SOLUTION TO THE PROBLEM CONTROL OF A DISTRIBUTED PARAMETER PROCESS

Transcription

1 DAAA INTERNATIONAL SIENTII BOO pp HAPTER 5 SOLUTION TO THE PROBLE ONTROL O A DISTRIBUTED PARAETER PROESS JADLOVSA A.; ATALINI B.; HRUBINA.; AUROVA A. & WESSELY E. Absrac: The chaper eas wh he ssues of a appromao of he souo o he probem of opma coro of a srbue parameer process. A mahemaca moe of he probem s epresse b a wo-mesoa para fferea equao of hea rasfer wh bouar a a coos a a opma seece crero. I orer o oba a appromae souo o he efe probem he eas squares meho has bee appe hereb has bee prove ha he obae souo s he appromao of he ree s fuco eveua ha of a mpuse raso fuco. The possb of he appcao of he eas squares meho aorhm o he souo of he mahemaca moe of a srbue parameer process coro has aso bee cae. e wors: srbue parameer process eas squares meho he ree s fuco mpuse raso fuco eac souo appromao Auhors aa: Assoc. Prof. PhD. Jaovska A[a]* Uv. Prof. Dp.-I. Dr.h.c.mu. Dr.ech. aac B[rako]** Assoc. Prof. PhD. Hruba [am]***; Assoc. Prof. PhD. acurova A[a]****; Assoc. Prof. Sc. Wesse E[m]**** *Techca Uvers of osce Lea osce Sovaka** Uvers of Techoo arspaz 3 4 Vea Ausra ***Iformaech L. ošce Sovaka ****Uvers of Secur aaeme osce Sovaka aa.jaovska@uke.sk kaac@ma.f.uwe.ac.a kam.hruba@uke.sk aa.macurova@uke.sk em.wesse@vsbm.sk Ths Pubcao has o be referre as: Jaovska A[a]; aac B[rako]; Hruba; [am]; acurova A[a] & Wesse E[m]. Souo o he Probem oro of a Dsrbue Parameer Process haper 5 DAAA Ieraoa Scefc Book pp B. aac E. Pubshe b DAAA Ieraoa ISBN ISSN Vea Ausra DOI:.57/aaam.scbook..5 69

2 Jaovska A.; aac B.; Hruba.; acurova A. & Wesse E.: Souo o. Irouco The ssues of epor a usao of he rea wor processes a pheomea whch are of he space-me aure requre a more aequae escrpo of he sae varabes are whch are epresse as srbue parameers eveua fe vaues. The ssues of srbue parameer ssems coro were brouh o he aeo of a professoa pubc a he frs IA eraoa coferece 96 where Bema 967 a Pora 983 presee her frs resus. I he h of hese ear works srbue parameer ssems have bee efe as he ssems whose sae varabes are he ve srbue parameers or fe vaues. The ear pubcaos cue he eerase mehos of a amc proramm a a mamum prcpe. The probems souos are mos base o he appromao of he coroe ssems amcs b equaos of mahemaca phscs. The processes a ssems are escrbe mahemaca b para fferea equaos paraboc hperboc a epc as we as b era a fucoa oes. The souos aso make use of screzao mehos. Ths approach ea wh each specfc probem proves he resus whch are especa usefu for our oreao. The eveopme of umerca mehos a her aorhms for sov para fferea equaos was ae ma b he pubcaos Hrekoff 94 oura 943. Ow o a amc eveopme of formao echoo oa here are ma umerca mehos a aorhms appe o he souo of para fferea equaos a era equaos scopes of he efo of compe spaa 3D shapes. a works he heor of coro of srbue parameer ssems are base o he works b Los968 a Bukovskj975 as we as o he resus of mahemaca scpes: cacuus of varaos cassca a o-cassca fucoa aass a he heor of semroups: Bema967 Bukovskj98 Los968 Pora e a.983 Lasecka & Tra.The am of he arce s o prese he ese proceure appe o he souo of he probem of process opma coro of hea rassfer wh srbue parameers. To epress he mahemaca moe of a hea rasfer process b a b-mesoa para fferea equao wh bouar a a coos accor o he pe of he probem as we as b a crera fuco whch ca be epresse a fucoa form. To use he eas squares meho orer o sove he efe probem of opma coro of a srbue parameer process. To show ha he souo obae b he aorhm of he eas squares meho s basca he appromao of he ree's fuco or a mpuse raso fuco.. The probem formuao Le us coser he se of coroe srbue parameer processes whch ca be escrbe b ear o-homoeeous para fferea equaos wh varabe coeffces ak he form: a b A B f u v w 7

