Particular Solutions of Chebyshev Polynomials for Polyharmonic and Poly-Helmholtz Equations

Transcription

1 Copyrght c 2008 Tech Scence Press CMES, vo.27, no.3, pp , 2008 Partcuar Soutons of Chebyshev Poynomas for Poyharmonc and Poy-Hemhotz Equatons Cha-Cheng Tsa 1 Abstract: In ths paper we deveop anaytca partcuar soutons for the poyharmonc and the products of Hemhotz-type parta dfferenta operators wth Chebyshev poynomas at rghthand sde. Our soutons can be wrtten expcty n terms of ether monoma or Chebyshev bases. By usng these formuas, we can obtan the approxmate partcuar souton when the rght-hand sde has been represented by a truncated seres of Chebyshev poynomas. These formuas are further mpemented to sove nhomogeneous parta dfferenta equatons (PDEs) n whch the homogeneous soutons are compementary soved by the method of fundamenta soutons (MFS). Numerca experments, whch ncude eghth order PDEs and three-dmensona cases, are carred out. Due to the exponenta convergence of the Chebyshev nterpoaton and the MFS, our numerca resuts are extremey accurate. Keyword: Partcuar souton, Chebyshev poynomas, poyharmonc equaton, product of Hemhotz equaton, boundary eement method, method of fundamenta soutons, Trefftz method, rada bass functon 1 Introducton There are a number of numerca methods that can be cassfed as boundary-type numerca methods [Cheng and Cheng (2005) because the numerca dscretzaton s performed ether on the souton boundary, or on a boundary-e geometry, whch resdes n a ower spata dmenson than the souton doman. Exampes ncude the boundary eement method (BEM), the MFS [Kupradze and 1 Department of Informaton Technoogy, Too Unversty, Cha-Y County, 61363, Tawan. E-ma: tsachacheng@ntu.edu.tw Aesdze (1964); Bogomony (1985), the Trefftz Methods (TM) [Cozzano, Rodríguez (2007); Lu (2007A); Lu (2007B), the meshess oca boundary ntegra equaton method [Zhu, Zhang, and Atur (1998), and et a. Boundary-type numerca methods generay requre the governng equaton to be homogeneous hence a speca agorthm s needed for nhomogeneous PDEs. One way to emnate the nhomogeneous term s by the method of partcuar soutons (MPS) addressed by Goberg and Chen (1999). Consder an nhomogeneous near PDE: Λu = f (x); x Ω (1) whch s subect to the boundary condtons B(u)=g(x); x Γ (2) In the above, Λ and B are parta dfferenta operators, Ω s the souton doman, and Γ s ts boundary. We can decompose the souton nto two parts, a partcuar souton part u p and a homogeneous part u h, such that u = u p +u h (3) We requre the partcuar souton to satsfy Eq. (1) as Λu p = f (x); x Ω (4) but not the boundary condton Eq. (2), such that the souton s easer to fnd. The soutons of Eq. (4) can be found by the MPS whch w be descrbed ater. Then, t s easy to show that the homogeneous souton must satsfy Λu h = 0; x Ω (5) and s subect to the modfed boundary condton Bu h = g(x) Bu p ; x Γ (6)

2 152 Copyrght c 2008 Tech Scence Press CMES, vo.27, no.3, pp , 2008 Eqs. (5) and (6) can then be formay soved by boundary-type numerca methods. The orgna dea of ths formuaton stemmed form the dua recprocty boundary eement method (DRBEM) nnovated by Nardn and Brebba (1982). Actuay, the dua recprocty procedure s equvaent to the MPS. From the above descrpton, t s cear that the appcabty of prevous formuaton depends on the avaabty of the partcuar souton u p assocated wth rght hand sde functon f and the operator Λ. It s obvous that anaytca expressons for u p are rare. Hence, n the same sprt of numerca soutons, approxmate expressons of the partcuar souton are sought. Ths can be accompshed by approxmatng the rght hand sde functon as a summaton of bass (tra) functons, f (x) α ϕ (x) (7) where ϕ are bass functons, and α are constants to be determned by sovng a near system resutng from coocatons or east squares. Once the bass functons n Eq. (7) are seected, the probem of fndng partcuar souton s reduced to Λψ = ϕ (x); x Ω (8) where ψ s the partcuar souton correspondng to ϕ.onceψ s found, the approxmate partcuar souton sought for becomes u p n α ψ (x) (9) Therefore, the success of the MPS depends on the avaabty of the exact expresson of ψ assocated to the bass functon ϕ and operators Λ. In the ast few decades, sgnfcant progress has been made n obtanng anaytca partcuar soutons for varous bass functons. Among these are the rada bass functons [Cheng, Lafe, and Gr (1994); Goberg (1995); Goberg, Chen, and Karur (1996); Goberg and Chen (1999); Mueshov, Goberg, and Chen (1999); Cheng (2000); Mueshov and Goberg (2007), the trgonometrc functons [Atnson (1985); L and, Chen (2004), the monomas [Janssen and, Lambert (1992); Cheng, Lafe, and Gr (1994); Cheng, Chen, Goberg, Rashed (2001); Goberg, Mueshov, Chen, and Cheng (2003), the Chebyshev poynomas [Goberg, Mueshov, Chen, and Cheng (2003); Reutsy and Chen (2006); Karageorghs, and Kyza (2007); Dng and Chen (2007) and others. In ths study, we consder the anaytca partcuar souton correspondng to the Chebyshev poynomas. In the orgna study of Chebyshev nterpoaton, Goberg, Mueshov, Chen, and Cheng (2003) utzed symboc software Mathematca to connect monomas and Chebyshev poynomas and used ther derved partcuar souton as foows Λα x y = x m y n (10), to mpement foatng number computng. However some boo eepngs are requred n ther study. Reutsy and Chen (2006) remeded the tedous boo eepng by usng two-stage approxmatons of trgonometrc functons and Chebyshev poynomas. On the other hand, Karageorghs, and Kyza (2007) studed the same ssue by drecty consderng Λβ T (x)t (y)=t m (x)t n (y) (11), where T m (x) s the Chebyshev poynoma of degree m. However matrx nverses are conducted to ther fna formuas. Thus they have to face the ssue of -condtonng. Recenty, Dng and Chen (2007) dscovered a recursve formuaton free from boo eepngs and matrx nverses. Thus ther formuaton can be mpemented by foatng number computng. It s we nown that systems nvovng the coupng of a set of second order eptc equatons are encountered n some engneerng probems, such as a mutayered aqufer system [Cheng and Morohunfoa (1993A); Cheng and Morohunfoa (1993B), or a mutpe porosty system [Cheng (2000). These couped systems can be reduced to a snge parta dfferenta equaton by usng the Hörmander operator decomposton technque [Hörmander (1963). The resutant parta dfferenta equatons usuay nvove the poyharmonc or the products of Hemhotz-type operators. Ths

3 Partcuar Soutons of Chebyshev Poynomas for Poyharmonc and Poy-Hemhotz Equatons 153 motvated us to generaze the stuaton by consderng anaytca partcuar soutons of Λ α x y z = x m y n z (12),, where the parta dfferenta operator Λ s n a very genera form Λ = (Δ ) A =(Δ 1 ) A 1 (Δ 2 ) A 2...(Δ ) A (13) wth A 1, A 2,..., A N, 1, 2,..., C, and Δ s the Lapacan. We notce that when 0 n Eq. (13), the operator s the product of the Hemhotz-type operators, and when = 1, Eq. (13) becomes the poy-hemhotz or poyharmonc operator. In our mpementaton, we use the expct formuas between monomas and Chebyshev poynomas [Mason and Handscomb (2003) as we as the expct formuas n Eq. (12), whch can be easy coded by mutpe oops. Our mpementaton s free from boo eepngs and matrx nverses. Compared to the recursve formuaton [Dng and Chen (2007), our formuaton s easer and more sutabe for hgher order PDEs and three dmensons. Furthermore, we sove the homogeneous souton by the MFS n our mpementaton [Kupradze and Aesdze (1964); Bogomony (1985); Tsa, Young, Cheng (2002); Smyrs, Karageorghs (2003); Chen, Fan, Young, Murugesan, Tsa (2005); Young, Ruan (2005); Young, Chen, Chen, Kao (2007); Hu, Young, Fan (2008). Ths two-stage procedure forms a boundary-type meshess numerca method, whch s opposte to doman-type meshess numerca methods sovng PDEs n one stage, such as Kansa s method [Emdad, Kansa, Lbre, Rahman, Shearch (2008); Kosec, Šarer (2008); Lbre, Emdad, Kansa, Rahman, Shearch (2008) and the meshess oca Petrov- Gaern method [Gao, Lu, Lu (2006); Han, Lu, Raendran, Atur (2006); Sade, Sade, Wen, Aabad (2006); Sade, Sade, Zhang, Soe (2007); Wu, Shen, Tao (2007) and et a. Our numerca resuts are very accurate even for eghth order PDEs and three dmensons due to the exponenta convergence of the Chebyshev nterpoaton and the MFS. In order to compete the mathematca consderaton, we aso ustrate by an exampe for obtanng expct partcuar soutons wth Chebyshev poynomas at rght hand sde Λ β T (x)t (y)t (z)=t m (x)t n (y)t (z) (14),, Λ γ x y z = T m (x)t n (y)t (z) (15),, by usng the expct formuas between monomas and Chebyshev poynomas [Mason and Handscomb (2003). We do not suggest drect mpementatons of Eqs. (14) and (15) because they are ess effcent. 2 Chebyshev nterpoaton We begn wth the trvarate Chebyshev poynoma nterpoaton f (x, y, z) for a functon f (x,y,z), n whch ower dmensona stuatons are ncuded. The Chebyshev nterpoant usng Gauss-Lobatto nodes for cubc doman [x a, x b [y a, y b [z a, z b taes the form: f (x,y,z)= where m n T ( 2y yb y a y b y a 8 a = mnc, c m, c n, wth ( ) 2x xb x a a T x b x a ) ( ) 2z zb z a T z b z a m cos π n f (x,x,x ) c, c m, c n, cos π π cos m n c,0 = c, = 2, c, = 1, 1 1 x = (x b x a ) cos π 2 + x b +x a 2 y = (y b y a ) cos π 2 m + y b +y a 2 (16) (17) (18a) (18b) (18c)

4 154 Copyrght c 2008 Tech Scence Press CMES, vo.27, no.3, pp , 2008 z = (z b z a ) cos π 2 n + z b +z a (18d) 2 Note that, m and n are the numbers of Gauss- Lobatto nodes n the x, y and z drectons, respectvey. Aso, t s we nown that the use of Gauss-Lobatto nodes ensure the spectra convergence for Chebyshev nterpoaton. Detas of Chebyshev nterpoaton can be found n a recent exceent revew boo of Mason and Handscomb (2003). In the appcaton of MPS, the rght hand sde f (x,y,z) n Eq. (4) s frst approxmated by the Chebyshev nterpoaton. In the foowng dervatons, we assume x a = y a = z a = 1, x b = y b = z b = 1, and [ 1, 1 [ 1, 1 [ 1, 1 bg enough to encose Ω. These assumptons do not ose the generaty. Then, Eq. (16) reduces to the foowng form: f (x,y,z)= m n a T T T (19) Eq. (19) can aso be rewrtten n terms of monoma bass by usng [Mason and Handscomb (2003) T n (x)= wth c (n) [n/2 c (n) x n 2 (20) =0 =( 1) 2 n 2 1n(n 1)!!(n 2)!, n > 2 (21a) c (2) =( 1), 0 (21b) In Eq. (20), the bracets [n the summaton mts ndcate tang the nteger part of the argument. Usng Eqs. (19) and (20) t s enough to obtan f (x,y,z)= m n b x y z (22) It shoud be noted that, we can actuay wrte b expcty, however that s not the best way for mpementatons. We w return to ths ssue n Secton 6. 3 Partcuar soutons of monomas poy- Hemhotz equaton Snce we have the approxmated rght hand sde n terms of monomas, we shoud fnd the partcuar soutons defned by Eq. (12) n order to appy the MPS. In ths secton, we consder Λ =(Δ ) L frst. In other words, we are gong to see P (L,,m,n) (x,y,z) so that (Δ ) L P (L,,m,n) (x,y,z)=x y m z n (23) Goberg, Mueshov, Chen, and Cheng (2003) had found the souton for L = 1, 0 P (1,,m,n) [ 2 [ m 2 [ n 2 (x,y,z)= =0 =0 =0 ( + +)!!m!n!x 2 y m 2 z n !!!( 2)!(m 2 )!(n 2)! or (24) ΔP (1,,m,n) P (1,,m,n) = x y m z n (25) Tang parta dervatve wth respect,we have Δ P(1,,m,n) P (1,,m,n) (Δ ) P(1,,m,n) (Δ ) 2 P(1,,m,n) P(1,,m,n) = P (1,,m,n) = x y m z n From Eq. (26), t s cear that P (2,,m,n) = P(1,,m,n) = 0 Repeatng the above dervatons we can obtan (26) (27) P (L,,m,n) 1 L 1 P (1,,m,n) = (L 1)! L 1 (28) Usng Eqs. (24) and (28) we can have the desred

