/2−r/2 (|v|Ar , + |jx v|Ar−1 , ). Proof. By Theorems 1 and 2 and Lemma 2, IL,M v − PM v, = IL,M (PM v − v), 1/2

1/2


c(PM v − v, + PM v − v, |PM v − v|1,,

+ (ln M)1/2 PM v − vA1 , )
c(M −r/2 |v|Ar , + M 1/2−r/2 |v|Ar , |jx v|Ar−1 , 1/2

+ (ln M)1/2 M 1/2−r/2 |v|Ar , ). Therefore IL,M v − v,
IL,M v − PM v, + PM − v,
c(ln M)1/2 M 1/2−r/2 (|v|Ar , + |jx v|Ar−1 , ).

1/2

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Next, for any  ∈ PM (), jx ,
cM, (see [2]). This with Theorem 2 leads to that |IL,M v − v|1,,
|IL,M v − PM v|1,, + |PM v − v|1,,
cM(IL,M v − v, + PM v − v, ) + |PM v − v|1,,
c(ln M)1/2 M 3/2−r/2 (|v|Ar , + |jx v|Ar−1 , ).

Finally we use space interpolation to complete the proof.  We now derive an important result on the modified Laguerre interpolation. We introduce the space B r (). For integer r  0, its norm is given by 1/2  r  x (r−1)/2 (x + 1)1/2 jrx v2 . vB r , = k=0

For any real r > 0, we define the space B r () by space interpolation. Theorem 5. For any v ∈ B r , r  1 and 0

1,
L,M v − v,
c(ln M)1/2 M +1/2−r/2 vB r , . I

(30)


L,M v(x) = e−x/2 IL,M u(x). By Theorem 4, Proof. Let u(x) = ex/2 v(x). Then I
L,M v − v = IL,M u − u,
c(ln M)1/2 M 1/2−r/2 (|u|Ar , + |jx u|Ar−1 , ) I
c(ln M)1/2 M 1/2−r/2 vB r , .


L,M v − v|1, similarly. Finally we use space interpolation to complete the proof.  We can estimate |I In the error estimations, we need an imbedding inequality. In fact, for any v ∈ H 1 (), we have that v(x) → 0, a e. as x → ∞. Therefore √ 1/2 1/2 sup |v(x)|
2v |v|1, . (31) ¯ x∈

4.2. Multidomain Legendre interpolation We first derive some results on the Legendre approximation, which are more precise than those in the existing literatures. Let P0N (Iˆ) = {v ∈ PN (Iˆ)|v(±1) = 0}. We introduce the orthogonal projections PˆN , PˆN1 and PˆN1,0 , defined by (PˆN v − v, )Iˆ = 0,

∀v ∈ L2 (Iˆ),

 ∈ PN (Iˆ),

(jx PˆN1 v − jx v, jx )Iˆ + (PˆN1 v − v, )Iˆ = 0, (jx PˆN1,0 v − jx v, jx )Iˆ = 0,

v ∈ H01 (Iˆ),

∀v ∈ H 1 (Iˆ),  ∈ P0N (Iˆ).

 ∈ PN (Iˆ),

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317

Lemma 3. If v, (1 − x 2 )r/2 jrx v ∈ L2 (Iˆ) and integer r  0, then PˆN v − vIˆ
cN −r (1 − x 2 )r/2 jrx vIˆ .

(32)

Moreover, if v ∈ H  (Iˆ), (1 − x 2 )(r−1)/2 jrx v ∈ L2 (Iˆ) and integer r  1, then for 0

1, PˆN1 v − v,Iˆ
cN −r (1 − x 2 )(r−1)/2 jrx vIˆ .