3 DAAA INTERNATIONAL SIENTII BOO pp HAPTER 5 where s a spaa varabe s me. I he equao u v a w are he coro fucos. The sae of he coroe ssem s characerse b he fuco. The coeffces a b A B a he fuco f are cosere o be kow. I a eera case he bouar coos for he equao f a we ca wre f u f u 3 where are he ve fucos of he varabe. or he uqueess of he souo o he equao wh he bouar coos 3 s ecessar o efe he a coos: 4 5 The fucos u u are he coro fucos for he bouar vaues. The coros u v w u u efe he vecor of coro varabes u u v w u u 6 whch ca be vesae as a eeme of some orme space. The vecor of coro varabes u ca be mpose aoa resrcos o e..: u 7 where s he ve umber. Le a a parcuar me erva f be ve he fe saes of he process f 8 f 9 If we coser he maera hea smuao echques he he mahemaca moe of he process we are escrb has o ake o accou hea rasfer b emsso whch s epresse b he foow oear bouar coo: 4 4 u 73 / S 73/ u S 4 4 u 73 / 73/ u Where s he hea coucv of he maera he coeffce of hea rasfer b emsso he coeffce of hea rasfer b covecu he meum emperaure he furace. The subscrps a eoe he upper or ower surface of he heae maera respecv. urhermore for purpuses of smuao for mmzao of osses ue o oao ur hea we ma efe a era crero as foows: J p f R S H R f S H ; R f S < H P s he surfasce overbur of he mea S S he surface emperaure of he mea H he m emperaure for scae formao S he aressv coeffce of combuso prouc H a S are cosere as ve for he purposes of smuao. I opmum coro probems for ssems wh srbue parameers 7

4 Jaovska A.; aac B.; Hruba.; acurova A. & Wesse E.: Souo o ma appear rec or rec he era of he fucoa o be mmze hrouh he bouar coos Hruba 99. Probem I s ecessar o f such a vecor of coro varabes u so ha he coos 8 a 9 are sasfe he owes possbe me erva f. The vecor of coro varabes ca be mpose o he resrco of he pe 7. Probem. Le be ve he me erva f. I s ecessar o f a vecor of coro varabes u for whch he coos 8 a 9 are sasf. Thereb here ca be pace a coo so ha he saar of he vecor of coro varabes u obae he mmum vaue. Probem 3. I hs case he probem s o propose a aorhm for sov he mahemaca moe epresse b he ve a a bouar coos accor o he probem pe whch ma be cosere e.. as ve b a he seece opma crero he era form.the resu of hesouo a he coro fuco u he meum emperaure se he furace as we as he me epeeces of he emperaure of surfaces a ceer of he heae maerahruba &Jaovská Hruba The heoreca souo o he efe probems a presee he pubcao b Bukovskj 975 a he aorhm for sov he probem. he work b Hruba 99. Hruba 7 a s possbe o show he meho of equaos rasformao o a suabe form for he appcao of he eas squares meho. The possb of such a rasformao sems from he fac a ve me erva w be he me-epee coro fuco u ha he fuco whch sasfes he equaos o 4 ca be epresse he form of eras whch cue he fucos f f a : N N f f u v w f f u f u oser he eera heor he eres fac s ha a he cores.e. he fucos N N ca be epresse he fe form b a se fuco whch s cae eher a mpuse raso fuco or he effec fuco a fuamea souo o he ssem o 5 or he ree s fuco Bukovskj 98; Huko Appromao of he souo o a para fferea equao b he eas squares meho The eas squares meho cosss search for a vecor of ukow parameers so ha he wehe sum of squares of he resuas s mma. The eas squares meho ca be formuae for boh a couous meho a a scree oe 7

