MATHEMATICAL MODELS AND THEIR SOLUTIONS FOR DOMAINS OF COMPLEX FORM



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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

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

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

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

33 Dffrc Sc for F 4 33 Dffrc Sc for Rg Pr of s 4 333 Dffrc Sc for Lf Pr of s 43 334 ougo of Souos 45 33 UERIL RESULTS 46 34 PPROXITE SOLUTIO OF TE PROLE WIT PRTIL OILI 48 34 urc Souo for F 48 34 urc Souo for s 49 34 Dffrc Sc for Rg Pr of s 49 34 Dffrc Sc for Lf Pr of s 5 343 ougo of Souos 5 4 TETIL ODEL FOR WILLOW FLUTE 53 4 PROLE FORULTIO 54 4 PIT D FREUEY 56 43 EERY OF TOES 56 44 UERIL RESULTS 56 44 Op- sc 57 44 os- sc 57 443 Ir sc 58 444 Erg of Tos 59 ILIORPY 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

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 48-53 IS: 978--684-83-8 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: 978--684-83-8 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 978--684-8- 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 978--684-6-8 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 384-389 IS 978--684-6-8 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 79-544 6

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 978-96-474-58-8 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 88-9 4 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 384-389 5 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

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

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

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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: 35 36 coos: 37 z w 38 T souo of w pro 34 38 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

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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 34-38 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

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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 45 47 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

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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: 3 3 3 4 3 5 ss quos foowg our coos r pos W fu :

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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 46 48 s c fou o wr c c s uow cos Trfor 46 47 48 49 5 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 54 55 ogr w 5

g g 56 Pug s wo prssos 55 56 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

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w c cos c c 3 s gro cos c 3 69 3 ougo of Souos W us fou souo o g pro u w r s f w fg uow coss forus 49 63 69 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 7 7 7 c ppro c souo o D pro s uqu r 5; 6 63; 57 ogr w 64 69 4 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: 448 4 K 7 4 W W K K for sco \ 3 4 5 98 98 98 98 979 978 9 999 998 998 998 997 996 8 34 34 34 34 33 3 7 88 88 88 88 87 86 6 6 6 6 6 59 58 5 5 5 5 5 49 48 4 358 358 358 358 357 356 3 485 485 484 484 483 483 69 69 69 69 68 67 79 79 79 79 79 79 974 974 973 973 97 97 T Tprur sruo f 8

\ 3 4 5 974 974 973 973 97 97-5 38 38 38 37 37 36-39 39 39 39 39 38-5 79 79 79 79 78 78-458 458 458 458 458 458-5 678 678 678 679 679 679-3 943 943 943 943 943 943-35 353 353 353 353 354 354-4 363 363 363 363 364 364-45 46 46 46 46 46 47-5 4496 4496 4496 4496 4496 4497 T Tprur sruo f pr of s \ 5 6 7 8 9 97 97 97 97 97 97-5 36 36 37 37 37 37-38 38 38 38 39 39-5 78 78 78 79 79 79-458 458 458 458 458 458-5 679 679 679 679 679 679-3 943 943 943 944 944 944-35 354 354 354 354 354 354-4 364 364 364 364 364 364-45 47 47 47 47 47 47-5 4497 4497 4497 4497 4497 4497 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 73 74 73 74 9

3 3 75 76 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 73-76 rsfor o 77 3 Pr og or 3 3 78 79 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

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 84 85 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 87 88 89 F 84 cosus s of r 9 9 9 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

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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 98 97 99 63 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 98 63 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 93 99 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

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

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

36 3 I s s o sow prsso ogr w gs s sc for f pr of s: 3 T prsso uss ros 3 D F ougo of Souos g scs 9 8 3 cos w c s p o us of Ts wo sou ccu for rg prurs o f cou coos s foowg qu 4 u s s usg pproo 5 I orr o u rs w us quos 3 8 w rspc Susug os 4 5 w g fro wc w f gg poss o ccu fro 4 Eu urc souo of g pro c o usg pc scs 9 8 3

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: 3 3 33 34 3 3 3 3 35 wr c c c c o spcfc u s for s r w us s our coos 6 9 3 r coc coos 5 s s pr s s coos sou pro s w: 36 37 38 39 3 3 3 3 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 36 37 our coos 3

38 3 33 34 35 36 37 cougo coos 38 39 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 33 36 3 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 34 35 s cos cos s cos s

39 w fuco s uow I orr o f frs w us fo 3 gr quo 3: ccorg o our coos 36 33 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

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 38 39 Fro 38 35 w r g 34 Fro 39 35 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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4 35 w coffcs Usg w oos 35 35 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

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

44 358 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

45 36 w coffcs D F for 334 ougo of Souos W s scs 355 357 36 c o us oc us of r fou T cou o fro s rqurs: 36 363 s prurs rg fus us coc coc po w f rg pr of w So fro 35 336 foows r r If rprss pproo o po 36 cos I or wors 364 u for coo 363 w 365 ccorg o forus 355 357 36 w rspc s s ru: 3

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ropsc proprs: 7 448 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 \ 3 4 5 98 98 98 979 979 978 9 998 998 998 997 997 996 8 34 34 34 33 33 3 7 88 88 88 87 87 86 6 6 6 6 59 59 58 5 5 5 5 49 49 48 4 358 358 358 357 357 356 3 484 484 484 484 483 48 69 69 69 68 68 67 79 79 79 79 79 79 973 973 973 973 97 97 T 3 Tprur sruo f w =75 s \ 3 4 5 973 973 973 973 97 97-5 37 37 37 37 37 36-39 39 39 38 38 38-5 78 78 78 78 78 78-457 457 457 457 457 457-5 678 678 678 678 678 678-3 94 94 94 94 943 943-35 353 353 353 353 353 354-4 36 36 363 363 363 364-45 45 45 45 46 46 47-5 4495 4495 4495 4496 4496 4496 T 3 Tprur sruo f pr of s w =75 s \ 5 6 7 8 9 97 97 97 97 97 97-5 36 36 36 36 36 36-38 38 38 38 38 38-5 78 78 78 78 78 78-457 458 458 458 458 458-5 678 679 679 679 679 679 47

\ 5 6 7 8 9-3 943 943 943 943 943 943-35 354 354 354 354 354 354-4 364 364 364 364 364 364-45 47 47 47 47 47 47-5 4496 4497 4497 4497 4497 4497 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: [ ] 37 37 373 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 33 344-346 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

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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

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 3 343 ougo of Souos Sc o r o ow w o o ow s o pp coos 36 363 Ts rsu J J

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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

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 43 45 c rwr foowg for p p u coos r p p 45 46 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 47 48 3 r gfucos of ssoc oogous pro Our our coos 45 46 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 47 48 Fourr srs So T w ssf u pro T c T F T T c souo of pro 4 4 46 48 s p w r s fuco F F 49 55

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 656 5 o prrs p p 34 R c coos

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 44 49 pop sou cp of cg ffrc f s s s z o Po Wow fu 4 663 663 5 535 537 5 78399 7849 6 465 4653 E6 385 386 6 56798 5698 6 86466 8343 7 93 936 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

T 4 sows wow fu s pcs o o qu cofor o os prouc po o Po Wow fu 3 38 38 4 39 3945 E5 6596 6548 5 9333 957 D6 7466 7735 F 6 47998 43898 6 66 76 6 97553 965 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

= 38 3945 6548 957 7735 43898 76 965 =/ 3457 39373 65485 967 7778 43933 79 965 =/ 3897 39533 6558 9696 783 43977 78 9683 =3/ 44 39735 6575 9784 79 4433 776 9634 =4/ 558 4 65868 99 799 449 74 96379 =5/ 5846 4365 6695 965 8 443 738 96456 =6/ 685 489 6643 939 83 4437 746 9657 =7/ 875 475 66974 979 866 4463 768 96763 =8/ 999 4349 67998 93484 944 4543 79 9744 =9/ 578 4657 748 9553 95 46593 7375 9849 = 663 537 7849 4653 386 5698 8343 936 T 43 Effc of prr o frqucs z of os sc 444 Erg of Tos Fg 48 ppro oco of pcs w = 5 7 9 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

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

ogrp [] -Ess O Dso F E Souo of Rcgur F w Rcgur Prforos WSES Trscos o ss Trsfr o o 6 pp 76-768 [] 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 84-845 [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 39-395 [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 384-389 [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 48-53 [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 4-46 6

[3] u Suo of S-S Procss for Rcgur F-og Ss c og ss o4 Tc 999 pp 33-43 [4] u us ppro c Two-Dso Souo for Logu F of Rcgur Prof c Urss Lss o6 998 pp 33-44 [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 5-7 7 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 64-69 [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 35-44 [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

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 74-746 [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 375-39 [33] R W K Josć T cs of usc Isrus rc c o o8 o4 pp 347-357 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 48-56 [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 95-98 [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 IEE4-66 4 7 p [43] Y S Woo us ppro c Souo o Fr oug Trsfr Pro Iro Jour of Pur pp cs o o 4 pp 89-5 63

[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