(33)

1 ˆ Proof. The first result comes from  x [1,5]. Next, due to [5, Lemma 2.3], for any v ∈ H (I ) with v(0)=0, we have vIˆ
c|v|1,Iˆ . Let (x)= −1 PˆN −1 jy v(y) dy + , where is chosen in such a way that v(0)= (0). Then by projection theorem and (32),

PˆN1 v − v1,Iˆ
 − v1,Iˆ
c| − v|1,Iˆ
cPˆN −1 jx v − jx vIˆ
cN 1−r (1 − x 2 )(r−1)/2 jrx vIˆ . This leads to (33) with  = 1. Moreover, we can derive (33) with  = 0 by the previous result and a duality argument as usual. Finally, result (33) with 0 <  < 1 follows from space interpolation.  Lemma 4. If v ∈ H01 (Iˆ), (1 − x 2 )(r−1)/2 jrx v ∈ L2 (Iˆ) and integer r  1, then for any 0

1, PˆN1,0 v − v,Iˆ
cN −r (1 − x 2 )(r−1)/2 jrx vIˆ . Proof. We first prove the desired result with  = 1. Let    x  1 0 PˆN −1 jy v(y) − PˆN −1 jz v(z) dz dy. vN (x) = 2 Iˆ −1 0 ∈ P0 (Iˆ). By the Poincaré inequality, projection theorem and (32), Clearly vN N 0 0 − vN |1,Iˆ PˆN1,0 v − v1,Iˆ
c|PˆN1,0 v − v|1,Iˆ
c|vN      
c PˆN −1 jx v − jx vIˆ +  (PˆN −1 jx v(x) − jx v(x)) dx  Iˆ


cPˆN −1 jx v − jx vIˆ
cN 1−r (1 − x 2 )(r−1)/2 jrx vIˆ .

(34)

Next, using (34) and a duality argument as in the proof of Lemma 3.16 of [7], we reach that PˆN1,0 v − vIˆ
cN −r (1−x 2 )(r−1)/2 jrx vIˆ . Finally, the desired result follows from the above statements and space interpolation.  Lemma 5. If v ∈ H 1 (Iˆ), (1 − x 2 )(r−1)/2 jrx v ∈ L2 (Iˆ) and integer r  1, then ˆ L,N v − v ˆ
cN −r (1 − x 2 )(r−1)/2 jrx v ˆ . I I I Proof. Let ∗ (x) = PˆN1,0 (v(x) − v ∗ (x)) + v ∗ (x), vN

v ∗ (x) =

1−x 1+x v(−1) + v(1). 2 2

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B.-Y. Guo, L.-L. Wang / Journal of Computational and Applied Mathematics 190 (2006) 304 – 324

1 ∗ (x) ∈ P (Iˆ) and v ∗ (±1) = v(±1). Moreover, |v ∗ | Clearly, vN N N 1,Iˆ = 2 |v(−1) − v(1)|
c|v|1,Iˆ . Thus, we have from Lemma 4 that for 0

1
r, ∗ − v,Iˆ = PˆN1,0 (v − v ∗ ) − (v − v ∗ ),Iˆ vN


cN −r (1 − x 2 )(r−1)/2 jrx (v − v ∗ )Iˆ
cN −r (1 − x 2 )(r−1)/2 jrx vIˆ .

(35)

∗ = v ∗ , we use Lemma 4.9 of [7] and (35) to obtain that ˆ L,N vN Since I N ∗ ∗ ∗ ∗ ˆ L,N (vN ˆ L,N v − vN Iˆ = I − v)Iˆ
c(vN − vIˆ + N −1 (1 − x 2 )1/2 jx (vN − v)Iˆ ) I


cN −r (1 − x 2 )(r−1)/2 jrx vIˆ .

The above with (35) leads to the desired result.  Finally, we present the main results of this subsection. Define the affine mapping v(x) = v( ˆ x), ˆ

x=

hkj 2

xˆ +

djk−1 + djk 2

,

xˆ ∈ Iˆ, x ∈ Ijk .

