MATHEMATICAL MODELS AND THEIR SOLUTIONS FOR DOMAINS OF COMPLEX FORM

Transcription

1 UIERSITY OF LTI FULTY OF PYSIS D TETIS T os TETIL ODELS D TEIR SOLUTIOS FOR DOIS OF OPLEX FOR PD Tss F: cs Suf: c og Scfc suprsor: Er prof Dr rs uķs Rg 4

2 Ts wor s suppor Europ Soc Fu w proc Suppor for Docor Sus Urs of L proc o 9/3/DP//9/PI/I/8

3 src I PD ss org c os r souos for os of cop for r cosr Is s qucg procss for s w fs s scr 3D proc proc couco quo For rg c souo for pro r s fuco o s ofco s us For sss w ou w ou fs r po or copurs D sor rs rsfr pros r g Our ppro souo s cosruc usg cosr rgg o f ffrc sc s ofcos for our coos w c o for wow fu s prs wc cosrs D r w quo w sourc fuco r p our coos Kwors: couco quos w quo r s fuco cosr rgg f ffrc o o of spro of rs L-sp sp ou w w ou fs wow fu

4 os ITRODUTIO 5 IPORTE OF TE SUJET 5 I OF TE TESIS 5 RESER ETODOLOY 5 SIETIFI OELTY D I RESULTS 6 PULITIOS REPORTS O OFEREES 6 Pucos 6 Rpors o Iro Scfc ofrcs 7 Rpors o Loc Scfc ofrcs 7 YPEROLI D PROLI ET ODUTIO EUTIOS FOR L-SPE SPLES 8 TE PYSIL PROLE 8 O-DIESIOLIZTIO 9 3 USI YPEROLI ET ODUTIO EUTIO TO DESRIE I PROESS 3 r S of 3D Pro for L-sp Sp 3 Foruo of Drc Pro D 3 Souo for s 3 3 Souo for Foo 5 33 ougo of o Souos 6 33 Foruo of Irs Pro D 7 4 USI PROLI ET ODUTIO EUTIO TO DESRIE I PROESS 7 4 r S of D Pro 7 4 Ec Souo of D Pro 8 5 OPRI TE RTE OF TE TEPERTURE 9 STTIORY ET TRSFER I SYSTE WIT DOULE WLL D DOULE FIS EOETRY D TETIL STTEET OF TE ORIIL PROLE PPROXITE SOLUTIO TO ET TRSFER WIT ET EXE 3 Souo for F 4 Souo for s 5 Souo for Rg Pr of s 6 Souo for Lf Pr of s 7 3 ougo of Souos 8 4 urc Rsus 8 3 PPROXITE SOLUTIO TO ET TRSFER WIT OILI 9 3 Pr og 3 3 Souo for F 3 3 Souo for s 3 33 ougo of Souos 3 3 Fu og 33 3 urc Souo for F 33 3 urc Souo for s 34 3 TRSIET ET TRSFER I SYSTE WIT DOULE WLL D DOULE FIS 37 3 TETIL STTEET OF TE PROLE 37 3 PPROXITE SOLUTIO OF TE PROLE 37 3 Ruco of D Pro for F 38 3 Ruco of D Pro for s 39 3 T Rg Pr of s 4 3 T Lf Pr of s 4 33 osruco of Dffrc Sc 4

5 33 Dffrc Sc for F 4 33 Dffrc Sc for Rg Pr of s Dffrc Sc for Lf Pr of s ougo of Souos UERIL RESULTS PPROXITE SOLUTIO OF TE PROLE WIT PRTIL OILI urc Souo for F urc Souo for s Dffrc Sc for Rg Pr of s Dffrc Sc for Lf Pr of s ougo of Souos 5 4 TETIL ODEL FOR WILLOW FLUTE 53 4 PROLE FORULTIO 54 4 PIT D FREUEY EERY OF TOES UERIL RESULTS Op- sc os- sc Ir sc Erg of Tos 59 ILIORPY 6

6 Irouco Tr r prcc ppcos por uss for o sg r sss w fs for sss w ou w ou fs T r us s rsfr qup for s qucg c To scr s procsss w foru pros 3D D prs prcs urc os for og souos T cs of o-r our coos s so cosr S frs r cprs of ss T s cpr s c o cosrucg c o for wow fu W cosrucg o for s sru o c su upo fw qusos s oug o us r w quo ow o scr rsr pgs g opg ss fow pssgw c Ts fcs r cosr our o Togr w co-uors uor s 8 pucos o suc of PD ss T uor s rpor o 6 ro 3 oc cofrcs Iporc of Suc Is qucg procsss r cosr frs cpr of ss sgfc pc rg usr Org Dr Koso s wrqucg coogs r ro op Sc s scor r urous prs sus couc Dr Koso coopro w scss fro US Ur co O of os ro o rsrc procs s Ds for coog cpcs of rous qucs o op w or copuo pr cqus WSES Sss w ou w ou fs r us or copurs coog sss u r o p o ro css fro copur copos coog w r s rpc qu coog s o og rsfr croc pssgs pros ffc w o ro g fu s rs Ts rsus o-r pro w so usg ffr of c os or w scrg usc srus c o cos w quo s so o of spro of rs I ss w us s o w og wow fu of Tss T ror go of s ss s o op c os for os of cop for I cus foowg ocs: scr s s qucg for 3D L-sp sp o proc proc couco quo; propos w pproc for rg souos for sor rs pros D sss w ou ou f; op c o for wow fu Rsrc ooog c urc os: cosr rgg o; r s fuco o s ofco for rgur o-coc os; spro of rs; f ffrc o s ofco 5

7 Scfc o Rsus w c os r propos for s qucg w coosg proc quo o scr procss 3D L-sp sps Ec souos for proc proc quos r cosruc os for D sor rs pros for sss w ou w ou fs r g o r or cs w ssgg og coos s cosr W propos c o for orwg fu usg r w quo W us sourc fuco o scr procss wr rsr s o for orcs w pgg g opg u ss fow pssgw s ppro 3 r p our coo Pucos Rpors o ofrcs Pucos T os u us Trsfr w Fu og Ss w Dou W Dou Fs Procgs of WSES Iro ofrc o Trsfr Tr Egrg Ero oug rc WSES Prss 3 pp IS: fro Ir: p://wwwwssus/-rr/cofrcs/3/oug/flue/flue-4pf s cc 8//3 T os u us Trsfr w Pr og Ss w Dou W Dou Fs Procgs of WSES Iro ofrc o Trsfr Tr Egrg Ero oug rc WSES Prss 3 pp 3-37 IS: fro Ir: p://wwwwssus/-rr/cofrcs/3/oug/flue/flue-pf s cc 8//3 3 T os u us o Sor Trsfr Ss w Dou W Dou Fs Procgs of WSES I of o D wors oucos opurs DOO 5 WSES I of o rs Scc TERILS S WSES Prss pp 6-65 IS fro Ir: p://wwwwssus/rr/cofrcs//s/dt/dt-43pf s cc 8//3 4 T os us c o for Wow Fu Procgs of 9 ISE/WSES I of o ET TRSFER TERL EIEERI EIROET TE Forc I WSES Prss pp 88-9 IS fro Ir: p://wwwwssus/rr/cofrcs//forc/eflu/eflu-9pf s cc 8//3 5 T os u us oprg Souos of proc Proc ouco Equos for L-sp Sps Procgs of 9 ISE/WSES I of o ET TRSFER TERL EIEERI EIROET TE Forc I WSES Prss pp IS fro Ir: p://wwwwssus/rr/cofrcs//forc/eflu/eflu-66pf s cc 8//3 6 T os u us proc Equo s c o for S ucg of L-sp T-sp Sps Drc Irs Pros WSES TRSTIOS o ET SS TRSFER 53: 63-7 ISS

8 fro Ir: p://wwwwssus/-rr/rscos///89-66pf s cc 8//3 7 T os u us proc Equo s c o for S ucg of L-Sp Sps Pr Irs Pro I S Sor J rs Koso Es Procgs of 5 ISE/WSES I of o OTIUU EIS rg UK WSES Prss pp -6 IS fro Ir: p://wwwwssus/rr/cofrcs//rg/f/f-pf s cc 8//3 8 us u T os Souo of proc quo ruco o gr quo for rg corr Procgs of I of Dffr Equos Tr ppcos DET 9 Pzs Lu 9 KTU 9 pp 4-6 Rpors o Iro Scfc ofrcs T os uķ us o Sor Trsfr Ss w Dou W Dou Fs WSES I of o D wors oucos opurs DOO 5 WSES I of o rs Scc TERILS S Spr 7-9 Procgs pp 6-65 T os u us Dou W w Dou Fs 7 Iro ofrc c og ss T Eso Ju 6-9 srcs p 9 3 T os us c o for Wow Fu 9 ISE/WSES Iro ofrc o ET TRSFER TERL EIEERI D EIROET Forc I ugus 3 5 Procgs pp T os u us oprg Souos of proc Proc ouco Equos for L-sp Sps 9 ISE/WSES Iro ofrc o ET TRSFER TERL EIEERI D EIROET Forc I ugus 3 5 Procgs pp T os us c o for orwg Fu 6 Iro ofrc c og ss Sgu L 5-8 srcs p 7 6 T os u us oprg Souos of proc Proc ouco Equos for L-sp Sps 5 Iro ofrc c og ss Drus Lu 6-9 srcs p 8 7 T os u us proc Equo s c o for S ucg of L-Sp Sps 5 ISE/WSES Iro ofrc o OTIUU EIS rg UK Frur 3 5 Procgs pp -6 Rpors o Loc Scfc ofrcs T os u us c o for Dou W w Dou Fs 9 L c ofrc Jg L rc 3-3 srcs p 9 T os uķ Prcīzs u uās rsāus scoār u scoār proē L- pg T 69 Scfc ofrc of Urs of L Sco of c og urc ss Rg Frur 7 3 T os u us O Drc Irs Pros for proc Equo for L-sp T-sp Sps 8 L c ofrc r L pr 9- srcs p 7

9 proc Proc ouco Equos for L-sp Sps O of os por r procsss urg s qucg wr s us for rg o pro cc proprs of r prouc uc ougr ur w srogr srucur I s procss prs r o rqur prur coo rp r o wr or or qu of s os for rg s prs r w ow s s qucg I oug r rous of -ow I procsss sg sc 9 s I coog ws gr fuc op Dr Koso Ur I cors w coo qucg I procss s couc g g wr or r w coog r s sgfc fsr qucg o r c [] [3] [38] T c os op o couc ss o I os cu rs couco quos u s I s procss wr prur of sp s cgg r quc ffr of os sou cosr I [8] Prof us propos o scr procss proc couco quo s s sow sr pucos s w c os g goo rsus r rso r-wor suo co g [4] [8] [7] [8] c ccorg o prous sus rsrc [5] [8] [9] [3] r w cosr our u pros IPs for o proc proc couco quos scrg I procss for L-sp sp Usg r s fuco o cosr rgg w cosruc c souos of rs rc pros for of gr quo of fr f cssr s poss o copr r of cg of prurs s gouroo of fr rg souos for s pros T psc pro s s w ow couco so o c scr w-ow Fourr quo T wr T f T c c os prur of o w r couc f s s c of sourcs s spcfc cpc s of o s r ffus of r T quo c r cog w of cosro of rg s w of rocs T c q w Fourr s w q -T f I ur of psc suos quo ps rrr g r propgo sp O of suc css s I W rsg pr o quc sp of propgo of s o f u cu s f s [39] So Fourr s w s o ogr su o scr propgo I s css 3 c rpc or gr w propos o ro: 3 q q -T 4 8

10 r s r propr s c ro Iroucg quo 4 o s us o proc couco quo or T T T f T T T f c f f 5 Ts so-c Tgrpr s quo or p w quo s f sp of propgo for T r s crcrsc sp w or foro o proc quos c fou r: [5] [3] [3] c r gr p o T o r oo u rougou ppr w ssu s r coss o-sozo L s g w w rcgur fs s Fg fro [7] s coo rp su fu g wr or r Sc fgur c o sr src L-sp prs w c us quos 5 o scr I procss for o pr o g s rsus s f w quc r fgur Fg E w fs I orr o spf pro w us o-sozo T sp rs r o-soz sg ˆ R R ˆ R L R z zˆ R R ˆ T * 6 R wr s w of f R spc w fs Ω g of w Δ g of w L g of f If os prur sruo w prur sruo f s sg z w c f soss prurs rs of w rs 6: ˆ z ˆ ˆ ˆ zˆ ˆ ˆ ˆ zˆ ˆ ˆ ˆ ˆ zˆ ˆ z z ˆ ˆ ˆ zˆ ˆ 7 wr r so crcrsc ro prurs Sr w prurs of surroug u r 9

11 z z ˆ ˆ ˆ ˆ ˆ 8 z z r us of prur of fu f s of w or ss of sp w prrs r ˆ R ˆ R R ˆ * ˆ T * ˆ R T T qus s g s prcg sco u rsfr coffc 3 Usg proc ouco Equo o Dscr I Procss I s sco w r gog o cosr quo 5 for scrg I procss so IP usg r s fuco o s g [8] [9] [] [6] [9] [] To of o o o cos-for r s fuco for so c rgur o-coc o w r gog o rprs org o s f uo of coc su-os w ppropr our coos og ps or s D cocg wo gour os 3 r S of 3D Pro for L-sp Sp W rfor suppos g L-sp sp s up fro wo rcgur prpps rcgs D s sos c w foo: z z z z o og surfc s of w o f I procss for c pr spr Tus rs of soss rs 6 7 roppg s proc quos of couco foowg for z S z r 9 z S z r wr z f z f c S r z f z f c S r If r r o sourcs of S S To s IP for rg prur of sp w o foru o our coos og ps sr coos us pp:

12 u or ss of sp cg s pc w surroug u wos prur s g 8 Tus z z z z For foo w our coos s w: z z z z For fuur w w s soss ro prur o zro for w cpo wr I s so poss o cosr o-r our coos s [4] I cs souos of proc proc quos ffr ss s s s r coc w foo cou of prur fu r pos rfc w c prs of sp: z L us so ss coos: z W z z z z 3 W z 4 I I coos 4 r ursc: -r of prur cg c sur pr I sou ccu o copr w crc coog r o prc rsfr os s coog r c ffr rgs s [39] Trfor w c ssu prur sruo sp of prur cg r g of procss:

13 z T T z T T 5 z W T T z W T T 6 Pug ogr pr ffr quos our coos w rc pro or rs pro w usg f coos Sc fg souo of 3D pro s qu sr s D cs w w cosr o r 3 Foruo of Drc Pro D L s suppos os r z rco for D o so ssug prur s os cos s rco ows us o us sps for of cosr rgg o pproo cos c o ruc 3D pros 9 cosr for o D os w w rouc rg us of fucos us for or r For p z z 7 z z 8 W us s pproos pp our coos o o wo D proc -couco quos s r 9 s r 3 o w w us s our coos s foru s of org pro prous susco Usg cou coos our coo for rg- s orr of s c wr for 3 f- s orr of foo w g 3 r p our coo s w: ] [ 3 u coos r w

14 3 w 3 Souo for s T pro for 9 c so rsforg pro w o-oogous our coo 3 o or o w zro our coos To o w ssu souo of for s [46] wr 33 ssfs o-oogous our coo Susug 33 o org pro w g rsfor pro : r 34 s r w s our coos: coos: 37 z w 38 T souo of w pro c wr s 3 r 39 r s souo of 34 ; s souo of 34 ; 3 s souo of 34 T fuco s f r Frs s op fuco s o g our coos w p souo s Y X T

15 wr gfucos foowg prssos X s w or squrs X ; Y s cos cos ; Y u gus r F co wr T fucos T c fou for r pos zro-pos of w r T p r s cos s s cos cos cos s r s o u Sog for gs r r r 4 4 r 4 4 r r r r ur coss ppg coo X Y s F w X Y 4

16 5 4 w r fuco g s p r Y Y X X Fro g fo 39 4 w s r 4 r 3 4 Trfor souo of s r w r s r r r r r 43 3 Souo for Foo T souo o sco pro 3 s qu sr: r

17 6 w r s r r r 44 wr Tus r fuco s s p r Y Y X X Fucos s s g s prous susco o X s ; Y cos cos s ; s r r pos zro-pos of f co u r pos zro-pos of g co 33 ougo of o Souos Upo coupg quo 44 w foru 3 w rprso for coo of souo for s s r orr w o prs - I orr o r sr rprso for w susu prsso 43 o 3 s fuco s p of w pug

18 7 r o forr o o-oogous orr-fro gr quo of rfc w o prs of L-sp sp: K L fr fg souo o s gr quo w c ccu prur sruo sp f r of prur cg 33 Foruo of Irs Pro D s ws o for r of prur cg s o ow u sos us r So w ow rss rs pro of rg r g f : prur sruo r of cg of prur T Ts corrspos o sog g quos cwrs W c rsfor s pro o rc pro roucg w r T 45 so T u u 46 T u u 47 I rs of w quos 9 3 rsfor o s r 48 s r 49 T our coos o o cg u coos co T T 5 w T T 5 w T T 5 Ts rc pros c so s for 4 Usg Proc ouco Equo o Dscr I Procss 4 r S of D Pro I s sco w ou ow o f r of cg of prur w usg proc couco quo o scr prur sp s [7] [8] for or Proc quos of couco 3D r g z S z 53 z f c S

19 8 z S z 54 z f c S usg rg u of fucos or r ] [ ppg ppropr our coos 5 6; 9 quos r rsfor o s 55 s 56 w wo sp rs o coos r s s prcg sco 4 Ec Souo of D Pro ow w pp cqu us Sco 3 s susco 3 o r souos o IPs Tus souo for s s for s 57 wr W us gfucos gus fro Sco 3 o p r fuco: p Y Y X X u fuco ssfg s pro for 56 ssus foowg for:

20 9 s 58 w r r fuco s f p Y Y X X Usg s o s prcg sco w o orr-fro gr quo g souo sp -r of prur cg 5 oprg R of Tprur W p souos of proc proc couco quos w r fou w c ffr s prssos w rspc o copr rs of cg of prurs s gouroo of sg p wr T T s prssos c r for fucos p

21 Sor Trsfr Ss w Dou W Dou Fs Tr urous sus o sss w surfcs wr r srucur s of s r I rsrc rs g o or copurs or cop s o po Usu s r srucurs of cror scs sr rous cqus r prs for pug s crosc s o coro surfc rougss I o of os s [9] [35] [36] p surfc s roug g s sru rc sco owrs cor so of cog g fuor cro 4F8 s Fg Suc cro/o srucurs r of op o coro c prforc of og rsfr Fg Scc of frco procsss for crsg ropc corg ropc/suprropc surfcs o ropoc/suprropoc surfcs [35] I s cpr o foows w cosr couco s of sss w c os sss w ou w ou fs L s ssu ss s D w srg fs of rcgur prof Fg Fg D ss w fs I frs pr of s cpr w focus o sps cs w procss s sor r oogous ss s cos proprs s rsus r pus [6] [7] [] T r cs ss s foow ss of sor couco g ss w r s og occurrg so ss of f [6] [7] Jus pucos [6] [7] [] [3] - [5] [] [] [4] [5] cosr rgg o for L-p os s po o ruc sos for g pro For r cs ppro c souo s cosruc I o-r cs w cosruc ppro souo usg f ffrc o s ofco for our coos Our c os r w qu ffr os wr r sp f sss r cosr g [] [3] [4] [5] [34] [4] [44] [5] [53] [54]

22 or c S of Org Pro cus of gorc r sr of o w c o sr src prs I s suffc o scr s pro for o o of os L- sp prs s Fg 3 Fg 3 L-p o Suc o c rprs s f uo of coc o-orppg suos w ppropr cougo coos og s cocg wo gour os W rfor suppos s L-sp sp s up fro f rcgs s Fg 4 Fg 4 Dfo of gorc prrs for sp L us o prurs of os sos T sc proprs suc s r couc rsfr coffc r cos o rspc For spc rsos w r gog o ssu 3 s procss s sor prur fs r scr Lpc s quos: ss quos foowg our coos r pos W fu :

23 6 s gor of rs s rror sr og s sr our coos r pp: T prcur coc of our coos o us or ss of ss ps o suo s g o og s cocg wo gour os cou of prur fu r sur cougo coos:

24 5 s uppr r s qu r susr w ssu prur ros cross r css r s s s o gg I s w prur c cos r c fro cou coos w g s ppro c prssos for ccug prurs uppr r: Trfor o os s-s couco pros r f for sc r o so So w Lpc s quos T rqur our coos o ss of w o wc souo s o o W s o sco ppro Souo o Trsfr w Ecg ssug or ss r s cg w sp s surrougs r p our coos o spcf r: wr To r prssos for our coos w o s us our coos ppropr cougo coos foru 6 8 For p w wou 3 3 or qu u :

25 35 I o r r cougo coos 3 s f w r r coc rfc of cours our coos 6 8 Usg cosr rgg o w r gog o rsfor g c o o or us for cosruc ppro souo for s pro T of o s o rpc uow fuco cr coo of fucos pg o wc rco our of souo s prc Souo for F So s us po pproo -rco for D prur f f T gr for of fuco s g f f f f 36 w r uow fucos To ccu s fucos w us coos r f for W posg sr coo 8 o 36 w f f f So 36 ow cos: cos f f L s rouc w fuco r [ ] : 37 f s gr rg u of or 38 ow grg prsso 37 w rspc o w c f fuco f f f 39 s L s susu 39 37: s cos cos s s f ppg our coo 34 o 4 s o 4 s cos f s s cos c our ccuo s foru w f : 4 s cos 4 s cos s osqu wr 43 4

26 s cos cos s cos s 44 Tus fuco s ruc o for 43 cog o o uow W r gog o r s fuco rqurg ssfs cr ffr quo W grg pr ffr quo fro o w g 45 T ffrc of rs fou our coos 34 8 prsso 43 Trupo ffr quo cos wr W pp opror 38 o 33 o g our coo for 46: T souo o pro s c fou o wr c c s uow cos Trfor c 5 Souo for s W c os qu for w usg s o of cosr rgg o scr D prur f r L s rouc po prur sruo rco g g g 5 w Oc g w us proprs of fuco o so for uow fucos for w proc s o rg u fuco gr: g 53 fr grg 5 or sg g g g gs 54 s fuco ssfs our coo 6 w pp o 5 o f g : g g 55 T cog ogr w 5

27 g g 56 Pug s wo prssos foru 5 cos g g 57 Equo 57 s cos wo uow fucos for s r r s co o s o wo prs rg pr of wc occups o u f o s for T gro s o prfor o or r o g D quo fro wc w wou f : Souo for Rg Pr of s T fuco g coo 35 o 57: g wc s w 58 for rg pr of s s o posg our g Susug 59 o rprso 57 s 59 6 Ts sows fuco ow ps o o o uow fuco W f sog 58 Owg o our coo 6 6 quo for rg pr of w rsus orr ffr quo wr 6 6

28 For spc rsos w cfor ssu fuco Igrg w o our coo: T souo of 6 6 s w c c s uow cos o cos s 6 63 Souo for Lf Pr of s g W r s f w wo uow fucos o r for prur f pr of s To g os w r gog o us cougo coos 3 Fro frs of s coos prsso 5 for w g u g c 64 s fou sog D quo 58 frs ofg so s for f pr of s s fucos ssf 3 5 foows So ow w c ro r fro 3 c 65 6 quo 58 cos c wc c rwr s wr 58 usg 65 our coo 66 c c cos 67 s cos s s cos I o w g D our coo w grg 7: T souo of pro s 68 7

29 w c cos c c 3 s gro cos c ougo of Souos W us fou souo o g pro u w r s f w fg uow coss forus To r os w sr rqurs o fuf Frs prurs us coc coc po w f rg pr of w So c c Sco prur us w cou c 7 c cos 3 c Tr w c fus so coc c c s : : 7 7 coss c fou fro ss c ppro c souo o D pro s uqu r 5; 6 63; 57 ogr w urc Rsus To g so of oo f s o cou scr cu suo copur coog sss for L-sp cro s w us foowg gorc prrs: 5 5 u for ropsc proprs w cos: K 7 4 W W K K for sco \ T Tprur sruo f 8

30 \ T Tprur sruo f pr of s \ T 3 Tprur sruo rg pr of s W c cocu o scrs r-f suo qu prcs 3 ppro Souo o Trsfr w og I so ppcos og s ocz cr ss of g srucur I s sco s pro s usr cs of o-r our coos L r cs prur fs ssf Lpc s quos 5 ur s our coos s pos Sco w cpo of o-r coos f ours wr r s og occurrg So w 6 fs fu sr coos 7 of cours cougo coos 5 For pro w pr og w our coos 9 3 so w foowg our coos f : 3 3 u for fu og w wou

31 s ws o puco [5] for cs of og u of sou og o r 3;3 3 I s usu o qu o 3 T o s o-r cus of og coo Trfor u ws o prous sco w r gog o sg ffr pproc o so s pro I w of prous ssupo prur s ufor cross susr css s foru 6-8 w spf org pro So ow s w o our coos rsfor o 77 3 Pr og or L s sr w cs of pr og Frs cosr rgg o w us for g D pro o rsfor o D for Sco r o for ppro sog o-r quo for f w prs 3 Souo for F For prur f w us sps pproo ssug s cos w rspc o rgu W 8 wr s rg u of 38 T fuco s fou fro D quo 45 for f Usg our coos 77 8 s quo cos wr 8 T our coo for 8 foows fro gro of coo 33 s g 48 for coug o our ss of o us rwr orr ffr quo 8 s foows: 8 s quo s o-r w r gog o so usg f ffrc sc ros So s s up rgur gr 3

32 rpc couous pro s scr pproo: 83 r s ffrc pproo o u of s po suprscrp os - ro I s cs w c rrrg o quo 83 s s [48]: w coffcs F F u To o sco orr pproo for our coo 48 frs fuco s op s Tor srs o g O 86 g us of ffr quo 8 sco orr r 86 s rpc T w ppropr us of s 83 pproo of coo 48 s o prsso or w coffc o Tg o ccou 88 for g u of quos: w F 84 cosus s of r Fro 85 9 c s for 3 Souo for s s o coos for s r s s r cs souo for w s o s r s susco Tus souo of D pro for rg pr of s s 63: 3

33 c W rg D quo for f pr of w s por o rr for fucos us fuf cougo coos 3 So fro w g g T s fuco u s qu o g Wrs fro 3 w g g w cosrg f pr of w s cos s Isrg 9 o s prsso fr g w c wr Fro s prsso w s w sou us ro procss o so quo for f pr of w r s w for w f s quo s rwr s prsso s r 95 ow f w us coos 95 6 gr quo 58 cos wr s f So souo of pro s: c 98 I s cs w g prsso s p of prous ro 33 ougo of Souos I orr o sur cou of souo of g pro o s cssr for prurs o coc coc po w f rg pr of w So fro ssupo 8 prssos w g:

34 33 c 99 oug ppro souo o pro s fou r r s wo coss f o ccu c c I s rqur I ccorc w s ss cos c c c c c W sog s w f coss g s r ro-p c c : c c ; c T ro gor c orgs s w: frs w us coos so u for s f coss c c fr w prfor frs ro f ccug fro 99 Kowg s w c us 9 o ccu frwrs w c proc w ro procss To g souo of D pro for s w us foru I w g D prur ss wc c ccu rc fro 8 for f fro 57 for s r fuco g for f pr of s s fou fro foru u for rg pr of w w us 59 3 Fu og W fg souo o rsfr w fu og prcp rs s s prous susco 3 3 urc Souo for F r pro s foru s w s susco 3 w cpo of our coo Rwrg our coo 78 s foows grg s r s [4] 47: s o o-r D our coo

35 34 I w g s ffrc sc s 9-9 u r coffc 89 cgs o 3 urc Souo for s s D quo for s s o o-r w cosr scr rzo of r procss for s s w us r ffrc sc So s f foowg gr pos: T Rg Pr of s Upo ppg our coo 79 o 57 w g g g 3 W c rwr s prsso s g 4 wr g g So D quo 6 for rg pr of s cos 5 w our coo for quo s 6 W scrz quo 5 s Usg s oos

36 35 F ffrc sc s qu o F 6 W pprog our coo 6 w us s pproc s susco 3 o g 7 w og 6 7 gs 8 wr F T Lf Pr of s frs w f D quo usg cou coo 3 fo 8: 9 So 58 cos For fg urc pproo o souo of pro 68 for f s of s quo s rpc or F D 3 wr coffcs r prss D F

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38 37 3 Trs Trsfr Ss w Dou W Dou Fs Ts cpr foows o rc fro pr I s w c spcs of rs couco for ou w w ou fs D gor T o-sor couco pro s w ssgg r p r our coos pr og coos o f s surfc osr rgg f ffrc os r pp o g pro o cosruc urc souo 3 c S of Pro Usg s oos ssupos s pr w r gog o scr s osor prur fs foowg pr ffr quos: wr c c c c o spcfc u s for s r w us s our coos r coc coos 5 s s pr s s coos sou pro s w: ppro Souo of Pro Jus w prous cpr w sps pproo for uppr r g prur o cos ppropr rcos Ts s o pro for sc r o wc s rprs quos 3 3 coos our coos 3

39 cougo coos Usg cosr rgg o for rcgur fs [6] [7] [] [3] [4] [5] [6] w r gog o ruc g D quos o D quos To urc pproo o souo of pro f ffrc o c us 3 Ruco of D Pro for F Frs w ppro prur f f f f f gr rg u or r T uow fucos f 3 : 3 ppropr coos Ts s w r fou s w s pr ppg f f 3 f f s 33 f s cos s cos s Susuo of 3 34 o 3 gs w s cos cos s cos s

40 39 w fuco s uow I orr o f frs w us fo 3 gr quo 3: ccorg o our coos prsso 35 s c wr s 36 w T our coo for 36 c so o sp ppg opror 3 o 35 36: 37 u 38 3 Ruco of D Pro for s For prur pproo w w us foowg rprso: g g g 39 rg u of 33 Fro fo 33 our coo 3 s poss o ccu s fucos: g g 33 g g 33 T foru 39 rucs o g 333 for coug w w o wo prs og gr quo 3 w rspc o : 334

41 4 3 T Rg Pr of s r w pr of s occups o I orr o copu g us pp our coo 37 o 333 o copu: g 335 wr T prsso 335 s pugg o rprso 333 so w o 336 T w ppropr us of our coos 37 3 foru 336 quo 334 for rg pr of w s for 337 w W grg 34 or our coo for 337 cos 338 u D coo u 339 s o gro of 37 3 T Lf Pr of s To f fucos g for prur f pr of s w us cougo coos Fro w r g 34 Fro w r 34 Usg 34 our coo 3 procg s w w ro of 66 quo 334 cos 34 w o coos o ssf:

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43 4 35 w coffcs Usg w oos cos 35 L s so 348 for : F 353 : F Susug 35 s rrrgg rs s 354 w F I sp w prsso 353 for rpc 354 So w g prsso for Dog s procur for cs 3 foows 355 w coffcs 35 F g us rcurrc ros o f coffcs us of r s o fro 355 T o pro r s w o ow W f u cours 33 Dffrc Sc for Rg Pr of s T ffr quo for rg pr of s 337 s ppro sc wr wc w rwr s

44 43 F usg foowg oos: F 356 W pprog our coo 338 w or w O crrg roug sr ss s for f ffrc sc for rg pr of s c prss s 357 w F for 333 Dffrc Sc for Lf Pr of s T sc for 34 s s w o g

45 I ps o us of T r c copu fro 355: 359 Susug 359 o 358 ffrc sc rsus F D 36 wr D F T our coo 343 s foowg f ffrc rprso: Ts c rwr s 3 usg 3 fr so susuos grc puos ffrc sc 36 for quo 34 cos

46 45 36 w coffcs D F for 334 ougo of Souos W s scs c o us oc us of r fou T cou o fro s rqurs: s prurs rg fus us coc coc po w f rg pr of w So fro foows r r If rprss pproo o po 36 cos I or wors 364 u for coo 363 w 365 ccorg o forus w rspc s s ru: 3

47 46 L us sr s o 365 rfr copu J 366 wr J I s s o fro 366 w 364: J J 367 Sog s quo for w g J J 368 Susuo of s o prsso 364 rsus J J 369 T g rsug us for quos ogr w pproos of coos g us for os frs u u cosu ppro souo of g ss D 33 urc Rsus r w us s prrs s pr T gorc prrs: 5 5

48 ropsc proprs: W K 4 c 7Jg 4 K W 5 3 K for sco for sco 33 g for sco o coos: 3 K u 5 T 8s u 35 \ T 3 Tprur sruo f w =75 s \ T 3 Tprur sruo f pr of s w =75 s \

49 \ T 33 Tprur sruo rg pr of s w =75 s Fro s s T T 3 T 3 T 33 w c s of procss r s uc ffrc w rs cs sor o cu fro urc rsus w go o css w c cocu rs prur s o sor o 34 ppro Souo of Pro w Pr og Jus sor cs for pr og w s our coo og : 37 u rs of foruo of pro rs ucg 34 urc Souo for F Oc g usg cosr rgg o pprog fuco s u or r w g foowg D pro: [ ] s pro s o-r for fg ppro souo o s pro w r gog o us ros W us s scr gr s f susco So quo 37 s ppro u 374 W procss - s co s w o us ro ur or Oous 374 c rwr s w F 48

50 49 F T our coo 37 s uc s ws o r cs T s 375 Tg s o ccou ppro scrpo of pro for f s g ss of quos 376 w F wr of cours us u Oc us fou s susco 343 pproos c ccu rcurs ppg foru urc Souo for s T pro for s pr og s s s suscos 3 3 w our coos wr r So foru o o cg W us sg ofco g 34 Dffrc Sc for Rg Pr of s L s cosruc scr pproo of ffr quo for rg pr of w: 377 for our coo 338:

51 5 378 I sur ffrc pproo o pro for rg pr of s s for 379 wr F for F u for 339: u 34 Dffrc Sc for Lf Pr of s T quo for f pr of w rucs o scr quo 38 for Wrs scr our coo s

52 5 38 T o scrzo of pro for f pr of s proucs 38 w 3 D F for r w us oo D F ougo of Souos Sc o r o ow w o o ow s o pp coos Ts rsu J J

53 5 wr foowg oos r us: J Tus pproo gor w us scr s sco proucs scr souo o g D couous pro

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55 op fu proucs o fu s oros pg w cos proucs or roc sc 4 Pro Foruo I gr procss s qu cop s r r gs sou o ccou oug o-r quo cou cosr w g w ss-rsfr w r gog o us r o Sco r s quso ou roc our of of r o pr orc rsus suggs s g [5] rsr s o for orcs w pgg g opg I cs of o so p w w r gog o us sourc fuco o scr s procss Tr ss fow pssgw s Fg 43 c ppro 3 r p our coo T wow fu c cosr s frs pproo s crc u op cos or pr op o To r c o w r gog o foowg ssupos: r of fu s s oug coprso o s g wg of sou; r of cr s rg oug so ffcs of scos c gc; c ws of pp r rg soo r sug so cosur s o ffc o w propgo ssug sou w fu s p w of sou ro c rprs wr p c p 4 p F c os cousc prssur fuco 4 F s so r sourc s of r L s suppos r s o pprc couco of rfor our of sou ws s c So s c u ouus of r wc s ppro g p r g 3 s of r cos prssur cos ou prssur 4 rfrs o ro of spcfc p 35 P s Fg 43 Dfo of gorc prrs for fu L s ss our coos w 3 r p our coo p p 4 T our coo fr of u ps upo wr s op cos or pr op For op of u o prssur us ppro qu o osprc prssur I or wors cousc prssur s zro: p 43 cos r s o of prssur s os of sou s rfc: p 44 p 54

56 I rsrc pprs wors wr w srus cosr s g [3] [33] [37] [47] o s wo ps of our coos r us u o sou r pr c so cos of u o fw I cs Ro coo p p us pp s [49] T u of ps o ow uc of s cos oos c rwr foowg for p p u coos r p p ooogous quo 4 c so usg o of spro of rs w oo for souo prss foowg o-cos for: wr p X T X r gfucos of ssoc oogous pro Our our coos p gfucos of pro r w X s X s u corrspog gus co co op; cos; To r fucos T cos r roos of s rsc quos: pr op w p sourc fuco coos Fourr srs So T w ssf u pro T c T F T T c souo of pro s p w r s fuco F F 49 55

57 cos c X X c X 4 Pc Frquc O of fu proprs of sou s s pc Pc s suc ru of sou r o frquc of sou w Icrsg frquc cuss rs pc u w crsg frquc pc of o ss T o of ows frquc s ow s fu or frs roc or poss pcs wos frqucs r gr ups of fu frquc r c uppr rocs or oros Usg ro 49 w o frqucs of ro o of wow pp: f c If prr rocs: c f f c 4 3 s suffc rg poss frqucs for fu r cos o 3 op 35 cos s s p o w c cg pc cgg g of u If pp s fgr os o c so cg ffc g of sru opg ffr os rg prssur of r ow o sru w cg u of s w up w souos rsug scr ffrcs pc T pc c so cg ofg our coo 43 Erg of Tos s sou of wow fu pg s r roc rg of sou s cocr cr frqucs of ro T or o cr pc grr cocro of rg frquc prso pg s prcur sru c g ffr pcs roug puo of supp r wc s cgg coos Foru for ccug rg sruo s E p p p 56 p 4 If coos r f s Sco 44 ou c of sruo of rg w fu s oros cgg 44 urc Rsus rsus sco wr o for foowg prr us: gorc prrs R o prrs p p 34 R c coos

58 wr s pos cos: sourc fuco F 44 Op- sc For g prrs fu for op fu s or 4 T sco roc s oc o fu; r roc s 5 so o Fg 44 os of op- sc O c oc s T 4 so of s frqucs ffr fro frqucs of os o po w-o qu pr -TET w 49 ff c 4 f s 44 z T frquc of -TET c fou fro quo f pop sou cp of cg ffrc f s s s z o Po Wow fu E T 4 Frqucs z of wow fu s op- sc copr w frqucs of -TET sc 44 os- sc osg rops fu oc ow pc of pp s op T poss o 4 s ppro r s frquc of fu 3; o E5 f s so o Ts s o o rocs r prs Fg 45 os of cos- sc 57

59 T 4 sows wow fu s pcs o o qu cofor o os prouc po o Po Wow fu E D F T 4 Frqucs z of wow fu s cos- sc copr w frqucs of -TET sc 443 Ir sc W c g furr ffcs w corg o o fw Ts ps w sou cg prr our coo 46 I s cs fu proucs r s of pcs f w os prouc w cos os w s op Fg 46 ppro oco of wow fu s pcs o po sc Pcs prouc w cos Pcs prouc w op Pcs prouc w pr op I Fg 48 T 43 ou c s ow prr ffcs oco of pcs p fu w g pr op Fg 47 Effc of prr o frqucs z of os sc 58

60 = =/ =/ =3/ =4/ =5/ =6/ =7/ =8/ =9/ = T 43 Effc of prr o frqucs z of os sc 444 Erg of Tos Fg 48 ppro oco of pcs w = Usg foru 4 w c f rgs of ffr rocs scor wc roc s o w cgg w of r or wc ow s r T ps g ow Fg 49 Fg 4 sow s r cosr ffc o rg of oros You us ow s sof s poss o prouc fu s s r o g ow rr ou g frs oro so o T rr ou ow gr rocs ou g o o Ts s prsc of g oros crss w crsg ε 59

61 Fg 49 Erg sruo J cross frqucs w cgg ε Op Fg 4 Erg sruo J cross frqucs w cgg ε os 6

62 ogrp [] -Ess O Dso F E Souo of Rcgur F w Rcgur Prforos WSES Trscos o ss Trsfr o o 6 pp [] roo I Koso J Pow Iusr-Sc Is ucg Procss for Too Proucs orczo ETE T-76-II F Rpor ugus 5 [3] roo Koso J Pow Is ucg of rurz S Prs ISE Trscos Issu 9 o or 5 pp [4] S o u us Sr Is S ucg os for Rcgur Sprc Sps Procgs of 9 ISE/WSES Iro ofrc o ET TRSFER TERL EIEERI EIROET TE Forc I WSES Prss pp [5] T os u us oprg Souos of proc Proc ouco Equos for L-sp Sps Procgs of 9 ISE/WSES Iro ofrc o ET TRSFER TERL EIEERI EIROET TE Forc I WSES Prss pp [6] T os u us Trsfr w Fu og Ss w Dou W Dou Fs Procgs of WSES Iro ofrc o Trsfr Tr Egrg Ero oug rc WSES Prss 3 pp [7] T os u us Trsfr w Pr og Ss w Dou W Dou Fs Procgs of WSES Iro ofrc o Trsfr Tr Egrg Ero oug rc WSES Prss 3 pp 3-37 [8] T os u us proc Equo s c o for S ucg of L-sp T-sp Sps Drc Irs Pros WSES TRSTIOS o ET SS TRSFER o5 o3 pp 63-7 [9] T os u us proc Equo s c o for S ucg of L-sp Sps Pr Irs Pro Procgs of 5 ISE/WSES Iro ofrc o OTIUU EIS rg UK WSES Prss pp -6 [] T os u us o Sor Trsfr Ss w Dou W Dou Fs Procgs of WSES Iro ofrc o D wors oucos opurs DOO 5 WSES I of o rs Scc TERILS S WSES Prss pp 6-65 [] T os us c o for Wow Fu Procgs of 9 ISE/WSES Iro ofrc o ET TRSFER TERL EIEERI EIROET TE Forc I WSES Prss pp 88-9 [] u Ec Souo of Fu 3-D S-S Pro for O E of Ss w Rcgur F w spcs of Trsfr Tr Egrg Ero Pr II WSES Prss 8 pp

