Applied Mathematics and Computation 182 (2006) 269–282 www.elsevier.com/locate/amc
Spectral method for linear elasticity Jeong Ho Chu a
a,*
, Hi Jun Choe b, Man-Hoe Kim
c
KIS Pricing, Inc., Samchully B/D 2F 35-6 Yoido-Dong, Youngdeungpo-Ku, Seoul 150-885, South Korea b Department of Mathematics, Yonsei University, Seoul, South Korea c Department of Mechanical Engineering, KAIST, Taejon, South Korea
Abstract We find a spectral method for the solutions to linear Lame´ system in the spherical domain. We represent the solutions in terms of spherical harmonics and find an algorithm deciding each coefficients of the given degree. A concrete power series expansions using the associated Legendre functions for three dimension and complex variables functions for two dimension are derived. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Spectral methods; Lame´ systems; Spherical harmonics
1. Introduction We let D Rn be a bounded smooth domain. We let u: D ! Rn be the deformation vector. We assume u is smooth in D and satisfies the linear Lame´ system lDu þ mr div u ¼ 0
ð1Þ
for some positive constants l, m > 0 in D. The positivity assumption of the Lame´ constants l and m is just for convenience of proofs. We observe that u is biharmonic. Hence considering the power series expansion of biharmonic functions, we find the structure of homogeneous solutions of the fixed degree and they form a finite dimensional vector space. In this paper we want to develop a spectral method which fully utilizes the biharmonic structure of the deformation vector. The most interesting feature is a characterization of biharmonic functions in terms of spherical harmonics. After finding a power series expansions of solutions, we can derive a precise structure of homogeneous solutions. Then using the orthogonality of spherical harmonics on the sphere, we develop a spectral method. To find explicit formulas we assume that our domain D is unit ball B1 Rn. For the Stokes system, Lamb [4] has derived a power series expansions of the velocity using the fact that the pressure is harmonic. Then Lamb’s general solution method has been extended to the adjoint method by Schmitz and Felderhof (see: [3,5,6]). Also, Choe and Chu [1,2] have studied a spectral method for the Stokes systems. *
Corresponding author. E-mail addresses:
[email protected] (J.H. Chu),
[email protected] (H.J. Choe),
[email protected] (M.-H. Kim).
0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.01.093
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If there is no confusions, we use the same expressions for vectors and scalars. Double indices mean summation. 2. Characterization of homogeneous solution The characterization of biharmonic function in the punctured domain is essential in series expansions for biharmonic functions. Lemma 1. Suppose that v is a smooth biharmonic function in Dn{0}. Then, v can be written 2
2
vðxÞ ¼ hðxÞ þ jxj kðxÞ þ jxj k 1 ðxÞ þ d2n K x log jxj; where dij is the Kronecker delta function, K is a constant vector, h(x) and k(x) are harmonic in Dn{0} and 8 > < c log jxj for n ¼ 2; 2 k 1 ðxÞ ¼ cjxj log jxj for n ¼ 4; > : 0 for n 6¼ 2; 4 for some c. Suppose v is biharmonic in Dn{0}. We let mk be the dimension of spherical harmonics of degree k and denote n1 {Skm : m = 1, . . . , mk} as the orthonormal basis of spherical harmonics of degree k in L2 ðS n1 is the 1 Þ. Here S R n sphere of radius R in R . Hence taking power series expansions of h and k at the origin we find X mk mk 1 X 1 X X x x 2 knþ2 knþ4 vðxÞ ¼ v1 ðxÞ þ jxj v2 ðxÞ þ akm jxj S km bkm jxj S km þ jxj jxj k¼0 m¼1 k¼0 m¼1 þ d2n ðc1 log jxj þ c2 jxj2 log jxj þ K x log jxjÞ þ d4n ðc3 log jxjÞ; where v1 and v2 are harmonic in whole D. Now considering the power series expansions of harmonic functions v1(x) and v2(x), we can write that with homogeneous harmonic polynomials H ijm ðxÞ of degree j x x akm S km þ bðkþ2Þm S ðkþ2Þm jxj jxj k¼0 m¼1 k¼0 m¼1 m m 0 1 X X x x nþ4 nþ3 þ b0m jxj S 0m b1m jxj S 1m þ jxj jxj m¼1 m¼1 0
vðxÞ ¼
mk 1 X X
0
H 1km ðxÞ
þ jxj
2
H 2ðk2Þm
þ
mk 1 X X
jxj
knþ2
ð2Þ þ d2n ðc1 log jxj þ c2 jxj2 log jxj þ K x log jxjÞ þ d4n ðc3 log jxjÞ. We know that the spherical harmonics S jm jxjx can be written Sjm(x) = jxjjPjm(x) for some homogeneous harmonic polynomial Pjm(x) of degree j. Therefore we have 0
mk 1 X X k¼0 m¼1
0
knþ2
jxj
ðakm S km þ bðkþ2Þm S ðkþ2Þm Þ ¼
mk 1 X X
jxj
2kn
2
ðjxj P 1km þ P 2ðkþ2Þm Þ;
k¼0 m¼1
P jkm ðxÞ
are homogeneous harmonic polynomials of degree k. where Since u is biharmonic, u has a power series expansion of the form (2). Now we decide the structure of homogeneous polynomial solutions to linear Lame´ system (1). In the following expressions we use the same notations for vectors and scalars. Lemma 2. Let k P 2. Suppose v(x) = Pk(x) is a solution to linear Lame´ system (1), where Pk(x) is a homogeneous polynomial of degree k. Then v(x) can be written as m 2 jxj rðr H k ðxÞÞ vðxÞ ¼ H k ðxÞ 2lðn þ 2k 4Þ þ 2mðk 1Þ for some homogeneous harmonic polynomial Hk(x) of degree k.