(36)

ˆ x)= ˆ 21 hkj jx v(x). Let
kj (x)= 41 (x −djk−1 )(djk −x)(hkj )−2
1. By (36), we have that 1− xˆ 2 =
kj (x) and jxˆ v( Thus we have that ˆ Iˆ
c(hkj )r−1/2 (
kj )(r−1)/2 jrx vI k . (1 − xˆ 2 )(r−1)/2 jrxˆ v

(37)

j

For description of the multidomain Legendre approximation, we introduce several piecewise Sobolev spaces. Let v k,j (x) = v(x)|I k , and define Hˆ 1 (k ) = {v|v k,j (x) ∈ H 1 (Ijk ), 1
j
Jk }, equipped with the j following norm and semi-norm ⎞1/2 ⎞1/2 ⎛ ⎛ Jk Jk   v k,j 2H 1 (I k ) ⎠ , |v|Hˆ 1 (k ) = ⎝ jx v k,j 2I k ⎠ . vHˆ 1 (k ) = ⎝ j =0

j

j

j =0


r (k )(r  1). For integer r, their semi-norms are given by We also define the spaces H¯ r (k )(r  0) and H ⎞1/2 ⎞1/2 ⎛ ⎛ Jk Jk   |v|H¯ r (k ) = ⎝ (
kj )r/2 jrx v k,j 2I k ⎠ , |v|H
r (k ) = ⎝ (
kj )(r−1)/2 jrx v k,j 2I k ⎠ . j

j =1

j

j =1

For any real r  0, we define these spaces by space interpolation.
N : L2 (k ) → VN (k ) and P
1 : H
1 (k ) → VN (k ), defined Next, we introduce the operators P k k k Nk by
N v)k,j = Pˆ k vˆ k,j (x),
N v)| k (x) = (P ˆ (P k k I N j

j


1 v)| k (x) = (Pˆ 1 k vˆ k,j )(x), (P ˆ Nk I N j

j

1
j
Jk .

Let h¯ k = maxj ∈Jk {hkj · (Njk )−1 }. We have the following important result. Theorem 6. For any v ∈ H¯ r (k ) and integer r  0,
N v − v
ch¯ r |v| ¯ r P k k k H (k ) .

(38)

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319


r (k ) and integer r  1, For any v ∈ H
1 v − v + h¯ k |P
1 v − v| ˆ 1
ch¯ r |v| ¯ r P k k Nk Nk H (k ) . H ()

(39)

Proof. By using (32), (36) and (37), we verify that
N v − v2 = P k k

Jk  j =1


c


N v)k,j − v k,j 2k
c (P k I

J  j =1


c

J  j =1

j

Jk  j =1

hkj PˆN k vˆ k,j − vˆ k,j 2Iˆ j

hkj (Njk )−2r (1 − xˆ 2 )r/2 jrxˆ vˆ k,j 2Iˆ 2 hkj (Njk )−2r (hkj )2r−1 (
kj )r/2 jrx v k,j 2I k
ch¯ 2r k |v|H¯ r ( ) .

(40)

k

j

We can use (34) to prove the second result similarly.  The main result on the multidomain Legendre interpolation IL,Nk is stated below.
r (k ) and integer r  1, we have IL,N v − v
ch¯ r |v|
r Theorem 7. For any v ∈ H k k H ( k ) . k k,j ˆ L,N k vˆ k,j . In view of this = I
Proof. By Definitions (6) and (10), we find that (I
L,Nk v) L,Nk v|I k = I j j fact, we can use Lemma 5 and a similar argument as in the derivation of (40) to obtain the desired result.



In numerical analysis, we also need the following results. Lemma 6. For any  ∈ VNk (k ), √ k
Nk ,k
3k .

(41)


r (k ) and integer r  1, Moreover, for any v ∈ H |(v, )Nk ,k − (v, )k |
ch¯ rk |v|H
r (k ) k ,

∀ ∈ VNk (k ).

(42)

Proof. The proof of the first result is simple. Next, by (11), (38), Theorem 7 and (41),
N −1 v, ) | + |(P
N −1 v, ) − (IL,N v, )N , | |(v, )k − (v, )Nk ,k |
|(v, )k − (P k k k k k k k
N −1 v − v + P
N −1 v − IL,N vN , )
(P k

k

k

k

k


N −1 v − v + IL,N v − v )
(2P k k k k k r 
ch ¯ k |v|H
r (k ) k .

k

k

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B.-Y. Guo, L.-L. Wang / Journal of Computational and Applied Mathematics 190 (2006) 304 – 324