5 DAAA INTERNATIONAL SIENTII BOO pp HAPTER 5 accor o he wehe sum of he resues he ve oma Hruba & ajercak. The cassfcao of he eas squares meho ca aso be carre ou accor o he seeco of he form of a para fferea equao appromae souo. B Easo 976 we sush he wo approaches: oba approach where we search for he appromae souo he shape of he sum of he seres Loca approach or he approach of he fe eemes meho Le be he boue area of he space R of he varabes wh smooh o-ercepe borer. We vesae f f he ask of ef of he appromae souo of he equao vl ra 3 where a ; ; 4 L a L s he emperaure posve me a he po. Le he souo 3 sasf o he borer he bouar coo P P 5 a he a coo for = 6 where he ve fuco P S S f a for = s P a. I hs paper we w prese some supposos specfcao of whch requres rouc of he coceps from he fe of absrac fferea equaos. Le us eoe b L a Hber space of rea fucos erabe wh he quarae he area where he scaar prouc s efe b he equa Wh hs space e us choose he se of fucos V he eemes of whch are wce couous ffereabe fucos. Le us search for he appromae souo he foow form v f Where bass fucos v V represe he frs fucos of he whoe ssem of fucos v a sasf he bouar coo. Le us eoe: E vl ra 7 8 We w eoe ukow fucos f so ha he fuco mmses he fucoas wh weh fucos Leras 97. E I E a I 73

6 Jaovska A.; aac B.; Hruba.; acurova A. & Wesse E.: Souo o If we eerme he souo of he homoeeous ask sasf he equaos 3 5 a 6 where P a he souo of he o-homoeeous ask sasf he equaos 3 5 a 6 where he b her superposo we w oba he souo o he ve probem. 3. Souo homoeeous probem Le v f v f 9 where v s e a f s coum mar. oorae fucos v are chose so ha v P ; P. Le f be ear combao f f k k k f Le us eoe he mar A α k ; k supposo 9 ca be wre ow he form of he mar equao f A f B mmsao of he fucoa I E a I we efe he eemes of he mar a he mar f.e. a vaues of he mar f. These wo resus efe A ukow f because from foow f e f. If we kow f he ca be epresse he foow form A v e f 3 If we subsue 7 for 9 a use he supposo we w oba: E v L ra v v A f ε f. 4 ε s e mar he -h eeme of whch has he foow form L rav ε v vk α If E f f s eouh o m I w oba he upper appromao of he mmum k k I E. mum. or each we I w be I efe b + of he equao k or compee b k equaos 5 from whch we ca cacuae of he ukow eemes of he mar A. vv vv vv v vl ra v 5 where for he sake of brefess we w use he eoao v v j vv j 74

7 DAAA INTERNATIONAL SIENTII BOO pp HAPTER 5 L ra v v vl ra v v v If we efe he marces B a D so ha j B v v j j 6 D v v L ra v j 7 j The he equaos of he ssem 5 ca be wre ow he form of he mar equao BA D whea B D. We mmse he fucoa I where v f. rom he I coos for he mmum I we w oba + f equaos for cacuao of he eemes of he mar f. v v f v v f v v f v 8 Us he mar B efe 6 he ssem 8 ca be epresse he foow formbf v where he mar v s raspose o he e mar v hus f B v 9 If we wre ow 3 us 9 a afer be epresse b he era we w oba he souo of he homoeeous bouar ask 3. Souo o-homoeeous probem A ' v e B v' ' ' '. Le us search for he foow form Where ω P P for ω v f 3 P. Le f α f k k k j O H or he mar form f Af H 3 where A s he mar efe H s he ukow coum mar. B mmsao of he fucoa I E a I we w efe he marces A H a f hus aso he ukow mar f f from 3 A A τ f e f e H ττ 3 If we subsue 7 for 3 a 3 we w oba: E vl ra v v A f v H where 75

8 Jaovska A.; aac B.; Hruba.; acurova A. & Wesse E.: Souo o vl ra msao of he fucoa I E w be performe wo seps v L ra v v A f 33 Epresso 33 was cue v I E for he homoeeous bouar ask a b s mmsao he mar A B D w be efe. a Le us mmse he fucoa I v H wh he same weh fuco as wh 33. rom coos for mmsao of I H I we w oba he ssem of he foow equaos: whece v v H v v H... v v H v H B v' 34 hus 6 s Le us rem ha he o-homoeeous ask reuce o v f I a smar wa o homoeeous ask from he coos for he we ca cacuae mmum f B v' 35 Afer ef H from 34 f from 35 f he reaoshp 3 ca be wre ow he foow form f e e A A B B ' ' v' ' ' v' ' ' Souo of he o-homoeeous bouar ask base o he foows: A v e B v' ' ' ' v e A B v' ' ' ' w be as eera souo of he probem w be obae b he superposo of souo of he homoeeous a he o-homoeeous asks hus 76

