Hyperbolic Type Approximation for the Solutions of the Hyperbolic Heat Conduction Equation in 3-D Domain

Transcription

1 Mhemicl d Compuiol Mehods i Applied Scieces Hperbolic pe Approimio or he Soluios o he Hperbolic He Coducio Equio i 3-D Domi BUIKIS ANDRIS KAIS HARIJS vi Uiversi Isiue o Mhemics d Compuer Sciece Ri bulv9 Rig V459 AVIA buiis@lelv; lis@lelv hp://www llv/scieiss/buiishm Absrc: - I his pper we develop mhemicl models or 3-D -D d -D hperbolic he equio d cosruc heir pproimed licl soluios b he mehod o coservive vergig or he deermiio o he iiil he lu or oe domi Impor is h i his pper we propose o-polomil orm o pproimio ucio Approime soluios were obied b iie dierece mehod Ke-Words: - Hperbolic Equio Approimed Soluio Iverse Problem Fiie dierece Ordir diereil ssem Iroducio Rel processes e plce i url or echicl ssems wih compliced srucure Ver oe such ssems cosis o sepre lers wih diere hicess d diere phsicl properies I mes h o he surces bewee wo djce lers we hve jump i coeicies o diereil equios mhemicll describig correspode phsicl process his mes ddiiol diiculies or pplicios o rdiiol mhemicl mehods Oe o he uhors hs developed specil mehod coservive vergig mehod []-[5] - d he hs iroduced specil ew pe o splie [] [] []- [5] used or solvig such problems or wide clss o direc [6] [7] d iverse [8] [9] [5] problems or pril diereil equios wih discoiuous coeicies Proposed mehod diers rom mehods rdiioll used b mhemicl modellig o groudwer polluio or oher rspor processes i url or riicil porous medi he mi ide or he mehod o coservive vergig is o ulil i he simpliied problem ormulio he coservio o eerg or mss I his pper we give modiied descripio o hperbolic pproimio or oe domi d emplo his pproimio or some iesive seel quechig problems i hree dimesiol domis 3-D Problem Formulio he mi diereil equio o hperbolic he rser or ucio ( ) is i ollowig orm: r cρ ( ) () ( ) ( ) ( ) ( ] Boudr codiios (BC) re smmer codiios o he ero pois: () O oher side we hve he echge codiios: + α + α (3) + α Iiil codiios re i he orm: ( ) V( ) (4) Here c is speciic he cpci - he coducivi coeicie ρ - desi r - relio ime For some problems [7] [8] secod iiil codiio is uow I his cse we use ddiiol codiio i he il mome: ( ) (5) ISBN:

2 Mhemicl d Compuiol Mehods i Applied Scieces 3 Hperbolic Approimio For he coservive vergig mehod i ll siuios here were wo or more sub-domis Now we describe cse i which ol oe domi eiss I his cse he chrcer scle i oe direcio is smller h oher direcio I publicios [] [3] were ol irs cse described: wih wo or more sub-domis e i be give coiuous ucio U [ l] d posiive cos his ucio is uow bu we hve iormio bou verged vlue: l u l U d (6) Addiioll here re ow boudr codiios i pois l: U() U + β U l We will pproime he ucio U i he orm (prmeer is ree cos): l sih U u + ml l sih l sih + eg A (7) l 4sih sih ( l) l l A G 4 cosh ( l ) his orm ulills he equli (6) wo idepede coeicies me we c deermie rom boudr codiios 4 Reducio o 3-D Hperbolic He equio o he Ssem o Ordir Diereil equio 4 Reducio he 3-D problem o he -D problem he problem or ucio ( ) is give b seme ()-(4) For he rgume we use he pproimio (7) I his cse ucios u m e re ucios o he rgumes For brevi i he ollowig ormuls we wrie ucios wihou rgumes: sih + m + sih sih eg A (8) 4sih sih ( ) G A cosh ( ) BC gives: α m e g g α G 5 + A + / d ( ) ( ) α α d coh A A 4 d Fill we hve iiil-boudr -D problem: r + cρ + Bg ( ) + d B (9) + α + α ( ) V ( ) ( ) ( ) d ( ) ISBN:

