A Method for Solving the Three-dimensional Wave Equation

Transcription

1 Ueral Joural of Compuaoal Mahemac 4: 8-8 DOI: 89/ucm4 hp://wwwhrpubor Mehod for Sol he hree-dmeoal Wae Equao S Yakmo omk Sae Uery omk 644 Rua Correpod uhor: yakmoa@malru Copyrh Horo Reearch Publh ll rh reered brac O he ba of he aplace eral raform locally oe-dmeoal cheme of cleaae ad qua-learao mehod o oba a appromae aalycal oluo of he hree-dmeoal olear hyperbolc equao of ecod order he aeme of he accuracy of aalycal formula whe compared wh he eac oluo of he fr boudary alue problem ad umercal oluo by a kow mehod Keyword alycal Soluo hree-dmeoal eleraph Equao aplace Ieral raform Iroduco I mahemacal model of hea ad ma rafer [] he hea rafer hh-frequecy procee [] brao [] ad o o here a problem he oluo of eleraph ype a equao [-] If he oluo of a olear parabolc equao [] here are a umber of aalycal echque reewed [4] he eac aalycal oluo are obaed for he lear oe-dmeoal he abece of a ource [56] or mul-dmeoal [] a equao of he eleraph ype Howeer pracce mo ofe of ere o he oluo of olear boudary-alue problem [-47] For oe-dmeoal oluo of olear ordary dffereal equao [8] propoed a mehod of qua-learao Wh h mehod a deco alo he olear problem reduced o ol a equece of lear problem whch eeally a deelopme of he well-kow Newo' mehod ad eeraled ara propoed by V Kaoroch [9] Oherwe he qua-learao he applcao of a olear fucoal eeraed by he olear boudary alue problem he Newo-Kaoroch I he umercal oluo of problem of mahemacal phyc were effece pl mehod [] I parcular he locally oe-dmeoal cheme cleaae [] propoed o ole he mul-dmeoal hea equao combao wh he aalycal coa coeffce ad umercal mehod he purpoe of arcle wh he help of he locally oe-dmeoal cheme pl [] qua-learao [89] ad he aplace eral raform [] o fd a appromae aalycal oluo of he olear hree-dmeoal hyperbolc hea coduco equao a fe reo ad o ae he accuracy of aalycal formula Saeme of he Problem ad he lorhm of Mehod Suppoe you wa o fd a oluo o a hyperbolc equao of he ecod order [] wh ource C k Y he parallelepped Q:[ < < < < ] Q Q Г Q Q [ < ] Г boudary urface of he doma Q wh al codo p p ad o mplfy furher calculao wh he boudary codo of he fr kd Ψ Г Ψ co where co c / Y C w c he peed of propaao of hermal perurbao peed of oud he medum m/; w he rae of coece hea rafer m/; C he coeffce of olumerc hea capacy J/ m K ;

2 9 Ueral Joural of Compuaoal Mahemac 4: 8-8 he hermal coducy he hermal coducy a he al emperaure W/m K; he emperaure K; a of he Decare coordae yem m; me ; leh of he de of he parallelepped m ccord o [] he equao obaed aum ha C do o eplcly deped o Y me ad w << m/ dreard med derae w a compared / o he lef of equao Furhermore equao o he rh de are abe form med derae ad ummad τ r / τ r χ / c τ r relaao me χ coeffce hermal dffuy m / whoe alue for he me codered below: > τr τ 9 r elble We wll alway aume: Problem ha a uque oluo whch cououly Q ad ha couou derae / / / / Performed follow codo: l > C l > l > l co p p e o couou fuco Q а C Y couou fuco Q he coeffce C Y he eeral cae ca be o-learly depede o he oluo of problem [] Vew defed by he follow formula 6; Ψ e couou fuco o he boudary Г for < ha bouded paral derae of he fr order pplcable he locally oe-dmeoal cheme pl of he equao o he dffereal leel [] ad roduce he upercrp o deoe he oluo of he ermedae ae a well a here he dreco of wae deco ad here dreco he oluo of he parabolc par equao he we hae σ p < < 4 p < < G G 5 C Y 6 k σ[ ] < < G G ; 7 σ < < 8 Q Q 9 C Y k σ [ ] < < Q Q ; σ < <