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271
Proof. From the characterization of biharmonic function we know that 2
vðxÞ ¼ H k ðxÞ þ jxj H k2 ðxÞ for some homogeneous harmonic polynomials Hk and Hk2. Note that (x Æ $)Pk = kPk for homogeneous polynomial Pk. From direct calculations, we have r v ¼ div H k ðxÞ þ 2x H k2 ðxÞ þ jxj2 div H k2 ðxÞ; i
ð3Þ 2
i
rðdiv vÞ ¼ rdiv H k þ 2H k2 þ 2x rH þ 2xðdiv H k2 Þ þ jxj rðdiv H k2 Þ; Dv ¼ 2ðn þ 2k 4ÞH k2 . We know that div v is harmonic and hence Ddiv v ¼ ð2n þ 4k 8Þdiv H k2 ¼ 0. Since lDv + m$(div v) = 0, we have mrðdiv H k Þ þ ð2lðn þ 2k 4Þ þ 2mÞH k2 þ 2mxi rH ik2 ¼ 0. Now our crucial observation is that curl H k2 ¼ 0 and hence we obtain o j o i H ¼ H . oxi k2 oxj k2 Therefore we find that xi rH ik2 ¼ ðk 2ÞH k2 . Consequently, from the equation, we have m H k2 ¼ rðdiv H k Þ 2lðn þ 2k 4Þ þ 2mðk 1Þ and this completes the proof.
h
To have full description of solution to linear Lame´ system in a punctured domain, we need to consider homogeneous solutions of negative degree. The following characterization of homogeneous solutions of negative degree is important. Lemma 3. Let k P 0. Suppose that vðxÞ ¼ jxj
2kn
P kþ2 ðxÞ
is a solution to linear Lame´ system (1) for a homogeneous polynomial Pk+2 of degree k + 2. Then we have m jxj2 rdivðjxj2knþ2 H k ðxÞÞ vðxÞ ¼ jxj2knþ2 H k ðxÞ þ 2lð2k þ nÞ þ 2mðk þ n 1Þ for some homogeneous harmonic polynomial Hk(x) of degree k. Proof. From the power series expansion of biharmonic polynomials we know that vðxÞ ¼ jxj
2kn
2
ðjxj H k ðxÞ þ H kþ2 ðxÞÞ
for some homogeneous harmonic polynomial Hk and Hk+2 of degree k and k + 2, respectively. We set u0 ðxÞ ¼ jxj
2knþ2
H k ðxÞ
and
u1 ðxÞ ¼ jxj
2kn2
2
H kþ2 ðxÞ.
We note v = u0 + jxj u1 and u0 and u1 are harmonic. Hence from direct calculations Dv ¼ ðDjxj
2kn
ÞH kþ2 þ 2ðrjxj
2kn
rÞH kþ2 ðxÞ.
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From homogeneity we know that x rH kþ2 ðxÞ ¼ ðk þ 2ÞH kþ2 ðxÞ. Hence we have Dv ¼ 2ð2k þ nÞjxj
2kn2
H kþ2 ðxÞ ¼ 2ð2k þ nÞu1 ðxÞ.
From the equation, we have div v is harmonics and div u1 ¼ 0. Taking the curl of the equations, we have D curl v ¼ 2ð2k þ nÞ curl u1 ¼ 0. In other words, we have o j o u1 ¼ j ui1 i ox ox for all i and j. Since lDv þ mrðdiv vÞ ¼ 0; we have mrðdiv u0 Þ 2lð2k þ nÞu1 þ mð2u1 þ 2xi rui1 Þ ¼ 0. Since u1 is homogeneous of degree (k + n) and o j o u1 ¼ j ui1 ; i ox ox we obtain xi rui1 ¼ ðk þ nÞu1 . Thus we get u1 ¼
m rðdiv u0 Þ. 2lð2k þ nÞ þ 2mðk þ n 1Þ
From this we can express the velocity v in terms of Hk(x). This completes the proof.
h
To decide the structure of terms involving log, we consider the cases n = 2 and n = 4 separately. First, we let n = 4 and v = c4 log jxj be a solution to the linear Lame´ system for some constant vector c4 ¼ ðc14 ; c24 ; c34 ; c44 Þt . From the equation, we have oxoj vi oxo i vj is harmonic for all i, j = 1, 2, 3, 4. Hence we get c4 ¼ cð1; 1; 1; 1Þ
t
for some constant c. Furthermore, since div v = c(x1 + x2 + x3 + x4)jxj2 is harmonic only if c = 0, we conclude c¼0 and no log-term is involved in the power series expansion when n = 4. Secondly, we consider the case n = 2. Matching the degree of each homogeneous parts, we find a representation of solutions to linear Lame´ system (1) in terms of homogeneous harmonic polynomials. In the power series expansion of biharmonic function, we need to find a structure condition for 2
c1 log jxj þ c2 jxj log jxj þ K x log jxj; where c1 and c2 are two dimensional constant vectors and K is two by two matrix. From a direct computation, we get 2
2
2
lDðc2 jxj log jxjÞ þ mrdivðc2 jxj log jxjÞ ¼ 4lc2 ð1 þ log jxjÞ þ mð2c2 log jxj þ c2 þ 2c2 xjxj xÞ ¼ 0
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273
and this implies c2 = 0. Now we have lDððKxÞ log jxjÞ þ mrdivððKxÞ log jxjÞ 2
¼ 2lðKxÞjxj
2
2
2
4
þ m trace Kjxj x þ mjxj ðKxÞ þ mjxj ðxt KÞ 2mjxj ðxt KxÞx ¼ 0
and taking inner product with respect to x we get xt Kx ¼ cjxj2 for some constant c. Hence we have K ¼ cI; where I is the identity matrix. Hence if l þ m > 0; we have c¼0 and consequently we have K ¼ 0. Finally, we know that for any constant vector A m jxj2 ðA xÞx A log jxj 2l þ m is a constant multiple of the fundamental tensor. Therefore considering all the previous structure lemmas, we obtain the following theorem for punctured domain. Theorem 1. Let u be a solution to linear Lame´ system in Rnn{0}. Then u can be expressed as a power series X 1 m 1 2 jxj ðA xÞx A þ H k ðxÞ uðxÞ ¼ d2n A log jxj 2l þ m 2 k¼0 1 X m 2 2knþ2 jxj rðdiv H k ðxÞÞ þ jxj J k ðxÞ 2lðn þ 2k 4Þ þ 2mðk 1Þ k¼0 m 2 2knþ2 jxj rdivðjxj þ J k ðxÞÞ 2lð2k þ nÞ þ 2mðk þ n 1Þ
for some constant vector A and homogeneous harmonic polynomials Hk and Jk. the dimension all homogeneous harmonic polynomials of degree k is Recall that of the set of nþk1 nþk3 m . Here, we defined ¼ 0 for j 6 1. Thus we have the following corollary. k k3 j Corollary 1. Let n P 2 and Sðn; kÞ ¼ fvk ðxÞg be the set of all homogeneous solutions to linear Lame´ system in Rn of degree k. Then the dimension d(n, k) of Sðn; kÞ is " ! !# 8 nþk1 nþk3 > > > dðn; kÞ ¼ n for k P 1; > > > k k2 > > > > < dðn; kÞ ¼ n for k ¼ 0; > > > dðn; ðn þ 2ÞÞ ¼ n > " ! !# > > > n þ k 1 n þ k 3 > > > for k P 1; : dðn; ðn þ 2Þ kÞ ¼ n k k2 where d(n, (n + 2)) for n = 2 is the dimension of the terms involving d2n in the expressions of Theorem 1.