63 [3] u Suo of S-S Procss for Rcgur F-og Ss c og ss o4 Tc 999 pp [4] u us ppro c Two-Dso Souo for Logu F of Rcgur Prof c Urss Lss o6 998 pp [5] u us ppro Souos of ouco Pros u- Dso r Tp Do osr rgg o Pr & Procgs of 5 ISE/WSES Iro ofrc o Trsfr Tr Egrg Ero oug s ugus pp 5 pp -6 [6] us u Ec 3-D Souo for Ss w Rcgur F Pr Procgs of WSES Iro ofrc o PPLIED TETIS T 7-II ro Egp WSES Prss 7 pp [7] u us Ec Trs Souo for Ss w Rcgur F Proc of 5 WSES Iro ofrc o ET D SS TRSFER T 8 cpuco co WSES Prss 8 pp 5-3 [8] u us proc Equo s c o for S ucg of L-sp Sps Pr Drc Pro Procgs of 3 WSES Iro ofrc o PPLIED TETIS T 8 Puro D L ruz r Iss Sp WSES Prss 8 pp 98-3 [9] us osr rgg s ppro o for souo of so rc rs rsfr pros c opuo os Trsfr IX WIT Prss 6 pp 3-3 [] us Two-Dso Souo for Trsfr Rgur F ss L Jour of Pscs Tcc Sccs o5 995 pp 5-58 [] us u os Two-Dso Souo for Trsfr Proc Ss w F Procgs of L c of Sccs Sco o5 o5 998 pp 8- [] us u So c 3-D s-s souos for sss w rcgur f ISE Trscos Issu 7 o Spr 5 pp -9 [3] us u T os Souo of proc quo ruco o gr quo for rg corr Procgs of Iro ofrc Dffr Equos Tr ppcos DET 9 Pzs Lu 9 KTU 9 pp 4-6 [4] us u S uso c Two-Dso Souos for Trsfr Ss w Rcgur F c opuo os Trsfr III WIT Prss 4 pp [5] us u Uo c-urc Souo for Trs Procss Ss w Rcgur Fs Proc of 5 WSES Iro ofrc o ET SS TRSFER cpuco co WSES Prss 8 pp 3-36 [6] us u Uo Trs souo for rsfr proc ss w fs АБуйкис МБуйке НУланова Нестационарные решения для теплопереноса в периодических системах с ребрами Procgs of Ur ogrss of cs Sco 8 pp5-3 I Russ [7] us S uso Irs I Fu Ro T Fg Pros for proc Equo Procgs of WSES Iro 6

64 ofrc o PPLIED TETIS Ds Ts US WSES Prss 6 pp-4 [8] us S uso Souo o Rrs proc Equo for Is rurz S ucg Procgs of Iro ofrc o opuo & Epr Egrg Sccs IES 5 I 5 pp [9] o Pr cur 8 og Trsfr Ec Usg o cro Tcoog Rc cs Fu ccs & ss Trsfr Forc I ugus 3-5 p 3 [3] Fcr T D Rossg T Pscs of usc Isrus Sco Eo Sprgr Scc+usss Ic [3] DD Josp L Przos ws Rws of or Pscs o6 o 989 pp 4-73 [3] DD Josp L Przos u o ppr: ws Rws of or Pscs o6 o 99 pp [33] R W K Josć T cs of usc Isrus rc c o o8 o4 pp fro Ir: p://wwwsuu/r/wo/rcpf s cc 8//3 [34] D Kr D Krus E Surfc Trsfr crw- oo op w Yor 97 [35] S K S S SJ S K K o oro of Suprropc/Suprropoc usg Sco owrs Ecross Ecg o Fuor ro ogs Lgur o7 o6 pp [36] S K S S SJ S K K o cro-o r srucurs w pu w usg wo-sp sco cg o rg r osc Rsrc Lrs SprgrOp Jour o6 ID:333 [37] L E Ksr R Fr opps J Srs Fus of couscs Four Eo Jo W& Sos Ic [38] I Koso Is S ucg os Tor Tcoog of ucg Sprgr-rg w-yor Y 99 [39] Koso P Kruos D Yurco I rc Fu Dss Eu o ss of FD og Eprs urg Is ucg Procgs of 5 ISE/WSES Iro ofrc o ET TRSFER TERL EIEERI D EIROET s rc WSES Prss 7 pp [4] Kor T Kor c oo for Scss Egrs Dfos Tors Forus for Rfrc Rw crw-ill OOK OPY I 96 [4] D Krus ss Euo of E Surfc Tr Sss spr Pusg orporo 98 [4] L R Kr c Tr-Dso Souo for S Tprur Procgs of IEE p [43] Y S Woo us ppro c Souo o Fr oug Trsfr Pro Iro Jour of Pur pp cs o o 4 pp

65 [44] zoor Lcur os Egrg Fow roug E Surfcs Ecgrs E r S Orszg Sprgr-rg r rg w Yor Too 984 [45] YS uzc Yooc JR u Tr Sprg Rssc opou Ororopc Sss Jour of Tropscs Trsfr o8 o 4 pp 45-5 [46] D Po oo of Lr Pr Dffr Equos for Egrs Scss p&/r Prss oc Ro [47] T D Rossg E Sprgr oo of couscs Sprgr Scc+usss LL w Yor 7 [48] Srs T Tor of Dffrc Scs Pur pp cs o4 w Yor rc Dr [49] L Soč our oos Effc o usc Isrus Tr Fcu of Pscs cs Urs of L cor s Tss L [5] P Soros og fu pps: susoc fow c oz pr sru copurs Dpr of Ecrc Egrg opur Scc IT PD Dssro Jur 995 fro Ir: p:///7/36534 s cc 8//3 [5] L Wg X Zou X W ouco Sprgr-rg r rg 8 [5] S Woo E Tupo I PJ ggs Prforc Icors for S-S Trsfr Troug F sss SE Jour of Trsfr o8 o 996 pp 3-36 [53] S Woo E Tupo I PJ ggs S-S Trsfr roug E P Surfcs Iro ouco ss Trsfr o o 995 pp 99-9 [54] Z Zg o/crosc Trsfrcrw- 7 [55] p://f-og-ogspoco/p/gus--f-og- s cc 8//3 [56] p://wwwfuo/rff/orgssor/srfr/o//og [57] p://wporg/w/wow_fu s cc 8//3 64