5 Partcuar Soutons of Chebyshev Poynomas for Poyharmonc and Poy-Hemhotz Equatons 155 partcuar souton for 0 as foows [ 2 [ m P (L,,m,n) 2 [ n 2 (x 1,x 2,x 3 )= =0 =0 =0 [( 1) L ( + + +L 1)!!m!n!x 2 y m 2 z n 2 Itseasytoshowthat ( ) 2 ++L [! +2+2L x 2 x+2 ( L)! = x (33) /[ (L 1)! + ++L!!!( 2)!(m 2 )! (n 2)! (29) ( ) Usng Eq. (33) to emnate the 2 ++L x operator n Eq. (32), we obtan the partcuar souton 2 for poyharmonc operator gven as It shoud be notced that Eq. (29) reduce to twodmensona soutons by settng n = 0. Aso, Eq. (29) can be mpemented very easy by mutpe oops. If we compare our formua wth the recursve formuaton ntroduced by Dng and Chen (2007), t s cear that our formua s easer for mpementaton for hgher order PDEs and three dmensons whch s often occurred n engneerng appcatons. 4 Partcuar soutons of monomas poyharmonc equaton Then, we consder the partcuar soutons correspondng to poyharmonc operators as foows: Δ L P (L,,m,n) 0 (x,y,z)=x y m z n (30) In order to fnd the soutons,we consder Eq. (29) wth = 0 [ m P (L,0,m,n) 2 (x 2,x 3 )= [ n 2 =0 =0 ( 1) L ( + +L 1)!m!n!y m 2 z n 2 (L 1)! ++L!!(m 2 )!(n 2)! (31) Then substtute = 2 to Eq. (31) and mutpy x, we can x 2 obtan [ m Δ L 2 [ n 2 =0 =0 ( 1) + ( + +L 1)!m!n!x y m 2 z n 2 ( ) (L 1)! 2 ++L x!!(m 2 )!(n 2)! 2 = x y m z n (32) [ m P (L,,m,n) 2 0 (x,y,z)= [ n 2 =0 =0 [ ( 1) + ( ++L 1)!!m!n!x L y m 2 z n 2/[ (L 1)!!!( L)!(m 2 )!(n 2)! (34) For L = 1, the resut obtaned n Eq. (34) appears to be smper as compared to those found n Goberg, Mueshov, Chen, and Cheng (2003) for 3D Lapacan operator wth monoma rght hand sde. 5 Partcuar soutons of monomas Λ = (Δ ) A As we have mentoned that many probems n engneerng and scence nvove the product of Hemhotz-type and harmonc operators. Therefore we are gong to derve partcuar soutons n Eq. (12) wth Λ = (Δ ) A. Baneree (1994) ntroduced the dfference trc to fnd the fundamenta souton for the product of Hemhotz and harmonc operators. Here we fnd the dfference trc can be ned to the parta fracton [O Ne (2002). Therefore we use the partcuar soutons of poy-hemhotz and poyharmonc operators, Eqs. (29) & (34), as we as the parta fracton to derve the desred partcuar souton. Consder we have the foowng parta fracton 1 (ξ ) A = =1 A 1 =0 Θ (ξ ) A (35) where the coeffcents Θ can be fnd formay by the decomposton of parta fracton [O Ne

6 156 Copyrght c 2008 Tech Scence Press CMES, vo.27, no.3, pp , 2008 (2002). Then, the partcuar souton correspondng to [ (Δ ) A P (,m,n) (x,y,z)=x y m z n (36) s P (,m,n) (x,y,z)= A 1 Θ P (A,,m,n) (37) =0 where P (A,,m,n) are gven n Eqs. (29) and (34). To compete the dervaton we frst compare Eqs. (23), (30), and (36) to obtan (Δ ) A P (,m,n) (x,y,z) =(Δ ) L P (L,,m,n) (x,y,z) (Δ ) A P (,m,n) (x,y,z) (Δ ) L = P (L,,m,n) (x,y,z) (38) Then, we substtute [ ξ = Δ to Eq. (35), mutpy both sde by (Δ ) A P (,m,n) (x,y,z), and use Eq. (38) to obtan 1 (Δ ) A P (,m,n) (x,y,z) =0 = =0 A 1 =0 [ A 1 Θ (Δ ) A =0 Θ (Δ ) A = (Δ ) A P (,m,n) (x,y,z) P (,m,n) (x,y,z) = A 1 =0 =0 Θ P (A,,m,n) (x,y,z) (39) These fnsh our dervatons. Snce we have the partcuar souton n Eq. (12) we are ready to appy the MPS. We can use Eqs. (29), (34), and (37) wth Eq. (22) to get the approxmated partcuar souton correspondng to F(x,y,z)= f (x,y,z) (40) [ (Δ ) A as foows F(x,y,z)= m n b P (,,) (x,y,z) (41) These compete the MPS formuatons. Exampe 1: Let s fnd the partcuar soutons of Δ 2 (Δ 4)(Δ +9) 2 P(x,y,z)=x y m z n (42) We frst consder the foowng parta fracton by the method of undetermned coeffcent [O Ne (2002): 1 ξ 2 (ξ 4)(ξ +9) 2 = 1 324ξ ξ (ξ 4) (ξ +9) (ξ +9) Then we have the partcuar souton as foows P = P(2,,m,n) P(1,,m,n) P(1,,m,n) P(2,,m,n) P(1,,m,n) (43) (44) where P (2,,m,n) 0 and P (1,,m,n) 0 are gven n Eq. (34) as we as P (1,,m,n) 4, P (2,,m,n) 9,andP (1,,m,n) 9 are addressed n Eq. (29). Exampe 2: Then we try to fnd the partcuar soutons of (Δ 2 +1) ˆP(x,y,z)=x y m z n (45) We frst consder the foowng parta fracton: 1 ξ 2 +1 = 1 (ξ +)(ξ ) = 1 2(ξ ) 1 2(ξ +) (46) Then we have the partcuar souton as foows ˆP = P(1,,m,n) 2 P(1,,m,n) 2 (47) where P (1,,m,n) and P (1,,m,n) are gven n Eq. (29). It s nterestng to note that ˆP s a rea vaued functon athough we are worng n compex numbers. Ths s another advantage over Dng and Chen (2007).