3 Mehod for Sol he hree-dmeoal Wae Equao D D C Y 4 k σ [ ] < < D D 5 where σ σ σ Whe C oled he yem of equao 4 5 for he eleraph equao: he preece of frco coduce medum Whe C oled he yem of equao for he wae equao: he lack of frco decay erome Idee: he al alue characerc alue a he boom he ulmae alue r alue of relaao boom par of he wae equao below he parabolc compoe of he equao We are alk abou he e model for eample he coduce-coece hea rafer for he hyperbolc hea equao a Fr he fr ae ured off by he coduce-coece hea rafer coordae dreco ha coder he problem 4 7 we e a emperaure drbuo he a ak her by he al ur off he coduce-coece hea rafer o dreco ad ole he problem 8 he a we hae he drbuo We ake for a al emperaure ur off he coduce-coece hea rafer o dreco ad coder he problem 5 we fally e he emperaure he mome whch cocde wh he rue alue ccord o h model he proce he coduce-coece hea rafer "reched" me ad ake place dur he me of he ap [] ad ead Such a approach o ole mul-dmeoal equao of paral dffereal equao wh coa coeffce propoed ad ufed [] For he wae equao C are ecluded by he coordae dreco wae elocy hear elacy of he medum ad o o Bu before ha o he yem 4 5 mu apply Krchhoff raformao [4] ad qua-learao [89] o oba he dffereal equao wh coa coeffce whch ca be oled wh aplace eral raform [] I he fuure o ue he ero formula ake he form m N m > We ue Krchhoff raformao [4] N N co 6 d 7 he ak o accou he relao [4]: 8 e ou 4 8 b p / φ φ / N m 9 σ a < < p < < c r σ [ a k a ] < < b ; 4 σ a < < 5 q q c r

4 Ueral Joural of Compuaoal Mahemac 4: 8-8 σ [ a k a ] < < b q ; q σ a < < 6 d d c r σ [ a k a ] < < d d where c C / r Y / b / c a / a / / φ G q / φ d / φ Q D h ermedae alue dreco: formula deermed from 9 he ero / φ / φ 7 he he fal oluo of he problem wre 8 Noe ha he rae of he depede arable ad he ype of boudary codo do o chae uder he raformao Krchhoff 7 ad he preece of he ero formula boudary codo of he fr kd of oe o he Drchle codo Our purpoe o recee a oluo of he olear boudary alue problem f e a he lm of a equece of oluo of lear boudary alue problem o do h we ue he reul [7-9] ume furher ha all he coordae dreco pace are equale e co here a al appromao [a a al appromao o ake for he fr formula of wh he fr equao 9] For mplcy coder he cae of a qua-oe dmeo aaly ad equece defed by he recurrece relao [8] do ad bar a he op dcae he paral derae wh repec o me ad pace y f 9 f f Ψ / Г φ where y here he ay coordae from y he re coordae 9 9 he for < < chae paramercally he rema coordae he preparao of epreo 9 ad occur crcular replaceme de f ead ubue y repecely Noe ha ol he hree-dmeoal boudary alue problem 6 whle he fr coordae he dreco of a al erao ac he ubeque erao obaed from he fal epreo [ee comme below o formula 56 6] he a qua-oe dmeo of equao 9 ca be rewre he coordae [7]: f f f Epreo mlar o 9 ca be wre oher coordae dreco I parcular he ecod coordae dreco eceary eerywhere o replace he op ad boom dee ad o ad ad he op de o hu for he ery codo he ecod co-ordae dreco hae Each fuco 9 he cae of qua-oe dmeo or oluo of he lear equao whch a ery mpora feaure of h alorhm he alorhm follow from he appromao mehod of Newo-Kaoroch [9] he fucoal pace I order o reduce furher record roduce he follow oao:

5 Mehod for Sol he hree-dmeoal Wae Equao f R σa c r Z φ k / / σ az a c / / f r k / Φ σ akz / ZN R Φ / k / σaz [ k / Z N] a h σ c R Oba a qua-oe dmeo oluo of he problem coordae dreco u equao h b R h < < 4 p p 5 6 aplace eral raform applcable o he dffereal equao 4 ecep for derae o ad replac wh a lear epreo relae o he mae of he dered fuco I wha follow we coder fuco for whch he aplace eral raform aboluely coere he real par of he comple umber p α β poe ha Re p > We deoe he mae b leer V H I aumed ha calcula he mae coordae we operae wh he fuco of he aalyc couao o he alue > o ha law whch hey are defed he eral I beleed ha he oluo ad derae equao 4 afy he codo for he eece of aplace eral raform o ad he deree of rowh o he fuco ad derae do o deped from For mplcy we om he de calculao a a he boom of he de a a well a he de of a he op ad a he H boom a V h dca / / he we hae from 4 []: p V p p H p < < V p p / H 7 U he ere aplace eral raform []: / p p [ H p / p] h y o reore he oral для from 7 [] y h y 8 he derae / 8 we fd u he ecod boudary codo of 6 y h y 9 herefore fd / 9 ad ubu o 8 we oba y h y 4 y h y raform he epreo o he rh-had de of 4 o a o e rd of he eral wh arable upper lm he by roduc he Gree' fuco E [78] y y / y y y / y E 4 epreo 4 rewre he reur of he upper de ad lower o ha b clearly o depede o ulke R b E p y 4 E y R F We apply aa Krchhoff raformao 7 o reur o he oral arable equao 4 d m N / φ 4

6 Ueral Joural of Compuaoal Mahemac 4: 8-8 raform he lef de of equao 4 u 4 / φ φ / N m Sce he ecod erm o he lef-had de of equao 4 ake he form / φ he eceary o reapply he qua-learao he we hae где B φ m [ Y F / φ B ] P Y b E y 44 B Y B > p E y > из 4 We ue he dea of Doech [] o he applcably of aplace eral raform o he paral dffereal equao a may me a dmeo he wh he al codo 5 we oba [7] of 44 p p τ P τ dτ 45 γ γ τ γ [ γ ] 46 τ co γ τ co[ γ ] τ m γ β β B / φ p γ p 47 τ P τ dτ Now we oba a aalycal oluo of he problem of he ermedae 4 u he oao Φ h r < < 48 Noe ha Φ of doe o deped eplcly of 48 [ Φ you ca alway ak a he boom of erao kow he alue of he al ad ubeque me from 45 ad from 9] For mplcy we om he de calculao a he boom of he de a ad a well a he de of a he op ad a he boom a V h H he a 4 4 we hae p V p p Φ V p r pv p ] H p [ < < p δ b δ / H V 49 p δ b b [ p δ b ] where δ r / b Φ δ U he ere aplace eral raform []: [ p / p b ] co b a b Φ δ > [ p / p b ] coh b a b < [ p δ ] ep δ [ H p / p] of 49 [] h y o reore he oral for ep δ [ u δu ] u / } 5 { δ ep ep δ u y h y u < < ; y b b u y b [ b y] 5 u co b Φ δ b ; > u coh b u b h b u y b h[ b y] b < We fd he derae / 5 a 9 u he ecod boudary codo 48 ad ubue o equao 5 we oba