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3. Spectral method for interior domain In this section we want to develop a concrete algorithm solving linear Lame´ system in B1 ð0Þ
lDu þ mr div v ¼ 0
with the boundary condition on S 1n1
u¼f
for some L2 ðS n1 1 Þ function f. Here we consider only two and three dimensions. In case of two dimension, we take complex variable method and the computation is rather simple. Meanwhile we have to use the associated Legendre functions for the three dimension and the algorithm is more complicated. For the completeness, the explicit relations of the derivatives of the associated Legendre functions are listed in Appendix A. 3.1. Two dimensional case When n = 2, by Corollary 1, the dimension of Sð2; kÞ ¼ 4 for all k P 2 and hence independent of degree k. Employing complex variables and considering the structure lemma (Lemma 2), the basis of homogeneous polynomial solutions of degree k can be written as ! 1 1 kmjfj2 fk2 fk ð1Þ uk ðfÞ ¼ pffiffiffiffiffiffi þ pffiffiffiffiffiffi ; 2p 0 2 2pð2l þ mÞ 1 2 i 0 1 kmjfj fk2 ð2Þ þ pffiffiffiffiffiffi uk ðfÞ ¼ pffiffiffiffiffiffi k 2 2pð2l þ mÞ 1 2p f for k P 2 and ð1Þ uk ðfÞ
1 ¼ pffiffiffiffiffiffi 2p
fk 0
! ð2Þ uk ðfÞ
;
1 ¼ pffiffiffiffiffiffi 2p
0
fk
for k = 0, 1, where f = x + iy. We know that from Theorem 1 u can be represented by a power series of homogeneous polynomial solutions. Then, taking terms of degree lower than or equal to N, u can be approximated by the homogeneous polynomial solution N n o X ð1Þ ð1Þ ð2Þ ð2Þ uN ¼ c k uk þ c k uk ; k¼0 ð1Þ ck ,
ð2Þ
ck are complex coefficients to be determined. where By the completeness of spherical harmonics in L2 ðS 11 Þ, we can approximate f by fN so that ! ð1Þ N X eikh f^ k ih pffiffiffiffiffiffi fN ðe Þ ¼ ; 2p f^ ð2Þ k¼0 k
where Z 2p 1 ðjÞ f^ k ¼ pffiffiffiffiffiffi f ðjÞ ðeih Þeikh dh; 2p 0 f ðeih Þ ¼ ðf ð1Þ ðeih Þ; f ð2Þ ðeih ÞÞ. Since uN = fN on the boundary, we have N n X k¼0
ð1Þ ð1Þ c k uk
þ
ð2Þ ð2Þ c k uk
o
N X eikh pffiffiffiffiffiffi ¼ 2p k¼0
ð1Þ f^ k ð2Þ f^ k
! .
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275
From the orthogonality of the spherical harmonics, we can determine the coefficients cik in the following algorithm: ð1Þ
ck þ ð2Þ
ck þ
mðk þ 2Þ ð1Þ imðk þ 2Þ ð2Þ ð1Þ ckþ2 þ c ¼ f^ k ; 2ð2l þ mÞ 2ð2l þ mÞ kþ2 mðk þ 2Þ ð1Þ mðk þ 2Þ ð2Þ ð2Þ c c ¼ f^ k 2ð2l þ mÞ kþ2 2ð2l þ mÞ kþ2
for k = 0, 1, 2, . . . , N 2 and ð1Þ
ð1Þ
cN 1 ¼ f^ N 1 ; ð1Þ ð1Þ cN ¼ f^ N ;
ð2Þ
ð2Þ
cN 1 ¼ f^ N 1 ; ð2Þ ð2Þ cN ¼ f^ N .
ðiÞ Therefore we found an algorithm which determines cik from the values of f^ k . Note that the even degree coefficients of f^ decides the even degree coefficients of u and the odd degree coefficients of f^ decides the odd degree coefficients of u.
3.2. Three dimensional case When n = 3, by Corollary 1, the dimension of Sð3; kÞ ¼ 3ð2k þ 1Þ. The basis of homogeneous harmonic polynomial of degree k in spherical coordinate is H kl ðr; h; uÞ ¼ rk Y lk ðh; uÞ; Y lk ðh; uÞ ¼ P lk ðcos hÞeilu ; where P lk is the normalized associated Legendre’s function. To find the coefficients cik , we need to express $2Hkl in term of H(k2)l. Since {H(k2)l: l = (k 2), (k 1), . . . , k 2} form a basis for homogeneous solutions of degree k 2, $2Hkl can be written as a linear combination of {H(k2)j: j =(k 2), (k 2), . . . , k 2}. Furthermore, $2Hkl is a linear combinations of the terms H(k2)(l2), H(k2)(l1), H(k2)l, H(k2)(l+1) and H(k2)(l+2). We refer to Appendix A for the detailed proof. Consequently $2Hkl can be written as lþ2 X o2 H kl ðr; h; uÞ ¼ ai;j ðk; l; mÞH ðk2Þm . oxi oxj r¼1 m¼l2 From the structure lemma (see Lemma 2), the basis of homogeneous polynomial solution of degree k become 1 m a ðk; l; mÞY k2 C B m¼l2 1;1 0 k l 1 C B r Y k ðh; uÞ C B k lP þ2 C B mr B C ð1Þ m C B uk;l ðr; h; uÞ ¼ @ a1;2 ðk; l; mÞY k2 C; 0 A B 2lð2k 1Þ þ 2mðk 1Þ B m¼l2 C C B 0 A @ lP þ2 m a1;3 ðk; l; mÞY k2 0
lP þ2
m¼l2
1 m a ðk; l; mÞY k2 C B m¼l2 2;1 0 1 C B 0 C B k lP þ2 C B mr B k l C ð2Þ m C B uk;l ðr; h; uÞ ¼ @ r Y k ðh; uÞ A a2;2 ðk; l; mÞY k2 C; B 2lð2k 1Þ þ 2mðk 1Þ B m¼l2 C C B 0 A @ lþ2 P m a2;3 ðk; l; mÞY k2 0
lP þ2
m¼l2
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0
lP þ2
1 a3;1 ðk; l; mÞY mk2
C B C B m¼l2 C B k lP þ2 C B mr B C ð3Þ m C B uk;l ðr; h; uÞ ¼ @ a3;2 ðk; l; mÞY k2 C A B 2lð2k 1Þ þ 2mðk 1Þ B m¼l2 C rk Y lk ðh; uÞ C B lþ2 A @ P a3;3 ðk; l; mÞY mk2 0
1
0 0
m¼l2
for k P 2 and
0
rk Y lk
0
1
B C ð1Þ uk;l ðr; h; uÞ ¼ @ 0 A; 0
0
0
1
B C ð2Þ uk;l ðr; h; uÞ ¼ @ rk Y lk A; 0
0
1
B C ð3Þ uk;l ðr; h; uÞ ¼ @ 0 A rk Y lk
for k = 0, 1 and l 6 k 6 l. Then u can be approximated by the homogeneous polynomial solution N X k n o X ð1Þ ð1Þ ð2Þ ð2Þ ð3Þ ð3Þ uN ¼ ck;l uk;l þ ck;l uk;l þ ck;l uk;l ; k¼0 l¼k ð1Þ ð2Þ ð3Þ ck;l ; ck;l ; ck;l
are to be determined. where By the completeness of spherical harmonics, we can approximate f by fN so that 0 ð1Þ 1 f^ N X k B k;l C X B f^ ð2Þ CY l ðh; uÞ; fN ðh; uÞ ¼ @ k;l A k k¼0 l¼k ð3Þ f^ k;l
where ðjÞ f^ k;l ¼
Z pZ 0
2p
f ðjÞ ðY lk Þ sin h du dh;
0
f ðh; uÞ ¼ ðf ð1Þ ðh; uÞ; f ð2Þ ðh; uÞ; f ð3Þ ðh; uÞÞ. Since uN = fN on the boundary, we have
0 1 ð1Þ f^ N X l¼k B k C n o X ð1Þ ð1Þ ð2Þ ð2Þ ð3Þ ð3Þ B f^ ð2Þ CY l ðh; uÞ. ck;l uk;l þ ck;l uk;l þ ck;l uk;l ¼ @ k A k k¼0 l¼k ð3Þ f^ k
From the orthogonality of the spherical harmonics, ð1Þ
ck;l ð2Þ
ck;l ð3Þ
ck;l
3 lþ2 X X m ðpÞ ð1Þ c ap;1 ðk þ 2; q; lÞ ¼ fk;l ; 2lð3 þ 2kÞ þ 2mðk þ 1Þ p¼1 q¼l2 kþ2;q 3 lþ2 X X m ðpÞ ð2Þ c ap;2 ðk þ 2; q; lÞ ¼ fk;l ; 2lð3 þ 2kÞ þ 2mðk þ 1Þ p¼1 q¼l2 kþ2;q 3 lþ2 X X m ðpÞ ð3Þ c ap;3 ðk þ 2; q; lÞ ¼ fk;l 2lð3 þ 2kÞ þ 2mðk þ 1Þ p¼1 q¼l2 kþ2;q
for k = 0, 1, 2, . . . , N 2 and ð1Þ
ð1Þ
cN 1;l ¼ fN 1;l ; ð1Þ
ð1Þ
cN ;l ¼ fN ;l ;
ð2Þ
ð2Þ
cN 1;l ¼ fN 1;l ; ð2Þ
ð2Þ
cN ;l ¼ fN ;l ;
ð3Þ
ð3Þ
cN 1;l ¼ fN 1;l ; ð3Þ
ð3Þ
cN ;l ¼ fN ;l :
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277
4. Spectral method of exterior domain Once the spectral method for the interior domain of the ball has been derived, we develop a spectral method for the exterior domain of the ball. Here, our solutions u satisfy lDu þ mr div u ¼ 0 in Rn n B1 ð0Þ and satisfy the boundary condition u¼f
and
S 1n1
and the boundary condition at infinity lim uðxÞ ¼ Oðjxj
2n
jxj!1
þ d2n log jxjÞ.
Moreover we also assume that f 2 L2 ðS n1 1 Þ. For the exterior problems, we have to find a power series expansions involving homogeneous solutions of nonpositive degree. If we recall the structure lemma for the negative degree, we expect the algorithm is more complicated. In fact we have to consider the logarithmic functions for the two dimension cases and more complicated expressions for deformation vector. From the decay condition at infinity, we find the coefficient of the constant vector in the power series expansion is zero. From Corollary 1, we know that Sð2; f Þ ¼ 2 and Sð2; kÞ ¼ 4 for k P 1, where Sð2; f Þ is the set of all homogeneous polynomial solutions of fundamental solution degree. 4.1. Two dimensional case When n = 2, by corollary, the dimension of Sð2; kÞ ¼ 4. The basis of homogeneous polynomial solution of degree k can be written as ! ! fk 1 1 mkffk1 ð1Þ uk ðfÞ ¼ pffiffiffiffiffiffi ; þ pffiffiffiffiffiffi 2p 2pð4l þ 2mÞ i 0 ! ! 0 i 1 mkffk1 ð2Þ uk ðfÞ ¼ pffiffiffiffiffiffi þ pffiffiffiffiffiffi 2p fk 2pð4l þ 2mÞ 1 for k P 1 1 1 ð1Þ ; u0 ¼ pffiffiffiffiffiffi 2p 0 0 1 ð2Þ u0 ¼ pffiffiffiffiffiffi ; 2p 1 ð3Þ u0
ð4Þ u0
¼ ¼
14 12 log jfj 0 0 14 12 log jfj
! þ ! þ
1 8jfj
2
1 8jfj2
f2 þ ff þ f2 if2 þ if2
! ;
if2 þ if2 f2 þ 2ff f2
!
for k = 0, where f = x + iy. From the L2 stability of linear Lame´ system for the exterior domain, u can be approximated by the homogeneous polynomial solution N n o X ð1Þ ð1Þ ð2Þ ð2Þ ð3Þ ð3Þ ð4Þ ð4Þ ð1Þ ð1Þ ð2Þ ð2Þ uN ¼ c 0 u0 þ c 0 u0 þ c 0 u0 þ c 0 u0 þ c k uk þ c k uk ; k¼1
where
ð1Þ ck ,
ð2Þ ck
are complex coefficients to be determined.
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By the completeness of spherical harmonics, we can approximate f by ! ð1Þ N X eikh f^ k ih pffiffiffiffiffiffi ; fN ðe Þ ¼ 2p f^ ð2Þ k¼0 k
where Z 2p 1 ðjÞ ^ f k ¼ pffiffiffiffiffiffi f ðjÞ ðeih Þeikh dh; 2p 0 f ðeih Þ ¼ ðf ð1Þ ðeih Þ; f ð2Þ ðeih ÞÞ. Since uN = fN on the boundary, we have ð1Þ ð1Þ c0 u0
þ
ð2Þ ð2Þ c 0 u0
þ
ð3Þ ð3Þ c 0 u0
þ
ð4Þ ð4Þ c 0 u0
þ
N n X k¼0
ð1Þ ð1Þ ck uk
þ
ð2Þ ð2Þ c k uk
o
N X eikh pffiffiffiffiffiffi ¼ 2p k¼0
ð1Þ f^ k ð2Þ f^
! .
k
Now assume uðfÞ ¼ lim jfj!1 log jfj
Að1Þ
!
Að2Þ
;
we can determine the following: ð3Þ
c0 ¼ 2Að1Þ ;
ð4Þ
c0 ¼ 2Að2Þ .
From the orthogonality of the spherical harmonics, for k = 0, ð1Þ
ð1Þ
0
0
c0 ¼ f^ 0 ; ð2Þ ð2Þ c ¼ f^ for k = 1, ð1Þ ð1Þ c1 ¼ f^ 1 ; ð2Þ ð2Þ c ¼ f^ 1
1
for k = 2, pffiffiffiffiffiffi pffiffiffiffiffiffi 2p ð3Þ i 2p ð4Þ ^ ð1Þ c0 ¼ f 2 ; c0 þ p8ffiffiffiffiffiffi p8ffiffiffiffiffiffi i 2p ð3Þ 2p ð4Þ ^ ð2Þ ð2Þ c0 c ¼ f2 c2 8 8 0
ð1Þ c2
and mðk 2Þ ð1Þ imðk 2Þ ð2Þ ð1Þ c c ¼ f^ k ; 4l þ 2m k2 4l þ 2m k2 imðk 2Þ ð1Þ mðk 2Þ ð2Þ ð2Þ ð2Þ ck2 c ¼ f^ k ; ck 4l þ 2m 4l þ 2m k2 ð1Þ
ck
where k = 3, 4, 5, . . . 4.2. Three dimensional case For n = 3, the basis of homogeneous solution becomes more complicated than the above. First of all, note that $2Jk(x) defined in Theorem 1 is expressed as the combination of Y mkþ2 and Y mk2 . Hence, the basis of solutions are
J.H. Chu et al. / Applied Mathematics and Computation 182 (2006) 269–282
0
0 B ð1Þ uk;l ðr; h; uÞ ¼ @
rk1 Y lk ðh; uÞ 0 0
lþ2 P
b1;1 ðk; l; mÞY mk2
c1;1 ðk; l; mÞY mkþ2
b2;1 ðk; l; mÞY mk2
c2;1 ðk; l; mÞY mkþ2
b3;1 ðk; l; mÞY mk2
c3;1 ðk; l; mÞY mkþ2
279
1
þ C B m¼l2 C B C B lþ2 C B P C m m C b ðk; l; mÞY þ c ðk; l; mÞY A þ rk1 B 1;2 1;2 k2 kþ2 C; B C B m¼l2 C B A @ lP þ2 m m b1;3 ðk; l; mÞY k2 þ c1;3 ðk; l; mÞY kþ2 1
m¼l2
0
lP þ2
1
þ C B m¼l2 C B C B lþ2 C B P B C ð2Þ m m C uk;l ðr; h; uÞ ¼ @ rk1 Y lk ðh; uÞ A þ rk1 B b ðk; l; mÞY þ c ðk; l; mÞY 2;2 2;2 k2 kþ2 C; B C B m¼l2 C B 0 A @ lP þ2 m m b2;3 ðk; l; mÞY k2 þ c2;3 ðk; l; mÞY kþ2 0
1
0
m¼l2
0
lþ2 P
1
þ C B m¼l2 C B C B lþ2 C B B C ð3Þ k1 B P m m C 0 uk;l ðr; h; uÞ ¼ @ þ r b ðk; l; mÞY þ c ðk; l; mÞY A 3;2 3;2 k2 kþ2 C B C B m¼l2 C B rk1 Y lk ðh; uÞ A @ lP þ2 m m b3;3 ðk; l; mÞY k2 þ c3;3 ðk; l; mÞY kþ2 0
1
0
m¼l2
for k P 0. Suppose that u satisfies the linear Lame´ system and ujS 2 ¼ f . 1
Then u can be approximated by the homogeneous polynomial solution uN ¼
N X k n o X ð1Þ ð1Þ ð2Þ ð2Þ ð3Þ ð3Þ ck;l uk;l þ ck;l uk;l þ ck;l uk;l ; k¼0 l¼k
ð1Þ ck ,
ð2Þ
ð3Þ
ck and ck are real coefficients to be determined. where By the completeness of spherical harmonics, we can approximate f by fN so that 0 ð1Þ 1 f^ k;l N X k B C X B ^ ð2Þ C l fN ðh; uÞ ¼ B f k;l CY k ðh; uÞ; @ A k¼0 l¼k ð3Þ f^ k;l
where ðjÞ f^ k;l ¼
Z pZ 0
2p
f ðjÞ ðY lk Þ sin h du dh;
0
f ðh; uÞ ¼ ðf ð1Þ ðh; uÞ; f ð2Þ ðh; uÞ; f ð3Þ ðh; uÞÞ. Since uN = fN on the boundary, we have 0 n
ð1Þ ð1Þ
ð2Þ ð2Þ
ð3Þ ð3Þ
ck;l uk;l þ ck;l uk;l þ ck;l uk;l
o
¼
N X l¼k X k¼0
ð1Þ f^ k
1
B ð2Þ C l B f^ CY ðh; uÞ. @ k A k l¼k ð3Þ f^ k
280
J.H. Chu et al. / Applied Mathematics and Computation 182 (2006) 269–282
From the orthogonality of the spherical harmonics, ð1Þ
ck;l þ
3 lþ2 X X
ðpÞ
ðpÞ
ð1Þ
ðpÞ
ðpÞ
ð2Þ
ðpÞ
ðpÞ
ð3Þ
ckþ2;q bp;1 ðk þ 2; q; lÞ þ ck2;q cp;1 ðk 2; q; lÞ ¼ fk;l ;
p¼1 q¼l2 ð2Þ
ck;l þ
3 lþ2 X X
ckþ2;q bp;2 ðk þ 2; q; lÞ þ ck2;q cp;2 ðk 2; q; lÞ ¼ fk;l ;
p¼1 q¼l2 ð3Þ
ck;l þ
3 lþ2 X X
ckþ2;q bp;3 ðk þ 2; q; lÞ þ ck2;q cp;3 ðk 2; q; lÞ ¼ fk;l
p¼1 q¼l2
for k = 0, 1, 2, . . . , N. Acknowledgement The first author was supported by KOSEF, GARC and BSRI prg. of Korea Ministry of Educations. Appendix A Computation of ai,j(k,l,m) We know write
o ðrn Y mn Þ ox
is homogeneous polynomial of degree (n 1) as function of (x, y, z). Thus we can
n1 X o n m ðr Y n Þ ¼ ak rn1 Y kn1 . ox k¼ðn1Þ
Moreover, we know that frn1 Y kn1 g are orthogonal on the unit sphere S 21 and hence Z o n m ðr Y n ÞðY kn1 Þ dh du ak ¼ 2 ox S1 Z Z oY mn k 2 m k ðY n1 Þ cos h sin h cos u dh du ¼ n sin h cos uY n ðY n1 Þ dh du þ S 21 S 21 oh Z imY mn sin uðY kn1 Þ dh du. S 21
Using sin u ¼ e
iu eiu
2i
, cos u ¼ e
iu þeiu
and integration over u gives
2i