7 Partcuar Soutons of Chebyshev Poynomas for Poyharmonc and Poy-Hemhotz Equatons Partcuar soutons of Chebyshev poynomas In order to compete the mathematca consderaton, we aso ustrate by an exampe about how to obtan expct partcuar soutons wth Chebyshev poynomas at rght hand sde as defned n Eqs. (14) and (15). Consder partcuar soutons of two-dmensona poyharmonc operator as Δ L P(x,y)=T h (x)t (y) (48) By usng Eq. (20) we have T h (x)t (y)= [h/2 [/2 =0 =0 c (h) c () xh 2 y 2 (49) Then we can utze Eq. (34) wth n = 0tohave Δ L P (L,,m,0) 0 (x,y,z)=x y m (50) wth [ m P (L,,m,0) 2 0 (x,y,z)= e L,,m x +2 +2L y m 2 (51a) =0 e (L,,m) ( 1) ( +L 1)!!m! = (L 1)!!( +2 +2L)!(m 2 )! (51b) Combnng Eqs. (48) (50) we can obtan [ h 2 [ 2 P(x, y)= c (h) [ 2 =0 =0 =0 c (), 2 el,h 2 x h L y 2 2 (52) Eq. (52) s ust the desred partcuar souton n terms of monomas. The partcuar souton can aso be rewrtten n terms of Chebyshev poynomas. We frst ntroduce the expct formua to express monomas n terms of Chebyshev poynomas ntroduced by Mason and Handscomb (2003) as foows: x n = [n/2 d (n) T n 2 (x) (53) =0 wth d (n) = 2 1 n n!!(n )! Substtutng Eq. (53) to Eq. (52) we have [ h 2 [ 2 P(x, y)= c (h) =0 =0 c (), 2 el,h 2 [ 2 =0 d [ h L m=0 [ 2 n=0 h L m dn 2 2 (54) T h L 2m (x)t 2 2 2n (x) (55) Eq. (55) s ust the expct partcuar soutons n terms of Chebyshev poynomas. These dervatons can be extended to poy-hemhotz equatons and three dmensons. Aso, the partcuar soutons can be consdered as an aternatve to the formuatons derved by Karageorghs, and Kyza (2007), n whch they had to sove systems of near equatons. Here we provde these formuas n order to show the mathematca possbty of obtanng expct partcuar soutons wth Chebyshev poynomas at rght hand sde. In ths paper, we suggest to mpement the MPS by usng Eqs. (16), (22), and (37) n three stages nstead of drect utzng Eqs. (16) & (52) or Eq. (16) & (55) due to the computatona effcency. We can state the reason cearer that Eqs. (16) and (22) have to be executed once for a gven f (x,y,z) but the modfed boundary condton n Eq. (6) shoud be cacuated at a the dscrete ponts by usng Eq. (37). Thus the three-stage mpementaton s more effcent. 7 Numerca resuts Once we fnd an approxmate partcuar souton, we can sove the homogeneous probem (5) wth gven modfed boundary data (6) by the MFS. Consder [ (Δ ) A u h = 0 (56) Then we can approxmate u h (x) by the MFS as u h (x) = A 1 =0 K =1 γ G (A ) (x s ) (57)

8 158 Copyrght c 2008 Tech Scence Press CMES, vo.27, no.3, pp , 2008 where G L (x) s the fundamenta souton defned by (Δ ) L G L (x)=δ(x) (58) whch can be found n Cheng, Antes, and Ortner (1994). And s are K prescrbed source ponts outsde the computaton domans. The K A unnown coeffcent γ can be found by enforcng the boundary condtons (6) at K boundary fed ponts. It s cear that the order of (Δ ) A s 2 A. Therefore we need A boundary condtons whch are ust the rans of the vectored vaued parta dfferenta operator B n Eq. (6). In other words, the enforcement of boundary condtons at K boundary fed ponts resuts n K A near equatons, whch can be used to sove the K A unnown coeffcents γ. Detas of the MFS can be found n theoretca wor of Bogomony (1985). Aso Tsa, Ln, Young, and Atur (2006) dscussed the ocatons of source and boundary fed ponts. Exampe 3: Let s sove two-dmensona modfed Hemhotz equaton (Δ 900)u = 899(e x +e y ) (59) n [ 1, 1 [ 1, 1, Drchet boundary condton s set up correspondng to exact souton u = e x +e y (60) 40 source ponts are seected n the MFS. Tabe I gves the root mean square errors (RMSEs) for dfferent and m n whch exceent accuracy can be observed. Here and m are the numbers of Gauss-Lobatto nodes n the x and y drectons, respectvey. m 4 m 8 m 12 m 16 m 20 Exampe 4: Then we sove two-dmensona Posson s equaton Δu = e x +e y (61) aso n [ 1, 1 [ 1, 1 wth Drchet boundary condton. The exact souton s aso u = e x +e y (62) Tabe II gves the RMSEs. The accuracy s aso great. m 4 m 8 m 12 m 16 m 20 Exampe 5: Our expct partcuar soutons enabe us to sove three-dmensona probems very easy. Consder three-dmensona modfed Hemhotz equaton (Δ 900)u = 899(e x +e y +e z ) (63) n [ 1, 1 [ 1, 1 [ 1, 1. Drchet boundary condton s set up by usng the exact souton u = e x +e y +e z (64) 386 source ponts are seected n the MFS for ths three-dmensona case. Tabe III gves the RM- SEs for dfferent, m and n. The resuts aso perform we. Here, m and n are the numbers of Gauss-Lobatto nodes n the x, y and z drectons, respectvey. m n 4 m n 8 m n 12 m n 16 m n 20 Exampe 6: Aso, we can sove threedmensona Posson s equaton Δu = e x +e y +e z (65) wth Drchet boundary condton. The exact souton s aso u = e x +e y +e z (66) Tabe IV gves the RMSEs n whch exceent resuts can aso be observed.

9 Partcuar Soutons of Chebyshev Poynomas for Poyharmonc and Poy-Hemhotz Equatons 159 m n 4 m n 8 m n 12 m n 16 m n 20 where K 0 () s the zero order modfed Besse functon of second nd. We aso address the nce RMSEs n Tabe VI. Exampe 7: As we have mentoned that our formuaton can be extended to hgher order PDEs. We consder poyharmonc equaton n two-dmensona doman [ 1, 1 [ 1, 1 as Δ 4 u = e x +e y (67) The exact souton of the probem s set up as u = e x +e y (68) In ths probem, we have to set up four boundary condtons n Eq. (6) as [ T B = (69) n n 2 n 3 The MFS equaton n Eq. (57) n ths case reads u h (x) = K =1 ( γ1 r 6 nr +γ 2 r 4 nr +γ 3 r 2 nr + γ 4 nr ) (70) where r = x s. Tabe V gves the RMSEs where nce resuts are aso found. m 4 m 8 m 12 m 16 m 20 Exampe 8: Then et s consder a twodmensona probem n [ 1, 1 [ 1, 1 as Δ 2 (Δ 900)(Δ 100)u = 89001(e x +e y ) (71) The same four types of boundary condtons are set up correspondng to exact souton u = e x +e y (72) In ths case, the MFS equaton becomes u h (x) = K =1 ( γ1 K 0 (30r )+γ 2 K 0 (10r ) + γ 3 r 2 nr + γ 4 nr ) (73) m 4 m 8 m 12 m 16 m 20 8 Dscussons Many probems n engneerng and scence are governed by a system of couped near parta dfferenta equatons. Through Hörmander near parta dfferenta operator theory and agebrac factorzaton, they can be reduced to a snge equaton nvovng the products of Hemhotztype and poyharmonc operators. In ths paper we derved expct cosed-form partcuar soutons for these operators wth Chebyshev poynomas at rght-hand sde. Wth these partcuar soutons we can transform the nhomogeneous PDEs to homogeneous ones whch can then be soved by the MFS. Numerca experments ncudng eghth order PDEs and three-dmensona cases are carred out. Our numerca resuts are extremey accurate due to the exponenta convergence of both the Chebyshev nterpoaton and the MFS. Appcatons of these formuatons to probems governed by a system of couped near parta dfferenta equatons are currenty under nvestgatons. Acnowedgement: The Natona Scence Counc of Tawan s gratefuy acnowedged for provdng fnanca support to carry out the present wor under the Grant No. NSC E References Atnson, K.E. (1985): The numerca evauaton of partcuar soutons for Posson s equaton. IMA Journa of Numerca Anayss, vo. 5, pp Baneree, P.K. (1994): The boundary eement methods n engneerng, McGraw-H, London, UK.

10 160 Copyrght c 2008 Tech Scence Press CMES, vo.27, no.3, pp , 2008 Bogomony, A. (1985): Fundamenta soutons method for eptc boundary vaue probems, SIAM Journa of Numerca Anayss, vo. 22, pp Chen, C.W.; Fan, C.M.; Young, D.L.; Murugesan, K.; Tsa, C.C. (2005): Egenanayss for membranes wth strngers usng the methods of fundamenta soutons and doman decomposton, CMES: Computer Modeng n Engneerng & Scences, vo. 8, pp Cheng, A.H.-D. (2000): Mutayered Aqufer Systems Fundamentas and Appcatons, Marce Deer, New Yor/Base, USA. Cheng, A.H.-D. (2000): Partcuar soutons of Lapacan, Hemhotz-type, and poyharmonc operators nvovng hgher order rada bass functons, Engneerng Anayss wth Boundary Eements, vo. 24, pp Cheng, A.H.-D; Antes, H.; Ortner, N. (1994): Fundamenta soutons of products of Hemhotz and poyharmonc operators, Engneerng Anayss wth Boundary Eements, vo. 14, pp Cheng, A.H.-D.; Chen, C.S.; Goberg, M.A.; Rashed, Y.F. (2001): BEM for thermoeastcty and eastcty wth body force a revst, Engneerng Anayss wth Boundary Eements, vo. 25, pp Cheng, A.H.-D.; Cheng, D.T. (2005): Hertage and eary hstory of the boundary eement method, Engneerng Anayss wth Boundary Eements, vo. 29, pp Cheng, A.H.-D.; Lafe, O., Gr, S. (1994): Dua recprocty BEM based on goba nterpoaton functon functons, Engneerng Anayss wth Boundary Eements, vo. 13, pp Cheng, A.H.-D.; Morohunfoa, O.K. (1993): Mutayered eay aqufer systems: I. Pumpng we souton, Water Resources Research, vo. 29, pp Cheng, A.H.-D.; Morohunfoa, O.K. (1993): Mutayered eay aqufer systems: II. Boundary eement souton, Water Resources Research, vo. 29, pp Cozzano, B.S.; Rodríguez, B.S. (2007): The Trefftz boundary method n vscoeastcty, CMES: Computer Modeng n Engneerng & Scences, vo. 20, pp Dng, J.; Chen, C.S. (2007): Partcuar soutons of some eptc parta dfferenta equatons va recursve formuas, Journa of Unversty of Scence and Technoogy of Chna, vo. 37, pp Emdad, A.; Kansa, E.J.; Lbre, N.A.; Rahman, M.; Shearch, M. (2008): Stabe PDE souton methods for arge mutquadrc shape parameters, CMES: Computer Modeng n Engneerng & Scences, vo. 25, pp Goberg, M.A. (1995): The method of fundamenta soutons for Posson s equaton, Engneerng Anayss wth Boundary Eements, vo. 16, pp Goberg, M.A.; Chen, C.S. (1999): The method of fundamenta soutons for potenta, Hemhotz and dffuson probems, M. Goberg, M.A., edtor, Boundary ntegra methods: numerca and mathematca aspects, WIT Press and Computatona Mechancs Pubcatons, Boston, Southampton, pp Goberg, M.A.; Chen, C.S.; Karur, S.R. (1996): Improved mutquadrc approxmaton for parta dfferenta equatons, Engneerng Anayss wth Boundary Eements, vo. 18, pp Goberg, M.A.; Mueshov, A.S.; Chen, C.S.; Cheng, A.H.-D. (2003): Poynoma partcuar soutons for some parta dfferenta operators, Numerca Methods for Parta Dfferenta Equatons, vo. 19, pp Han, Z.D.; Lu, H.T.; Raendran, A.M.; Atur, S.N. (2006): The appcatons of meshess oca Petrov-Gaern (MLPG) approaches n hghspeed mpact, penetraton and perforaton probems, CMES: Computer Modeng n Engneerng & Scences, vo. 14, pp Hörmander, H. (1963): Lnear Parta Dfferenta Operators, Sprnger-Verag, Bern. Hu, S.P.; Young, D.L.; Fan, C.M. (2008): FDMFS for dffuson equaton wth unsteady forcng functon, CMES: Computer Modeng n Engneerng & Scences, vo. 24, pp Janssen, H.L.; Lambert H.L. (1992): Recursve