7 Mehod for Sol he hree-dmeoal Wae Equao 4 ep δ u δu ] 5 δ [ ep ep δ u y h y δ y ep ep δ u y h y u { ep y ep δ u u δ [ u δu ] }/ u raform he epreo o he rh-had de of 5 o a o e rd of he eral wh arable upper lm he by roduc he Gree' fuco G [78] G y y ep[ δ y ][ u u y / u u y] y ep[ δ y ] u u y / u y 5 epreo 5 u he formula ad he reur of a ubcrp a well a he op rewre U X S G y R W 54 S ep δ [ u δu ] u ep { ep δ δ G y c [ u δu ] }/ X U X u a reul he oluo of 54 ake he form [4]: W τep τu dτ 55 U < < ep where ake from he formula 45 u equao 9 Smlarly oluo are obaed qua oe-dmeoal problem 5 6 o he coordae dreco wh he oleme of ad he ero formula 7 he accord o he alorhm 7 55 a / derae were o pree we hae в 6 4 hee 56 τ P τ dτ P Y F B / φ 57 τ P τ dτ Y F B / φ P Y b E y B Y F q q q E y R F d d d E y R γ τ P τ dτ P Y F B / φ 58

8 5 Ueral Joural of Compuaoal Mahemac 4: 8-8 γ B [ F / ] φ τ P τ dτ P Y F B / φ 59 m 6 N W U W ep τu dτ 6 ep U W X S G y R X U X G y c S ep δ q[ u δu ] u ep δ q ep { δ [ u δu ] q }/ u S ep δ d[ u δu ] u ep δ d { ep δ [ u δu ] d }/ u where he formula are ued o ole he eleraph equao ad he formula are ued o ole wae equao C hu τ τ are ake from 46 wh he chae he de a γ β B order of equece aywhere o he de ad Smlarly obaed E y G y from 4 5 ad u u from 5 u y by replac all arume repecely order o follow 56 6 oher arable < < paramercally modfed a 55 mlar uao for h e a fal deco 8 eleraph equao wh he boudary codo he ero formula 7: > ca be ee from he alorhm fr ermedae alue are ecluded of he equao 9 ad ad formed of he formula 56 6 o he reame of he formula 7 fal epreo 8: he he erae proce cluded U he reul of [75] we fd he codo for he uque olably of he problem uder cera aumpo ad e a quadrac rae of coerece of he erae proce e Reul of Ipeco he accuracy of he aalycal formula check o he e fuco he oluo of boudary alue problem for paral dffereal equao Q : [ ] [ < ] a m N m > C 6 wh al codo ep y k F w ep y y ad he boudary codo of he fr kd w co w > ep τ ep τ ep τ ep τ ep τ ep τ / w 6 τ 64

9 Mehod for Sol he hree-dmeoal Wae Equao 6 I wa ake he eac oluo 6 64 ep τ y 65 he he ource of F 6 ha he form F ep τ y { / w / w C N ep[ m τ y]} ep[ k τ y] he follow referece alue were ued pu: m 5 w 5 σ 4 σ σ m N 5 W/m K c 4 m/ J/ m K ar erome [6]; N / N / M M emaed umber of u ad he ep pace ad me fd he eral equao by Smpo [7] he boudary alue problem 6 64 wa oled wh he help of formula he umber of erao wa moored for he fal epreo 56 6 for o he relae chae he ecor of error: V Q ma Numercal eample are e for V δ δ he ame me ook oly wo erao o achee h accuracy ad compuao me of ay ero of c he proram G-Forra he calculao wa performed o PC Peum 5 GH wh he double accuracy I able e he mamum relae error perce able he mamum relae error ε a he oluo of he eleraph equao C k ε % ~ % ε ma 66 Q where he eac eplc oluo 66 ~ a appromae aalycal oluo for ee mahemacal echoloy arcle for arou alue k C ca be ee from able preco aalycal oluo obaed h arcle afacory compared wh he umercal oluo of 6 64 ued he echoloy of calcula he lear eleraph equao [8] For he umercal compuao appled o-eplc ucodoally able dfferece cheme wh a abolue error of appromao for he fr ad ecod derae wh repec o pace O ad he hree-leel cheme for he me derae wh appromao error O[ ] For referece pu alue ad m m C o equao e ε 9% ad o he dfferece cheme [8] a N 4 e ε 86% ad c с pecfc eample he accuracy of aalycal formula wa foud wo me beer ha he accuracy of he umercal oluo [8] I able how he oluo of he wae equao by formula a dffere alue of m a ca be ee from able he oluo of he wae equao afacory accuracy ε < 6 % acheed for a mall me 5 able he mamum relae error ε a he oluo of he wae equao m a ε % Now compare he accuracy of he aalycal formula he oe-dmeoal cae he abece of erao for mplfed boudary alue problem y χ < 67 y w y 68 wh kow aalycal oluo of [5] a χ / C c y θ w θ u β δep δ 69