ak ¼ 0 except for k = m 1, m + 1. Therefore o n m ðþÞ ðÞ n1 m1 ðr Y n Þ ¼ aX ðn; mÞrn1 Y mþ1 Y n1 ; n1 þ aX ðn; mÞr ox where ðþÞ aX ðn; mÞ ðÞ
pn ¼ i
aX ðn; mÞ ¼
pn i
Z
p
P mn P mþ1 n1
0
Z
0
p sin h dh þ i 2
p 2 P mn P m1 n1 sin h dh þ
p i
Z
p
0
Z
0
p
oP mn mþ1 P cos h sin h dh mp oh n1 oP mn m1 P cos h sin h dh þ mp oh n1
Z Z
p
P mn P mþ1 n1 dh; 0 p
P mn P m1 n1 dh. 0
J.H. Chu et al. / Applied Mathematics and Computation 182 (2006) 269–282
By a similar calculation, we can have o n m ðþÞ ðÞ mþ1 þ aY ðn; mÞrn1 Y m1 ðr Y n Þ ¼ aY ðn; mÞrn1 Y n1 n1 ; oy o n m mþ1 ðr Y n Þ ¼ aZ ðn; mÞrn1 Y n1 ; oz where ðþÞ aY ðn; mÞ
pn ¼ i
Z
p
P mn P mþ1 n1 0
p sin h dh þ i 2
Z
p
0
oP mn mþ1 cos h sin h dh þ mp P oh n1
Z
p mþ1 P mn P n1 dh; 0
Z Z Z p pn p m m1 2 p p oP mn m1 P ¼ P P sin h dh cos h sin h dh þ mp P mn P m1 n1 dh; i 0 n n1 i 0 oh n1 0 Z p oP m aZ ðn; mÞ ¼ 2p nP mn P mn1 sin h cos h n P mn1 sin2 h dh. oh 0 ðÞ aY ðn; mÞ
Hence, taking one more differentiation and comparing with the definition of ai,j(k, l, m), we know that ðþÞ
ðþÞ
a1;1 ðk; l; l þ 2Þ ¼ aX ðk; lÞaX ðk 1; l þ 1Þ; ðþÞ
ðÞ
ðÞ
ðþÞ
a1;1 ðk; l; lÞ ¼ aX ðk; lÞaX ðk 1; l þ 1Þ þ aX ðk; lÞaX ðk 1; l 1Þ; ðÞ
ðÞ
ðþÞ
ðþÞ
a1;1 ðk; l; l 2Þ ¼ aX ðk; lÞaX ðk 1; l 1Þ; a1;2 ðk; l; l þ 2Þ ¼ aX ðk; lÞaY ðk 1; l þ 1Þ; ðþÞ
ðÞ
ðÞ
ðþÞ
a1;2 ðk; l; lÞ ¼ aX ðk; lÞaY ðk 1; l þ 1Þ þ aX ðk; lÞaY ðk 1; l 1Þ; ðÞ
ðÞ
a1;2 ðk; l; l 2Þ ¼ aX ðk; lÞaY ðk 1; l 1Þ; ðþÞ
a1;3 ðk; l; l þ 1Þ ¼ aX ðk; lÞaZ ðk 1; l þ 1Þ; ðÞ
a1;3 ðk; l; l 1Þ ¼ aX ðk; lÞaZ ðk 1; l 1Þ. ðþÞ
ðþÞ
a2;1 ðk; l; l þ 2Þ ¼ aY ðk; lÞaX ðk 1; l þ 1Þ; ðþÞ
ðÞ
ðÞ
ðþÞ
a2;1 ðk; l; lÞ ¼ aY ðk; lÞaX ðk 1; l þ 1Þ þ aY ðk; lÞaX ðk 1; l 1Þ; ðÞ
ðÞ
ðþÞ
ðþÞ
a2;1 ðk; l; l 2Þ ¼ aY ðk; lÞaX ðk 1; l 1Þ; a2;2 ðk; l; l þ 2Þ ¼ aY ðk; lÞaY ðk 1; l þ 1Þ; ðþÞ
ðÞ
ðÞ
ðþÞ
a2;2 ðk; l; lÞ ¼ aY ðk; lÞaY ðk 1; l þ 1Þ þ aY ðk; lÞaY ðk 1; l 1Þ; ðÞ
ðÞ
a2;2 ðk; l; l 2Þ ¼ aY ðk; lÞaY ðk 1; l 1Þ; ðþÞ
a2;3 ðk; l; l þ 1Þ ¼ aY ðk; lÞaZ ðk 1; l þ 1Þ; ðÞ
a2;3 ðk; l; l 1Þ ¼ aY ðk; lÞaZ ðk 1; l 1Þ. ðþÞ
a3;1 ðk; l; l þ 1Þ ¼ aZ ðk; lÞaX ðk 1; lÞ; ðÞ
a3;1 ðk; l; l 1Þ ¼ aZ ðk; lÞaX ðk 1; lÞ; ðþÞ
a3;2 ðk; l; l þ 1Þ ¼ aZ ðk; lÞaY ðk 1; lÞ; ðÞ
a3;2 ðk; l; l 1Þ ¼ aZ ðk; lÞaY ðk 1; lÞ; a3;3 ðk; l; lÞ ¼ aZ ðk; lÞaZ ðk 1; lÞ.
281
282
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