11 Partcuar Soutons of Chebyshev Poynomas for Poyharmonc and Poy-Hemhotz Equatons 161 constructon of partcuar soutons to nhomogeneous near parta dfferenta equatons of eptc type, Journa of Computatona and Apped Mathematcs, vo. 39, pp Gao, L.; Lu, K.; Lu, Y. (2006): Appcatons of MLPG method n dynamc fracture probems, CMES: Computer Modeng n Engneerng & Scences, vo. 12, pp Karageorghs, A.; Kyza, I. (2007): Effcent agorthms for approxmatng partcuar soutons of eptc equatons usng Chebyshev poynomas, Communcatons n Computatona Physcs, vo. 2, pp Kosec, G.; Šarer, B. (2008): Loca RBF coocaton method for Darcy fow, CMES: Computer Modeng n Engneerng & Scences, vo. 25, pp Kupradze, V.D.; Aesdze, M.A. (1964): The method of functona equatons for the approxmate souton of certan boundary vaue probems, USSR Computatona Mathematcs and Mathematca Physcs, vo. 4, pp L,X.;Chen,C.S.(2004): A mesh-free method usng hypernterpoaton and fast Fourer transform for sovng dfferenta equaton, Engneerng Anayss wth Boundary Eements, vo. 28, pp Lbre, N.A. ; Emdad, A.; Kansa, E.J.; Rahman, M.; Shearch, M. (2008): A stabzed RBF coocaton scheme for Neumann type boundary vaue probems, CMES: Computer Modeng n Engneerng & Scences, vo. 24, pp Lu, C.S. (2007A): A modfed Trefftz Method for two-dmensona Lapace equaton consderng the doman s characterstc ength, CMES: Computer Modeng n Engneerng & Scences, vo. 21, pp Lu, C.S. (2007B): A hghy accurate sover for the mxed-boundary potenta probem and snguar probem n arbtrary pane doman, CMES: Computer Modeng n Engneerng & Scences, vo. 20, No. 2, pp Mason, J.C.; Handscomb, D.C. (2003): Chebyshev Poynomas, Chapman & Ha/CRC, Boca Raton. Mueshov, A.S.; Goberg, M.A. (2007): Partcuar soutons of the mut-hemhotz-type equaton, Engneerng Anayss wth Boundary Eements, vo. 31, pp Mueshov, A.S. Goberg, M.A.; Chen, C.S. (1999): Partcuar soutons of Hemhotztype operators usng hgher order poyharmonc spnes, Computatona Mechancs, vo. 23, pp Nardn, D.; Brebba, C.A. (1982): A new approach to free vbraton anayss usng boundary eements, C. A. Brebba, edtor, Boundary eement methods n engneerng, Sprnger-Verag, New Yor, pp O Ne, P.V. (2002): Advanced Engneerng Mathematcs, Broos/Coe,USA. Reutsy, S.Y.; Chen, C.S. (2006): Approxmaton of mutvarate functons and evauaton of partcuar soutons usng Chebyshev poynoma and trgonometrc bass functons, Internatona Journa for Numerca Methods n Engneerng, vo. 67, pp Sade, J.; Sade, V., Wen, P.H.; Aabad, M.H. (2006): Meshess oca Petrov-Gaern (MLPG) method for shear deformabe shes anayss, CMES: Computer Modeng n Engneerng & Scences, vo. 13, pp Sade, J.; Sade, V.; Zhang, Ch.; Soe P. (2007): Appcaton of the MLPG to thermopezoeectrcty, CMES: Computer Modeng n Engneerng & Scences, vo. 22, pp Smyrs, Y.S.; Karageorghs, A. (2003): Some aspects of the method of fundamenta soutons for certan bharmonc probems, CMES: Computer Modeng n Engneerng & Scences, vo. 4, pp Tsa, C.C.; Ln, Y.C.; Young, D.L.; Atur, S.N. (2006): Investgatons on the accuracy and condton number for the method of fundamenta soutons, CMES: Computer Modeng n Engneerng & Scences, vo. 16, pp Tsa, C.C.; Young, D.L.; Cheng, A.H.-D. (2002): Meshess BEM for three-dmensona Stoes fows, CMES: Computer Modeng n Engneerng & Scences, vo. 3, pp

12 162 Copyrght c 2008 Tech Scence Press CMES, vo.27, no.3, pp , 2008 Wu, X.H.; Shen, S.P.; Tao, W.Q. (2007): Meshess Loca Petrov-Gaern coocaton method for two-dmensona heat conducton probems, CMES: Computer Modeng n Engneerng & Scences, vo. 22, pp Young, D.L.; Chen, K.H.; Chen, J.T.; Kao, J.H. (2007): A modfed method of fundamenta soutons wth source on the boundary for sovng Lapace equatons wth Crcuar and arbtrary domans, CMES: Computer Modeng n Engneerng & Scences, vo. 19, pp Young, D.L.; Ruan J.W. (2005): Method of fundamenta soutons for scatterng probems of eectromagnetc waves, CMES: Computer Modeng n Engneerng & Scences, vo. 7, pp Zhu, T.; Zhang, J.; Atur, S.N. (1998): A meshess oca boundary ntegra equaton (LBIE) method for sovng nonnear probems, Computatona Mechancs, vo. 22, pp