3 Mhemicl d Compuiol Mehods i Applied Scieces ( dv ) ( ) V( d ) I is es o prove h d A A Reducio o he -D problem o he -D problem As e sep we use coservive vergig or he rgume : () ( ) d I his cse ucios u m m e e re ucios o he rgumes Approimio is: sih α + m + sih α sih α e G A () 4sih α sih ( α ) α G A 4 cosh ( α ) Boudr codiio gives: m e g A A 4 α g α G 5 + A + / d ( ) ( α ) d α /coh / We hve iiil-boudr -D problem: r + cρ + Bg + Bg + ( ) B d / + α () ( ) () d ( ) d V V ( ) d () 43 he coservive vergig mehod i - direcio We use coservive vergig or he rgume : () ( ) d I his cse ucios u m m e e re ucios o he rgume Approimio is: sih α + m sih α sih α + e G A () 4sih α sih ( α ) α G A 4 cosh ( α ) I his cse boudr codiio gives: m e g g α α G 5 + A + / d α d 5α coh A A 4 We hve problem or ordir diereil equio (ODE): d d r + cρ + γ ( ) d d (3) d ( ) V d Here γ Bg + Bg + Bg d () ( ) d B ISBN:

4 Mhemicl d Compuiol Mehods i Applied Scieces d V V d 44 Alicl soluio o ODE We hve iiil problem or he problem (3): sih ( κ ) cosh ( κ ) e C + B + ξ (4) e sih κ( ξ) G( ξ) dξ κ Here r γ* γ κ B γ* cρ 4 cρ o () C V + G () κ cρ I 4γ> * he he hperbolic ucios i he (4) o eed o be replced b he rigoomericl γ * d he prmeer κ 4 5 Hperbolic Approimio or Oe Dimesiol Hperbolic Problem Here we emie such oe-dimesiol hperbolic he equio which is coeced wih wve eerg [] - [3]: + γ* + ( ) (5) r < < < cρ cρ he prmeer γ * is obied b vergig equio () wih he CAM: Bg + Bg γ * cρ Iiil d boudr codiios re s ollows: V o (6) α + σ σ 5 Reducio o Hperbolic Problem o he Ssem o Ordir Diereil Equios We use secod order pproimio or pril derivio o secod order respec o d or iiil problem or hperbolic he equio (5) we obi ssem o ordir diereil equios o secod order i he ollowig mri orm: U () + U () + AU () + γ* U () F () (7) U() U U () V Here A is he 3-digol o N + order i he orm: A h + hσ I ormul (7) we hve he colum-vecors o N + order U () U () U () V U F () wih elemes ( j ) uj() ( j) u j() ( j ) u j( ) v () j V ( j) (8) u () ( ) ( ) j N j j j j 5 Discree Specrl Problem he 3-digol mri A c be represeed wih ollowig dierece operor [8]: ( ) h j A ( j+ j + j) h j N (9) ( N N) h + σ N h j N m Usig wo vecors sclr produc: N m m m m m m h j j + ( + NN) j We c es prove: he operor A is smmeric d [ A ] [8] he correspodig specrl problem A µ N + hs ollowig soluio [8]: C N N () 4 ph si µ h Here si ph cos p / h j N j j p re posiive roos o he rscedel equio: ISBN:

5 Mhemicl d Compuiol Mehods i Applied Scieces ( ph ) ( ph) si co ( p) N hσ si + () he coss C c be obied i ollowig orm: C has + 5( A ( + cos ( p) )) si N ( ph ) A S cos ( p j) h j ( p ) si p( h) si ( ph) cos 5 N + Whe clculed ill we hve he m orhoorml eigevecors wih he sclr produc m δ m Here δ m is he Kroecer smbol I mes h we hve mri A which c be represeed i orm: A PDP () Where he colum o he mri P d he digol mri D cois M orhoorml eigevecors d eigevlues µ M correspod where M N + From PP E ollows h P P 53 he soluio o he discree problem We cosider he lic soluio o he problem (7) usig he specrl represeio o mri A PDP From rsormio W P U ( U PW ) ollows he sepre ssem o ODE s W () + W () + DW () + γ* W () G () (3) W() PU W () PV Here W( ) W ( ) W ( ) W( ) W ( ) G( ) P F( ) re he colum-vecors o M orders wih elemes w ( ) w ( ) w ( ) w ( ) w ( ) g ( ) M he soluio o ssem (3) is: sih w κ w ( ) e w + + κ + cosh κ w + ( ) } ξ e sih ( κ( ξ) ) g ( ξ) dξ κ (4) I he soluio (4) we hve used ollowig oio: d γ * κ δ µ d δ 4 + I 4d > he he hperbolic ucios re replced wih he rigoomericl d he prmeer d κ 4 I κ he w w ( ) e w + + w ξ + e ( ξ) g ( ξ) dξ he mri A rom (7) hs he irs d ls rows i orm: h ; h + σ h hereore he irs d ls equios o (7) re vlid whe he irs d ls compoes o vecor AU re divided b We c use lso he Fourier mehod or solvig he problem (5) i he orm: w g where w ( ) is he soluio (9) wih w ( ) g ( ) ( ) re he orhoormed eigevecors 54 he specrl problem or diereil equio d iie dierece scheme wih ec specrum Now we emie specrl problem or diereil equio d iie dierece scheme wih ec specrum [6] [7] he soluio o he specrl problem or diereil equio: + λ ( ) (5) () ( ) + σ Is i orm C λcos ( λ) σλ (6) C λ + λ + σ We hve ISBN:

6 Mhemicl d Compuiol Mehods i Applied Scieces ( ) d δ m m m he eigevlues λ re posiive roos o he rscedel equios: co ( λ λ ) σ (7) For he sclr produc m he iegrl is pproimed wih rpeoidl ormul m d i he limi cse i h he rom () (7) ollow µ λ For he dierece scheme wih ec specrum [6] [7] he mri A is i he orm () d he digol mri D cois he irs N + eigevlues δ λ N + rom he diereil operor correspodigl I δ µ he we hve he mehod o iie dierece pproimio wih ri-digol mri A We c use lso he Fourier mehod or solvig he problem (5) i he orm: w g where w ( ) is he soluio (4) wih w ( ) g ( ) ( ) re he orhoormed eigevecors 6 he Mehod or Solvig Iverse Problem For he iverse problem he secod iiil codiio i he ormul (4) is o ow We use he ddiiol codiio (5) I he ssem (7) ow we hve he codiio U ( ) u Here u is he vecor-colum wih elemes u M he licl soluio o his problem c be obied rom (3) We mus replce he secod iiil codiio W PV wih W ( ) Pu he soluio is ollowig: ( κ ) sih w ( ) e e w ( ) w sih ( κ ) ] ξ cosh κ e κ ξ g ξ dξ sih ( ) κ ( κ ) w + cosh + ξ e sih ( κ( ξ) ) g ( ξ) dξ κ (8) w re he compoes o I ormul (8) ( ) vecor W( ) he irs derivive o (8) is: w κ sih ( κ ) e w( ) w cosh ( κ) ξ w e sih ( κ( ξ) ) g ( ξ) dξ κ 7 Numericl Resuls Here we loo he iesive seel quechig or Crbo seel wih hese phsicl properies: W g J 65 ρ 787 c mc m gc α ;5 6 6; γ ;;3 N ;3;5 ble he vlues o V depedig o γ γ V γ V As we c see ol lrge vlues o γ iluece he iiil he lu ISBN:

7 Mhemicl d Compuiol Mehods i Applied Scieces 8 Numericl Esimig prmeer rom (7) We cosider he specil -D diusio problem i direcio or α α d source erm π π cos cos he sior soluio o () is i he orm: π π ( ) cos cos he ucio ( ) is he soluio o boudr problem: d d b ( ) d d d () d ( ) + α ( ) d d cos b π + he ec soluio is Ccosh ( b) b (9) he coss re: b α / b b C b sih b + α cosh b he verged vlue is: Csih ( b ) d b b As compriso we use he verged mehod o hperbolic (or epoeil) pproimio I his cse verged vlue is : m e g Bg + b I he ls equli cos g is give erlier i sub-secio 4 We hve solved problem (9) wih such prmeers: 8 6 b 8 8 α 3 I e ble he resuls re give Prmeer α mes he vlue give iegrl prbolic splie or N [] [] ble he miml error δ d verged m vlues depedig o d δ m We c see h miiml error δ 88 is prmeer 8 9 he coservive vergig mehod wih hperbolic pproimio wih wo prmeers We use pproimio (7) wih wo prmeers For -D problem i he segme [ ] we cosider pproimio: c cv + m + e A sih ( / ) cv cd ( ) sih ( / ) sih ( / ) (3 4sih / ( ) ( ) ( ) ( ) sih / A 4 cosh We c prove h i + + i he ormul (3) we obi iegrl prbolic splie [] [] or oe segme We loo ow o he boudr problem or ODE: β α c c + F c c C c + c C he soluio o his problem is: c C sih / + ( ) + Ccosh / F / Here α / ( C ) D β / ( C ) D C de D sih ( / ) + β / cosh ( / ) (3) (3) ISBN:

8 Mhemicl d Compuiol Mehods i Applied Scieces α D sih / + / cosh / F / ( αβ ) ( α β ) de sih + / + cosh + / ( ) + β ( ) α / C D / C D C de D3 cosh / + / sih / 3 4 β α D4 cosh / + / sih / he uow coss me we obi rom boudr codiios: dm d e β c 5m + ea C ( v ) ( 5 ) dm + d e + α c + m + ea C v From here e gc + C+ bc m g c + C+ b C v v α( 5β + ) + β( 5 α + ) / de α( 5 β + ) / de β( 5 α + ) / de ( β α) α( + β ) β( + α ) dd + ( α + β )( da+ d ) + αβ A 5 coh ( 5 ) d ( ) g d d d b d g d / de d A / de b d A / de A 5 A de 5 d 5 coh 5 de From ODE (3) ollows: cv F + Fill we obi: c v d bc+ C + F + dg We obi some resuls or coss β α C 3 C 5 3 For iegrl prbolic splie b he miml error δ 763 c v 847 For hperbolic (epoeil) pproimio b / error is δ 35 b error is δ 54 bu b / error is δ c v 497 Resuls re i ollowig wo igures: Fig Soluio c( ) (3) or prmeers / 5 3 Fig Soluio c( ) (3) or prmeers / 5 3 I is cler h pproimio wih hperbolic ucios is beer h polomil orm []-[5] Geerl resul: i / he or ll coss αβ C C we obi ec soluio We c prove his c b he usge o ollowig epressios: C m / sih ( / ) C e / 4 ( cosh ( / ) ) ( ) c e A + 4cosh / v ( ) ( ( )) I e wo igures we c see he grphics o : sih 5 / 5 cosh 5 / he ucios ISBN:

9 Mhemicl d Compuiol Mehods i Applied Scieces Fig 3 Fucios depeds o he prmeer [ 5 ] Fig4 Fucios depeds o he prmeer [ 5 ] s wo igures show h hperbolic pproimio c be used or boudr ler pproimio Coclusios he mi ide or he mehod o coservive vergig is o ulil i he simpliied problem ormulio he coservio o eerg or mss All o mi diereil equios ulil eerg or mss coservio he cojugios codiios ulil eerg echge bewee wo eighbourig subdomis or boudr codiios ulil he eerg coservio bewee objec d surroudig medi his ide is ver impor i url scieces [9] he 3-D diusio problem i mulilered domi is reduced o -D d -D problems used he iegrl prbolic []-[7] [] [4] hperbolic (epoeil) pproimio hese splies re obied rom geerl splie wih ied ucios he prmeers o hese ucios re chrcerisic vlues or correspodig homogeous ODE s o secod order hese prmeers re he bes prmeers or miiml error he -D diereil d discree problems re solved licll For hperbolic splie i D he bes prmeer or miiml error is clculed he soluios o rsie verged o sior - D iiil boudr problem re obied umericll wih he usge o lerig-direcio implici (ADI) mehod o Dougls d Rchord he umericl soluio is compred wih he licl soluio Reereces: [] A Buiis Ierpolio o iegrl vlue o piecewise smooh ucio b verge o prbolic splie vijsij memicesij jehegodi (vi Mhemicl Yerboo) Vol p (I Russi) [] A Buiis he clculio o he coeicies o he iegrl splie vijsij memicesij jehegodi (vi Mhemicl Yerboo) Vol p 8-3 (I Russi) [3] Buiis A Augbesellug ud ösug eier Klsse vo Probleme der mhemischer Phsi mi ichlssische Zusbediguge Rosoc Mh Kolloq S 53-6 (I Germ) [4] Buiis A Modellig ilrio processes i lered porous medi b he coservive vergig mehod Dr hesis Phsicsmhemics K987 (I Russi) [5] Buiis A Coservive vergig s pproime mehod or soluio o some direc d iverse he rser problems Advced Compuiol Mehods i He rser IX WI Press 6 p 3-3 [6] Vilums R Buiis A Coservive vergig mehod or pril diereil equios wih discoiuous coeicies WSEAS rscios o He d Mss rser Vol Issue 4 6 p [7] Buie M Buiis A Approime Soluios o He Coducio Problems i Muli- Dimesiol Clider pe Domi b Coservive Avergig Mehod Pr Proceedigs o he 5 h IASME/WSEAS I Co o He rser ISBN:

10 Mhemicl d Compuiol Mehods i Applied Scieces herml Egieerig d Evirome Vouligmei Ahes Augus p 5 [8] Bobis Buie M Buiis A Hperbolic He Equio s Mhemicl Model or Seel Quechig o -Shpe Smples Pr (Iverse Problem) Proceedigs o 5h IASME/WSEAS Ieriol Coerece o Coiuum Mechics (CM ) Uiversi o Cmbridge UK Februr 3-5 p -6 [9] Bloml S Buiis A He coducio problem or double-lered bll Progress i Idusril Mhemics ECMI Spriger 4 p [] M Buie A Buiis Modellig o hree dimesiol rspor Processes i Aisoropic lered srum b coservive vergig mehod WSEAS rscios o He d Mss rser Vol Issue 4 6 p [] M Buie A Buiis Ssem o vrious mhemicl models or rspor processes i lered sr wih ierlers WSEAS RANSACIONS o MAHEMAICS Issue 4 vol6 7 p [] M Buie A Buiis Alicl Approime Mehod or hree-dimesiol rspor Processes i ered Medi Proceedigs o 4 h IASME/WSEAS Ieriol Coerece o He rser herml Egieerig d Evirome Eloud Agios Niolos Cree Isld Greece Augus 3 6 p3-37 [3] M Buie A Buiis Ssem o Models or rspor Processes i ered Sr Proceedigs o 5h WSEAS Ieriol Coerece o SYSEM SCIENCE d SIMUAION i ENGINEERINGeerie Cr Islds Spi December p 9-4 [4] Buie M Buiis A Severl Iesive Seel Quechig Models or Recgulr Smples Proceedigs o NAUN/WSEAS Ieriol Coerece o Fluid Mechics d He &Mss rser Coru Isld Greece Jul -4 p88-93 [5] Buie M Buiis A Vilums R Oe- Dimesiol Iesive Seel Quechig Models Rece Advces i Mechicl Egieerig Proceedigs o he 5 h Ieriol Coerece o Fluid Mechics d He & Mss rser isbo Porugl Ocober 3- November 4 p 54-6 [6] A Buiis H Klis Hperbolic He Equio i Br d Fiie Dierece Schemes o Ec Specrum es reds o heoreicl d Applied Mechics Fluid Mechics d He & Mss rser WSEAS Press p 4-47 [7] H Klis A Buiis Mehod o lies d iie dierece schemes wih he ec specrum or soluio he hperbolic he coducio equio Mhemicl Modellig d Alsis Vol 6 No p-3 [8] Smrsii A A he heor o Dierece Schemes CRC Press [9] Sov G he Uiversl w he Geerl heor o Phsics d Cosmolog Sov s Uiversl w Press 998 Acowledgemes: his reserch ws prill suppored b vi Coucil o Scieces (gr 63/4) ISBN: