Joul of Applied Mthemtics d Physics, 5, 3, 75-8 Published Olie Juy 5 i SciRes. http://www.scip.og/joul/jmp http://dx.doi.og/.436/jmp.5.3 Expsio by Lguee Fuctio fo Wve Diffctio oud Ifiite Cylide Migdog Lv, Hui Li, Huilog Re, Xiobo Che College of Shipbuildig Egieeig, Hbi Egieeig Uivesity, Hbi, Chi Bueu Veits Resech Deptmet, Sigpoe City, Sigpoe Emil: 37378@qq.com Received Decembe 4 Abstct We coside veticl cicul cylide o which the veticl vitio of wte diffctio wves is to be epeseted by seies of Lguee fuctios ( ) = e L( ) usig Lguee Polyomils L ( ). The vitio is ssumed to be of the fom / ( + ) with the itege depedig o the dius of cylide. Geelly, the itege iceses fo cylide of lge dimete. The usul ppoximtio by Lguee fuctios is exteded by itoducig scle pmete. The covegece of Lguee seies is the depedet o the vlue of the scle pmete s. The lyticl d umeicl computtios of seies coefficiets e pefomed to study the umbe of seies tems to keep the sme ccucy. Ideed, the choice of itege depeds o the scle pmete. Futhemoe, diffctio wves geeted by semi-sphee iside the cylide e evluted o the cylide sufce. It is show tht the ppoximtio by Lguee seies fo diffctio wves o the cylide is effective. This wok povides impott ifomtio fo the choice of the dius of cotol sufce i the domi decompositio method fo solvig hydodymic poblems of body-wve itectio. Keywods Lguee Fuctio, Wve Diffctio oud Ifiite Cylide. Itoductio The Rkie souce pel method eeds lge umbe of pels due to peliig the fee sufce s well s dmpig oe voidig the eflected wve fom the sides of umeicl fluid domi. So cotol sufce c be itoduced to divide the fluid domi ito two subdomis by cotol sufce. This sufce septes the poblem ito two poblems: ) the iteio oe i which the ship is of y fom, the Gee fuctio is Rkie souce Gee fuctio; ) the exteio oe i which the shpe of the cotol sufce is kow d velocity potetil is ssumed to be kow. It bigs two impott beefits: e to be discetied becomes smlle; o eed to itoduce the dmpig oe []. A cicul cylide is dopted s cotol sufce. The veticl vitio of wte diffctio wve is ssumed to be the fom / ( + ) which is epeseted by seies of Lguee fuctios ( ). Lguee fuctios How to cite this ppe: Lv, M.D., Li, H., Re, H.L. d Che, X.B. (5) Expsio by Lguee Fuctio fo Wve Diffctio oud Ifiite Cylide. Joul of Applied Mthemtics d Physics, 3, 75-8. http://dx.doi.og/.436/jmp.5.3
M. D. Lv et l. ( ) which e defied by Lguee polyomils L ( ) e system of othogol fuctios o the itevl [, ] []. It plys impott ole i ppoximtio d itepoltio. The pupose of this ppe is to vlidte the ccucy d covegece of Lguee seies d ppoximte the veticl velocity potetil φ o ifiite cylide geeted by body. Sectio itoduces bsis of Lguee fuctios. Sectio 3 povides lyticl method to ppoximte the fuctio /( + ) d the velocity potetil φ by Lguee fuctios. Sectio 4 uses some exmples to ivestigte the covegece d ccucy of the method, d compes the esult of Compss-Wlcs-Bsic.. The Bsis of Lguee Fuctio.. Lguee Polyomils The Lguee polyomils e defied by the thee-tem ecuece eltio L ( ) = () L ( ) = () L( ) = L ( ) L ( ) (3) L ( ) is clled th degee Lguee polyomil [3]. The Lguee polyomils hve some useful eltios L () = (4) dl () = (5) dl ( ) dl ( ) By vitu3 of Equtio (6), we c obti Equtio (7) We defie ( m+ L ) ( ) s Equtio (8) ( m) [ ( )] dl + = L ( ) (6) dl [ ( )] = Lk ( ) (7) ( m+ ) k = By vitue of Equtio (9), we c obti Equtio () Ad the othogol eltio whee δ m is Koecke symbol. Futhemoe it c be esily show tht k is el umbe. = L ( ) m=,, (8) ( m) ( m) dl [ ( )] dl [ ( )] ( m) = L ( ) (9) dl [ ( )] = ( ) () ( m) ( m) Lk k = L ( ) Lm ( ) e = δm () k + e L ( ) = ( k ) k k () 76
M. D. Lv et l... Lguee Fuctios d Scle Pmete s We defie th degee Lguee fuctio ( ) s: ( ) = e L ( ) (3) The Lguee fuctios stisfy the othogol eltio [3]: ( ) m ( ) = δ (4) m whee δ m is Koecke symbol. It is impott to ote tht the Lguee fuctios e well behved. Ideed, the followig popeties e show [3]: d ( ) fo (5) 4 34 ( ) = π ( ) cos( π 4) + o( e ) (6) Fo =,, We c ppoximte fuctio by seies of Lguee fuctios with scle pmete s: f ( ) = c ( s) (7) The Lguee fuctios with scle pmete s lso stisfy the othogol eltio: The coefficiet c e defied ( s) m ( s) = m s δ (8) c = f ( s) ( ) (9) 3. Numeicl Appoximtio d Itepoltio by Lguee Fuctios As the velocity potetil ϕ is ssumed to f( ) = ( + ), we expd f( ) = ( + ) by Lguee fuctios to vlidte the covegece d ccucy. I dditio, we povide itepoltio method to ppoximte the velocity potetil ϕ. Fo fuctio f( ) = ( + ) [4] f ( ) = = c ( s) () + = c = + = + e () ( ) ( ) s s x ( xs + ) = ( ) () c e dx e L whee we use the Equtio () d Equtio (4) x x (3) x + c = e ( ) /( + ) dx s s x( + / s) s e dx / ( + / ) = (4) 77
M. D. Lv et l. The coefficiet c c be clculted by Equtio (3) i Guss-Lguee itegtio, see i Equtio (5) N x e f( x) dx Ak f( xk) (5) k = whee x k is the kth distict eo of th Lguee polyomil. Fo fuctio f() = /( + ) By vitue of Equtio (7) f ( ) = = ( ) c s (6) ( + ) = = ( + ) ( ) = ( + ) ( ) c s s s (7) ( ) c = s d + s + s dl ( ) c = s e L( ) + e s + s + s ( ) (8) (9) c = s e L( ) dx e Lk( ) s + s k = + s (3) The the itegtio i Equtio (3) c be clculted i Equtio (), f( ) = ( + ) c be expded 3 i the simil method. The f( ) = ( + ) is expded by Lguee fuctios s below k s + c = + s + ( e L( ) e Lk( ) e Ll( )) + + (3) 4 k= k= l= + s We suppose tht the veticl velocity potetil φ() is cotiuous fo, whee give fuctio φ() is oly kow umeiclly t evey poit. The Lguee-Guss itepoltio is pplied to ppoximte the φ(). 4. Numeicl Results φ( ) = c ( ) (3) = ( ) ( ) = ( ) ( ) c f e f L (33) N k k k k k = c A e f( ) L ( ) (34) I this sectio, we peset some umeicl esults. The lgoithm is implemeted by usig Itel Visul Fot Compose XE d ll clcultios e cied out i compute of CPU 3.3 GH. We fist use Equtio (3) to ppoximte fuctio ( + ), show s Tble. Fo desciptio of the globl eos, we itoduce the ottios. Appoximtio esults by Lguee fuctios use symbol f ( ). f( ) f ( ) E = (35) mx ( ) { f } The we use Equtio (3) to ppoximte fuctio ( + ), show s Tble. 3 At lst, we use Equtio (3) to ppoximte fuctio ( + ) show s Tble 3. A semi-sphee is dopted s body iside the cylide to geete diffctio wves. The dius of semisphee is m s well s cylide is.5 m. The icidet wve is i fequecy of.6 d/s, i height of m. We pefe to ppoximte veticl vitio with Equtio (34) t θ is d/s, show i Figue. 78