10 7 Ueral Joural of Compuaoal Mahemac 4: 8-8 β δ ] d δu β δ 5 I ep[ δ δ β δ β /χ 5 c δ cy / χ where u β δ here he u fuco u a u β δ u for β < δ I here modfed Beel fuco of he fr order [5] k k I / k k! 7 ca be ee from 7 he row coere lowly a > herefore by 69 7 for eample meal: eel c 57 m/ 47 m/ 6 χ 5 / 4 χ 6 / m copper c m lqud: waer c 7 5 m/ χ 4 m / [6] ar ad oher uder ormal codo you ca acually u oly rapd procee 8 o mcro dace 6 m decrbed by he oe-dmeoal problem of he form 67 ad 68 5 χ m / c 4 m/ 8 K w K 4 m 5 β 75 δ / N N 5 ulke he aalycal oluo he oe cae he eac 69 o eceeded 56% Cocluo O he ba of he locally oe-dmeoal cheme pl qua-learao ad aplace eral raform fd a appromae aalycal oluo of a olear hyperbolc equao of he ecod order whou u he heory of he ere [7] I he oe cae a comparo of he accuracy of aalycal formula arcle wh kow eac oluo of he eleraph ype [5] he mehod of he ral fuco e he comparo of aalycal oluo he deeloped echoloy wh he eac oluo of he boudary alue problem wh he umercal oluo by a kow mehod 4 he e calculao of 6 64 are codered coeoal fe pace-me eral: 5 m ecouered pracce [5] he work wa uppored by he Sle Order re 99 REFERENCES [] yko V Hea ad ma rafer HadbookEeryMocow 97 [] Zhou D Caa BakeX ebod Eeded rreerble hermoamc NI "Reular ad chaoc hermoamc"; Icomp e were performedmocow-ihek 6 [] choo N Samarky he equao of mahemacal phycscece Mocow977 [4] yko V Mehod for ol olear rae hea coduco equaonew cademy of Scece of he USSR Eery ad rapormocow [5] Baumeer K Hamll Hyperbolc equao of hea coduco he oluo of he em-fe bo Hearafer [6] Kudo V Kudo IV Preparao ad aaly of eac aalycal oluo of hyperbolc-parameer equao of hea coduco for a fla wallhermal phyc of hh emperauremocowol [7] Yakmo S he aalycal mehod for ol boudary alue problem Ed -d add Publh Houe of omk Uery Preomk [8] Bellma R Kalaba R Qua-learao ad olear boudary alue problem Sprer-Verla New York 968 [9] Kaoroch V Fucoal aaly ad appled mahemacsucce of mahemacal cecemocow [] Samarky O umercal mehod for ol mahemacal phyc Hea ad ma rafer Scece ad echoloy Mk ol [] Samarky Iroduco o he heory of dfferece chemescece Mocow97 [] Dk V Prudko P Operaoal calculu Hh chool Mocow 966 [] Doech G Gude o he praccal applcao of he aplace raform Phycal ad mahemacal Sae edo Mocow 96 [4] Maee NM Mehod for he erao of ordary dffereal equao Hh chool Mocow 967 [5] Yakmo S O a mehod of ol he couae hea rafer problem Par Joural of Eeer Phyc Mk ol [6] Kohk NI Shrkech MG Hadbook of elemeary phyc Scece Home edo of he phycal ad mahemacal leraure Mocow 988 [7] Broe NI Semeae Hadbook of mahemac for eeer ad ude of VU-callScece Home edo of hephycal ad mahemacal leraure Mocow 986 [8] Grh M Yakmo S he erae-erpolao mehod for ol hree-dmeoal wae equao Compuaoal echoloe Noobrk ol -4 7