5. Error estimation In this section, we prove the convergence of the mixed pseudospectral method proposed in Section 2. We first consider (9). Theorem 8. Let U (x) and u0M (x) be the solutions of (8) and (9), respectively. If U ∈ 0 H 1 () ∩ B s (), f ∈ B () and  1, then u0M − U 1, + sup |(u0M − U )(x)|
c(M 1/2−s/2 U B s , + (ln M)1/2 M 1/2− /2 f B , ). ¯ x∈


L,M and P¯ 1 be the same as in (3) and (26). Set U ∗ = P¯ 1 U . By (8) and (26), Proof. Let I 0 M 0 M M ∗ ∗ , ) = a (U, ) + ( − 41 )(UM − U, ) , a (UM

∀ ∈ S0M ().

Thus, by (2), (8) and (9), ∗ ∗
L,M f − f, ) , , ) = ( 41 − )(UM − U, ) + (I a (u0M − UM

 ∈ S0M ().

(43)

∗ in (43) and using Theorems 3 and 5, we verify that Taking  = u0M − UM ∗ ∗
L,M f − f  ) 1,
c(UM − U 1, + I u0M − UM


c(M 1/2−s/2 U B s , + (ln M)1/2 M 1/2− /2 f B , ).

This with (31) leads to the desired result.  We next deal with the convergence of (12). Theorem 9. Let U k (x) and ukNk (x) be the solutions of (12) and (14), respectively. If U ∈ 0 H 1 () ∩
r (k ), f ∈ B (), f k ∈ H
(k ) with integers r  1,  0 and real numbers s,  1, B s (), U k ∈ H then we have ukNk − U k k + |ukNk − U k |Hˆ 1 (k )
ck∗ (M 1/2−s/2 U B s , + (ln M)1/2 M 1/2− /2 f B , r−1/2

+ h¯ k

|U k |H
r (k ) + h¯ k |f k |H
(k ) ),

where ck∗ is positive constant only depending on  and the length of k .
1 U k , U k (x) = U k (x) − W k (x), and Proof. Let UNk k = P Nk Nk ,∗ Nk Nk WNk k (x) =

bk − x k x − ak k UNk (ak ) + U (bk ), bk − ak bk − ak Nk

x ∈ k .

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321

Then by (13) and (15), min(, 1)ukNk ,∗ − UNk k ,∗ 2Hˆ 1 ( )
aNk ,k (ukNk ,∗ − UNk k ,∗ , ukNk ,∗ − UNk k ,∗ ) k

= BNk ,k (ukNk ,∗ − UNk k ,∗ ) − aNk ,k (UNk k ,∗ , ukNk ,∗ − UNk k ,∗ ) k , ukNk ,∗ − UNk k ,∗ ) = (f k , ukNk ,∗ − UNk k ,∗ )Nk ,k − aNk ,k (wM

− aNk ,k (UNk k ,∗ , ukNk ,∗ − UNk k ,∗ ) + ak (U∗k , ukNk ,∗ − UNk k ,∗ ) + ak (W k , ukNk ,∗ − UNk k ,∗ ) − (f k , ukNk ,∗ − UNk k ,∗ )k . Thus, we have that min(, 1)ukNk ,∗ − UNk k ,∗ Hˆ 1 (k )


sup

|ak (U∗k , ) − aNk ,k (UNk k ,∗ , )|

∈V0N k  = 0

+ sup

∈VNk  = 0

Hˆ 1 (k )

+ sup

∈VNk  =0

k , )| |ak (W k , ) − aNk ,k (wM Hˆ 1 (k )

|(f k , )k − (f k , )Nk ,k | . Hˆ 1 (k )

(44)

We now estimate the terms at the right-hand side of (44). Firstly, by the definitions of U∗k and UNk k ,∗ , ak (U∗k , ) − aNk ,k (UNk k ,∗ , ) = ak (U k , ) − aNk ,k (UNk k , ) + A1 (),
1 U k , where A1 () = aNk ,k (WNk k , ) − ak (W k , ). Moreover, thanks to (11) and UNk k = P Nk aNk ,k (UNk k , ) = (jx UNk k , jx )k + (UNk k , )Nk ,k = (jx U k , jx )k + (U k , )k + (UNk k , )Nk ,k − (UNk k , )k = ak (U k , ) + A2 () + A3 (), where A2 ()=(UNk k , )Nk ,k −(U k , )k and A3 ()=(U k , )k −(UNk k , )k . The previous statements imply that |ak (U∗k , ) − aNk ,k (UNk k ,∗ , )|
|A1 ()| + |A2 ()| + |A3 ()|.

(45)

So it suffices to estimate |Aj ()|, j = 1, 2, 3. By the Sobolev inequality, for any v ∈ H 1 (a, b), 1/2  1 1/2 1/2 +2 vL2 (a,b) |v|H 1 (a,b) . max |v(x)|
x∈[a,b] b−a In view of (11), (39) and (46), a direct calculation gives that |A1 ()|
ck∗ (|(UNk k − U k )(ak )| + |(UNk k − U k )(bk )|)Hˆ 1 (k )


ck∗ UNk k − U k k |UNk k − U k | ˆ 1 Hˆ 1 (k )
ck∗ h¯ k H (k ) 1/2

1/2

r−1/2

|U k |H
r (k ) Hˆ 1 (k ) .

(46)

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By (11) and Theorem 6,
N −1 U k , ) | + |(P
N −1 U k − U k , )N , | |A2 ()|
|(U k − P k k Nk k k k
N −1 U k − U k  + P
N −1 U k − U k  )
ch¯ r |U k |
c(P k

k

Nk

k

k

k

k

H¯ r (k ) k .

Using Theorem 6 again yields that |A3 ()|
ch¯ rk |U k |H¯ r (k ) k . Substituting the estimates for |Aj ()| into (45), we assert that |ak (U∗k , ) − aNk ,k (UNk k ,∗ , )|
ck∗ h¯ k

r−1/2

|U k |H
r (k ) Hˆ 1 (k ) .

(47)

Next, by Theorem 8, k |aNk ,k (wM , ) − ak (W k , )|
ck∗ (|(u0M − U k )(ak )| + |(u0M − U k )(bk )|)Hˆ 1 (k )


ck∗ (M 1/2−s/2 U B s , + (ln M)1/2 M 1/2− /2 f B , )Hˆ 1 ( ) . k

(48) Moreover, (42) implies that

|(f k , )k − (f k , )Nk ,k |
ch¯ k |f k |H
(k ) k .

(49)

Inserting (47)–(49) into (44), we obtain that ukNk ,∗ − UNk k ,∗ Hˆ 1 (k )
ck∗ (M 1/2−s/2 U B s , + (ln M)1/2 M 1/2− /2 f B , r−1/2

+ h¯ k

|U k |H
r (k ) + h¯ k |f k |H
(k ) ).

(50)

Furthermore, ukNk − U k k
UNk k − U k k + ukNk − UNk k k k
UNk k − U k k + ukNk ,∗ − UNk k ,∗ k + wM − WNk k k


UNk k − U k k + ukNk ,∗ − UNk k ,∗ k + ck∗ (|(u0M − U k )(ak )|

+ |(UNk k − U k )(ak )| + |(u0M − U k )(bk )| + |(UNk k − U k )(bk )|).

(51)

Due to UNk k = P˜N1k U k , we can use Theorem 6 and (50) to estimate the first two terms at the right-hand side of the above inequality, and estimate the last four terms as in the derivations of (48) and the upper-bound of |A1 ()|. Accordingly ukNk − U k k
ck∗ (M 1/2−s/2 U B s , + (ln M)1/2 M 1/2− /2 f B , r−1/2

+ h¯ k

|U k |H
r (k ) + h¯ k |f k |H
(k ) ).

By replacing the norm  · k in (51) by | · |Hˆ 1 (k ) , we can prove the second result similarly.  Using the above theorem and (46), we have the same upper-bound for ukNk − U k L∞ (k ) .