M. D. Lv et l. We compe the ppoximte esults with esults clculted by Compss-Wlcs-Bsic (CWB, wve lod softwe is developed by CCS), show s Tble 4. 5. Coclusio I this ote we peseted umeicl method fo itepoltig veticl vitio of wte diffctio wves bsed o Lguee fuctios. The covegece d ccucy is vlidted by ppoximtig the fuctios Tble. Computed vlue d the eltive eos t diffeet vlues of. s =.5, N = 4 s =.5, N = 4 s =., N = 4 s =., N = 4 s =., N = 4 S =., N = 4 f( ) f ( ) E f ( ) E f ( ) E.E+ 9.999E.34E 4.3E+ 3.44E 3.E+.34E 5.E 5.E 4.6E 5 5.6E 6.478E 4 5.E.67E 5 3.333E 3.333E 9.655E 6 3.333E 3.443E 6 3.35E.669E 3 3.5E.5E.39E 6.495E 4.639E 4.59E 8.635E 4 4.E.E 3.E 5.5E 5.56E 4.987E.3E 3 3 3.58E 3.78E 3.368E 5 3.87E.39397E 4 3.4553E 8.8E 4 4.439E.438E 9.673E 6.44373E 4.758E 5.98E.48E 3 5.9678E.9547E 6.57E 5.967E 4.465E 4.475E.43969E 3 6.63934E.6789E.456E 4.59564E 4.378E 4.8588E.66537E 3 Tble. Computed vlue d the eltive eos t diffeet vlues of. s =.5, N = 4 s =.5, N = 4 s =., N = 4 s =., N = 4 s =., N = 4 S =., N = 4 f( ) f ( ) E f ( ) E f ( ) E.E+ 9.97588E.469E 3 9.9996E 4.65E 5.7E+.737E 4.5E.5385E 3.84964E 4.49997E 3.349E 6.5E.53E 7.E.86E.4895E 4.4E 7.485E 6.37E.53938E 5 3 6.5E 6.767E.6697E 4 6.4997E 3.378E 7 6.53E.345E 5 4 4.E 3.98377E.6343E 4 4.3E.97746E 6 3.998E.9848E 5 3.6757E 3.64E 3.5694E 3.377E 3.987E 3.875E 3.3883E 3 4 5.94884E 4 5.63398E 4 3.486E 5 5.9984E 4 4.9555E 6 5.76595E 4.8893E 5 5 3.84468E 4 3.649E 4.4587E 5 3.8795E 4 3.4847E 6 4.6387E 4 3.99E 5 6.68745E 4.655E 4 7.4E 6.8978E 4.38E 5 3.3363E 4 4.468E 5 Tble 3. Computed vlue d the eltive eos t diffeet vlues of. s =.5, N = 4 s =.5, N = 4 s =., N = 4 s =., N = 4 s =., N = 4 S =., N = 4 f( ) f ( ) E f ( ) E f ( ) E.E+ 9.99E 7.8E 3 9.99653E 3.4659E 4 9.99998E.53E 6.5E.688E.8848E 3.4953E 4.6693E 5.5E 3.339E 7 3.737E 3.6369E 6.6785E 4 3.789E.89E 5 3.737E.47E 8 3.565E.689E 5.55894E 4.56479E.8585E 5.5649E.66E 7 4 8.E 3 7.55455E 3 4.45448E 4 7.98476E 3.54E 5 7.99978E 3.7E 7 3 3.3567E 5 4.74463E 5 8.35E 5.984E 5 4.55883E 6 3.33679E 5.994E 7 4.4594E 5 7.7484E 5 9.978E 5.8399E 5 3.885E 6.48E 5 4.8583E 7 5 7.53858E 6 7.E 5 7.8548E 5 7.794E 6 4.64E 7 7.3789E 6.6386E 7 6 4.4566E 6 5.8496E 5 6.85E 5.6754E 6.734E 6 4.3944E 6 9.635E 8 79
M. D. Lv et l. Tble 4. Computed vlue d the eltive eos t diffeet vlues of. Velocity potetil (el pt) Velocity potetil (imgiy pt) φ ( ) ( ) CWB φ LAG E φ ( ) ( ) CWB φ E LAG 6.887E 6.956E.6E.448E 4.99E 4.448E 6.636E 6.68E.5998E 3.8357E 4.8364E 4.87E 3 6.3554E 6.34498E.4769E 3.4878E 4.4853E 4.776E 3 3 6.55E 6.36E.73354E 3.86E 4.763E 4.3739E 3 4 5.8798E 5.869E.44937E 3 9.655E 5 9.3975E 5.98463E 3 5 5.6573E 5.6678E 6.5445E 4 7.8996E 5 7.7E 5 8.856E 4 4.683E 4.6779E.37E 3.676E 6.5E 6.6677E 3 3.544E 3.76E.53E 3 4.855E 5 4.54E 5.54439E 3 3.333E.34E 5.8384E 4 4.47875E 5 4.4958E 5 7.3494E 4 4.5445E.5355E.56E 3 3.859E 5 3.7674E 5.7664E 3 5.6995E.66E.345E 3.9943E 5.9598E 5.84357E 3 6 7.4677E 3 7.3385E 3.43E 3.7477E 5.3748E 5.6673E 3 Figue. A semi-sphee i cylide. ( + ) ( =,, ). It is pplicble to ppoximte the veticl vitio oud cicle cylide by Lguee fuctios. Refeeces [] Te, I. d Che, X.B. () Zeo Speed Rkie-Kelvi Hybid Method. Bueu Veits Resech Deptmet, 6. [] Guo, B.-Y. d Wg, Z.-Q. (7) Numeicl Itegtio Bsed o Lguee-Guss itepoltio. Comput. Methods Appl. Mech. Egg, 96, 376-374. http://dx.doi.og/.6/j.cm.6..35 [3] Abmowit, M. d Stegu, I.A. (967) Hdbook of Mthemticl Fuctios. Dove Publictios. [4] Keilso, J., Nu, W. d Sumit, U. (98) The Lguee Tsfom. Cete fo Nvl Alysys, 6-8. 8