9 DAAA INTERNATIONAL SIENTII BOO pp HAPTER 5 ' v e v e ' v ' e A A B B A B v' ' v' ' ' ' v' ' ' ' ' If we eoe A ' ' v e B v' ' 38 ' ' ' ' ' ' ' ' ' ' 3.3 The ree s fuco I he foow par we are o o vesae he properes of he core a compare hem wh he properes of he heoreca core epresse b he ree s fuco. I he reaoshp 38 here occur he marces A a B - properes of whch we are o o vesae. uco w be cae he core. Accor o 6 s obvous ha he mar B s smmerc. If he ree s frs heorem s appe o he era epress he eeme j of he mar D v vl ra v j s obvous ha he mar D s aso smmerc. ar B s posve efe. Le Q Y BY' be he quarac form cojuae o he mar B Y s he o-zero e mar wh eemes. If b j ' v ' v ' ' j we ca wre ow Q Y ' u ' u ' Y ' ' j If we perform he mar prouc we w oba j Q ' Y vk ' k ' ha s wh for a o-zero vecors Y s Q posve efe hus he mar B s aso posve efe. ar D s eave efe. Le us coser he quarac form P X D X' cojuae o he mar D. I a smar wa o he foremeoe heorem we w come o 77

10 Jaovska A.; aac B.; Hruba.; acurova A. & Wesse E.: Souo o P L ' X k ra vk ' ' L ' k Thus he quarac form s eave efe for a o-zero vecors X coseque D s aso eave efe. Eevaues of he mar A = B - D are rea a eave. Le be he eevaue of he mar A a X he correspo eevecor he he foow s va: AX X Le us coser he wo possbes of he scaar prouc epresso BAX X BAX X BX X 4 BAX X X A* B* X 4 where A* B* are he marces cojuae o A a B. If B s rea a smmerc B* = B mar A s rea hus A* A' where A s mar raspose o A. rom he epresso 4 we w oba: BAX X X A' AX X BAX X BX X BX 4 rom 4 a 4 foows ha f X ess hus he eevaues are rea. Le us coser he scaar prouc BAX X BX X. If BA = D BAX X DX X whece DX X BX X BX X DX X. The core ' mmses he fucoa E I E. The core ' sasfes he homoeeous bouar coos. or he Drche coo v P for P P '.. The era ' s ermeae quarac appromao of he era ' ' where ' s he Drac ea fuco Bukovskj Appromao of he fuco a Drac s fuco Le be he fuco whch we e he area o subsue for he prouc av av av he co-orae fucos v are ve he ukow cosas whch we efe so ha he fucoa I a a v a v a v s mma. rom coos for he mmum I a I we w oba B a' ' ' v' ' '.Where he mar B s efe 6 a I ' v B a s he e mar. The ' ' v B ' v' ' ' ' v a' v' ' ' ' 78

11 DAAA INTERNATIONAL SIENTII BOO pp HAPTER 5 3. If we coser he Drac ea fuco ' he era I ' ' '. We ca see ha he era I s ermeae quarac appromao of he era I a hus ' s he appromao of '. rom he heorems a 7. foows ha ' s he appromae souo of he ssem ' v L ra ' P ' ; P 43 ' ' he eac souo of whch ca be epresse b he ree s fuco. ' ' ' 44 The smmerc of he core rear he pos ' ca be prove f we A show ha he prouc e B s smmerc as ' ' v e A B v' ' A ' ' ' v ' e B v' The proucs B DB a DB D are smmerc. If we uw he mar e A we w oba e I A A! A. If we subsue he mar A for he mar B D a mup e A s B o he rh we w oba e A B B B D B B D B D B from he above foows ha A e B s! smmerc mar Jaovska e a Appromao of he ree s fuco b he eas squares meho I he foow par of he paper we are o o show ha souo of he equao 3 efe b he eas square meho s he appromao of he eac heoreca souo epresse b he ree s fuco. Souo 3 uer he coos 4 5 a 6 f ess s he o souo. I ca be eas show ha f ' s he ree s fuco whch s cosere o be he souo o he ssem v L ra ' ' P ' ; P ' ' ; ' he he heoreca souo 3 uer he coos 4 5 a 6 s of he foow form * L P P ' ' P P s ' ' 45 79

12 Jaovska A.; aac B.; Hruba.; acurova A. & Wesse E.: Souo o where s he ervao he reco of he ouer orma o a he po P. p Theoreca souo of he homoeeous bouar ask for whch P w be obae from 45 * ' ' ' '.Le ' ' '. We ca eas f ou ha s aso he souo of he homoeeous bouar ask. The souo s uambuous hus we mus ' ' ' ' ' ' ' 46 Ie 46 s correc for he raom fuco hus ' ' ' 47 If specfc hea s cosa he he heoreca core ' s smmerc rear pos '. Us 47 we ca smp wre ow he heoreca souo 45 * ' ' ' L P P P P s P 3.6 omparso of eac a appromae souos omparso of he eac a appromae souo 3 for he Drche bouar ask wh oe-mesoa space. The souo of he ssem L ; f f 48 ; 49 ; efe b he eas square meho 39 s as foows: Le 8

13 DAAA INTERNATIONAL SIENTII BOO pp HAPTER 5 5 Seco era 5 epresse epc afer erao of he varabe b per pars meho w be as foows: L Thus L 5 Theoreca souo for he Drche bouar ask wh oe-mesoa space from he reaoshp 48 s as foows: * L 5 where s he heoreca souo of he ssem 53 L 53 8

14 Jaovska A.; aac B.; Hruba.; acurova A. & Wesse E.: Souo o If we mup he frs equao of he ssem 53 raua wh / / a erae wh he borers from o he for he epresso bewee he braces 5 we w oba: L L 54 Us he reaoshp 54 a ak o coserao oreao of he ormas he heoreca souo 5 ca be wre he foow form L 55 Theoreca souo 49 epresse b he reaoshp 55 has comparabe members wh he souo efe b he eas square meho 5 where here are wo era members I he 7h heorem we have show ha he era s ermeae quarac appromao ω. Thus he core ca be cosere o be appromao of he ree s fuco. If we choose c b a L a he co-orae fucos v he he aorhm of he souo e c v B v A of he homoeeous ask s base o he foow saes:. acuao of he marces B a B Eemes b j of he mar B are cacuae accor o he formua u u c b j j ; N N j N s he umber of he co-orae fucos. The verse mar B smp a eac cacuae s o compee smmerc bu f we perform cacuaos wce eac he verse mar B w be aso smmerc. 8

15 DAAA INTERNATIONAL SIENTII BOO pp HAPTER 5. acuao of he marces D a A B D Eemes j of he mar D are cacuae accor o he formua 3.We cacuae he era v 4.We eerme he eevaues a eevecors of he mar A. 5.We seec a cacuae he mar e A. 6.We mup e A b he mar B. 7.The mar e A B s mupe b he mar v '. 8.We seec a he fuco hus we oba: A v e B v' I s obvous ha for he me a he cosa umber of he co-orae fucos N we ca eerme he souo for he varous vaues of. If we chae he varabe whou cha he umber of he co-orae fucos cacuao mus be sare from he 5h sae. If for a N whch are cosa we chose he a coos s eouh o perform cacuao a he saes 3 7 a 8 keep oher resus. I he above escrbe aorhm he cacuao of he mar e A s he mos mpora ask. Smar we ca wre ou he aorhm of he souo of he ohomoeeous ask where cacuao of he era s he mos ffcu. e A bu rear he fuco we have o app appromae mehos. 3.7 Appromao of he souo o a fferea ssem wh cosa coeffces or he requremes of he process coro es he para fferea equaos 3 o 6 ca be rasforme b he meho of es.e hrouh he varabes screzao o he ssem of fferea equao ak he form: A f 57 where f f f A s a square mar of he -h erees whose eemes are epee of me. The fuco f are he ve fucos chose vaues are kow for ffere vaues.urhermore we assume ha he a vaues of he fucos are kow 83

16 Jaovska A.; aac B.; Hruba.; acurova A. & Wesse E.: Souo o. The am s o eerme he souo o he ssem of fferea equaos 57 b a umerca meho. We kow ha he eac souo o he probem formuae wh he a vaues s ef a mar represeao: A e e A f 58 The eac souo o he probem 57 ca be use he sep b sep meho whch aows he cacuao of he fucos me whereas us he fucos ; vaues; s he sep. Thus he eac souo w be he form: A As e e f ss or he use of he reao 59 he ework oma s ecessar o eveop A umerca mehos aorhms ha aow he cacuao of he mar meho e as we as ha of he marces era: As e f ss To oba he appromao of he efe era 6 wh a suffce accurae wh he appcao of a umerca meho we ca use a aorhm whose A s cacuao cues he eac vaues of he mar e a we erpoae he mar f s wh he s-eree pooma. I orer o mpeme he erpoao a of he umerca mehos ca be chose for eampe he oes Leras 978; Hruba a Jaovska ; Hruba Appcao of he eas squares meho o he sov of he probem of srbue parameer process coro Le us coser he mahemaca moe of he hea rasfer process hea of soae mea ro - a srbue parameer ssem o he erva L escrbe b he pe of a para fferea equao: Bouar coos: Y a Y U Y L Y Ia coo Y Y L a Saarze fuco: The ree s fuco: 84 U Y a a L R 59 6