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6. Concluding discussions In this paper, the modified Laguerre interpolation was first proposed for differential equations on the half-line. It keeps the natural weight as in continuous version, and so simplifies computation and numerical analysis. But like the standard Laguerre interpolation, the distance between the large interpolation nodes increases fast as the mode N increases. So the numerical solutions can not describe the character of exact solutions well, if the exact solution varies rapidly between the large nodes. To remedy this deficiency, we used the multidomain Legendre pseudospectral method to refine numerical solution. In other words, we reconstruct the numerical solutions by the multidomain Legendre pseudospectral approximation to recover the accuracy on certain subintervals where the exact solutions vary rapidly. These two techniques matched each other very well. Numerical results demonstrated the efficiency of this method, even for oscillated solutions. This method can be also regarded as a cascade multigrid pseudospectral method on the half line, which is also available for many other problems, such as nonlinear problems and exterior problems. We improved some results on the standard Laguerre interpolation, and first built up the results on the modified Laguerre interpolation. These results are applied successfully to analyzing the proposed method. In fact, they play important role in numerical analysis of pseudospectral methods for various problems on unbounded domains. We established some results on the multidomain Legendre pseudospectral approximation, which served as one of basic tools in the error estimates. In particular, in the expressions of norms appearing in the error estimates, there exist piecewise Jacobi-type weights which tend to zero as x goes to the endpoints of subintervals. Thus the conditions on the smoothness of unknown functions and numerical solutions at the endpoints of subintervals are weekened. The related results seem very appropriate for numerical analysis of domain decomposition spectral method. References [1] I. Babuška, B.Q. Guo, Direct and inverse approximation theorems for the p-version of the finite element methods in the framework of weighted Besov spaces, Part I. Approximability of functions in the weighted Besov spaces, SIAM J. Numer. Anal. 39 (2001) 1512–1538. [2] C. Bernardi, Y. Maday, Spectral methods, in: P.G. Ciarlet, J.L. Lions (Eds.), Handbook of Numerical Analysis, vol. 5, Techniques of Scientific Computing, Elsevier, Amsterdam, 1997, pp. 209–486. [3] O. Coulaud, D. Funaro, O. Kavian, Laugerre spectral approximation of elliptic problems in exterior domains, Comput. Mech. Appl. Mech. Eng. 80 (1990) 451–458. [4] D. Funaro, Estimates of Laguerre spectral projectors in Sobolev spaces, in: C. Brezinski, L. Gori, A. Ronveaux (Eds.), Orthogonal Polynomials and Their Applications, Scientific Publishing Co., Singapore, 1991, pp. 263–266. [5] B.-Y. Guo, Jacobi approximations in certain Hilbert spaces and their applications to singular differential equations, J. Math. Anal. Appl. 243 (2000) 373–408. [6] B.-Y. Guo, J. Shen, Laguerre–Galerkin method for nonlinear partial differential on a semi-infinite interval, Numer. Math. 86 (2000) 635–654. [7] B.-Y. Guo, L.-L. Wang, Jacobi interpolation approximations and their applications to singular differential equations, Adv. Comput. Math. 14 (2001) 227–276. [8] H.-P. Ma, B.-Y. Guo, A multidomain Legendre pseudospectral method for the Burgers-like equation, unpublished. [9] Y. Maday, B. Pernaud-Thomas, H. Vandeven, Une réhabilitation des méthodes spéctrales de type Laguerre, Rech. Aerospat. 6 (1985) 353–379. [10] G. Mastroianni, D. Occorsio, Lagrange interpolation at Laguerre zeros in some weighted uniform spaces, Acta. Math. Hungar. 91 (2001) 27–52.

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[11] J. Shen, Stable and efficient spectral methods in unbounded domains using Laguerre functions, SIAM J. Numer. Anal. 38 (2000) 1113–1133. [12] G. Szegö, Orthogonal Polynomials, Amer. Math. Soc., Providence, RI, 1959. [13] L.-L. Wang, B.-Y. Guo, Stair Laguerre pseudospectral method for differential equations on the half line, Adv. Comput. Math., in press. [14] C.-L. Xu, B.-Y. Guo, Laguerre pseudospectral method for nonlinear partial differential equations, J. Comput. Math. 20 (2002) 413–428.