17 DAAA INTERNATIONAL SIENTII BOO pp HAPTER 5 Trasfer fuco: L a s s ep L L L s s S s L L L a s L where s s he varabe of he Lapace rasformao. The souo o he para fferea equao wh bouar a a coos eas o he epresso of he coroe process oupu varabe whch s he covouo prouc of he saarze fuco a he ree's fuco: Y L R The ree's fuco coas compoes of he hea process amcs epe o me "" a he spaa varabe "". These compoes are he era fucos of a para fferea equao escrb he coroe process. 5.ocuso The corbuo of hs chaper es he preseao of he souo o he probem of he hea rasfer process coro whose mahemaca moe s epresse b he b-mesoa para fferea equao wh bouar a a coos. To eerme he appromae souo o a homoeeous a ohomoeeous probem of coro he eas squares meho has bee use. I has bee prove ha he souo of he mahemaca moe.e. ha of a para fferea equao of a paraboc pe eerme b eas squares meho s he appromao of a eac heoreca souo whch s epresse b he ree s fuco. The comparso of he eac a appromae souos of he mahemaca moe for he Drche mara probem s show oe-mesoa space. The paper preses he aorhm for sov a homoeeous probem as we as he possb of es he aorhm for sov a o-homoeous probem. 6. Ackoweeme Ths work has bee suppore b he Scefc ra Aec of Sovak Repubc uer projec Vea No./86/ Damc Hbr Archecure of he uae Nework oro Ssem. 7. Refereces Bema R Irouco o he ahemaca Theor of oro Processes I. Acaemc Press Bukovskj A ehos of oro Ssems wh srbue parameers Nauka oskva 85

18 Jaovska A.; aac B.; Hruba.; acurova A. & Wesse E.: Souo o Bukovskj A ree s fucos a rasfer fucos Habook. Es Horwoo Lme New York oura R Varaoa ehos for Souo of Equbrum a Vbrao. Bu.Amer.ah.Soc. Voume 49 Number p. -3 Easo D Noear eas square-he Leveber aorhm revse. J.Aus.ah.Soc.B. 9 p Hrekoff A. 94. Souo of Probems Easc b he rame-work ehos ASE J.App.ech.8 Hruba. 99. Sov Opma oro Probems for Ssems wh Dsrbue Parameers b eas of Ierave Aorhms of Aebrac ehos. berecs a Iformacs. Sovak Acaem of Sceces Brasava Hruba. a Jaovská A.. Opma oro a Appromao of Varaoa Iequaes. berees The Ieraoa Joura of Ssems a berecs. Vo 9/ Emera Ea pp ISSN 49X.368 Hruba. 7. Numerca opma coro probems for ssems wh srbue parameers b aorhms of aebrac mehos. aufacur Eeeer. No p ISSN Presov Huko. Aoova. Beav. Beask J. Szua J. & Véh P.998. oe oro a Des of Dsrbue Parameer Ssems Pubsh house STU Brasava ISBN Jaovska A. & Hruba.. Aorhms of Opma oro ehos for Sov ame Theor Probems. berees The Ieraoa Joura of berecs Ssems a aaeme Sceces. Vo..4 No. ½ pp Emera roup Pubsh Lme ISSN X Jaovska A. aac B. Hruba. acurovaa. & Wesse E.. Opma oro of Noear Ssems wh osras. DAAA Ieraoa Scefc Book aac B.E Vea Ausra pp.65-8 ISBN ISSN LaseckaI. & Tra R.. oro Theor for Parca Dfferea Equaos. ouous a Appromao Theores ambre Uvers Press ISBN Leras J. 97. ehoes e echques e aase umerque Duo Pars Los J. L oroe opma e ssemes ouveres par es equaos au erves parees Duo Quaher Vars Pars PoraL.. Boaskj V.. amkreze P.V. & seko E ahemaca Theor of Opma Processes Nauka oskva Trpah S.. 8. oer oro Ssemes A Irouco If Scece Pres LL Hham assachuses New Deh ISBN