Mathematical and numerical modeling of inverse heat conduction problem

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1 Mahemacal ad umercal modelg o verse hea coduco problem Sera DANAILA Ala-Ioaa CHIRA Correspodg auhor Faculy o Aerospace Egeerg POLITEHNICA Uversy o Buchares Polzu o.-6 RO-6 Buchares Romaa sera.daala@upb.ro INCAS Naoal Isue or Aerospace Research Ele Caraol B-dul Iulu Mau Buchares 66 Romaa chra.ala@cas.ro DOI:.3/ Absrac: The prese paper reers o he assessme o hree umercal mehods or solvg he verse hea coduco problem: he Alaov s erave regularzao mehod he Tkhoov local regularzao mehod ad he Tkhoov equao regularzao mehod respecvely. For all mehods we developed umercal algorhms or recosruco o he useady boudary codo mposg some resrcos or he useady emperaure eld he eror pos. Numercal ess allow evaluag he accuracy o he cosdered mehods. Key Words: drec problem verse problem Alaov s regularzao Tkhoov regularzao cojugae grade hea coduco.. INTRODUCTION A boudary value problem or a paral dereal equao s characerzed by mposg a vald equao wh a gve doma ad he al ad boudary codos o he boudary o he doma. Ial ad boudary codos are ormulaed o dey a gve soluo rom he mulude o possble soluos o he equao. I hese crcumsaces here s he cocep o well posed problem (Hadamard []). A well-posed problem acually provdes he sably o he soluo wh respec o small perurbaos o he daa makg he soluo o be vald eve case o small uceraes o he problem daa. Formulag a well-posed problem depeds o he ype o he problem (ellpc parabolc or hyperbolc). Cosequely he soluo ad he problem are cosdered as belogg o a ully deed ucoal space. Cosequely a gve problem ca be well-posed a parcular ucoal space bu may be ll-posed aoher ucoal space. I a codoal well-posed problem he Tkhoov sese we wll have o deerme a soluo ha belogs o a cera class o soluos. Resrcg he class o admssble soluos or a gve problem s possble o ormulae a well-posed problem eve appears o be a ll-posed problem he classcal Hadamard sese. From he physcal po o vew verse problems are characerzed prmarly by a lack o ormao eeded o solve as a drec problem. Ths lack o ormao has o be compesaed; cosequely solvg verse problems addoal ormao mus be INCAS BULLETIN Volume 6 Issue 4/ 4 pp ISSN 66 8

2 Sera DANAILA Ala-Ioaa CHIRA 4 ormulaed so ha we ca deerme a uque soluo. Iverse problems ca be classed respec o hs addoal ecessary ormao. I drec hermal coduco problem he dsrbuo o he emperaure sde a body s deermed respec o gve al ad boudary codos mposed o he body surace. However praccal applcaos he boudary codos are ukow alog he body surace (or o some poro o he surace) bu we ca measure he emperaures a umber o eral pos. Iverse coduco problem cosss o deermg he hea lux ad emperaure dsrbuo sde a body he hsory o chage body emperaure measured oe or more locaos wh he body s gve. Noe ha because he verse problem exrapolaesˮ he measured values wh he body o s surace eve small uceraes o measuremes ca be ampled leadg o mpora oscllag values o he surace. Brely drec problem he causes are kow ad he eecs are calculaed whle he verse problem a model o recosruc a pu rom he correspodg oupu should be used []. I he exsece o a soluo or a verse hea raser problem may be physcally argued he uqueess o he soluo o verse problems ca be mahemacally proved oly or some specal cases [3 4]. Moreover verse problems are exremely sesve o he al dsrbuos o he daa. So sesve are hese problems ha eve mue errors he daa ca wldly aec he compued soluo [5 6]. Geerally a ll-posed verse problem s solved as a approxmae well-posed problem supposg he soluo esmao he leas squares sese. Currely several echques o solvg he verse problems are proposed leraure []). The prese paper preses he soluo o he verse oe-dmesoal coduco problem o esmag he rgh sde useady boudary codo by usg wo echques: cojugae grade mehod wh adjo problem or grade uco esmao ad Tkhoov regularzao or hyperbolzao o he hea coduco equao respecvely. We exame he accuracy o boh approaches by usg rase smulaed measuremes o several sesors locaed sde he doma. The verse problem s solved or dere ucoal orms o he ukow boudary codo cludg hose coag sharp corers ad dscoues whch are he mos dcul o be recovered by a verse aalyss. For boh mehods he mahemacal ad umercal ormulaos are preseed ad ally he umercal resuls are comparavely dscussed.. INVERSE PROBLEM FORMULATIONS The physcal problem cosdered here s he oe-dmesoal hea coduco a sold wh cosa proprees. The mahemacal ormulao o hs drec problem dmesoless orm s gve by: T T x l. () I order o ormulae a well-posed problem or he parabolc equao () he ollowg boudary ad al codos are aached: T( l ) ( ) () T ( ) (3) INCAS BULLETIN Volume 6 Issue 4/ 4

3 5 Mahemacal ad umercal modelg o verse hea coduco problem T( x) x l. (4) I he verse problem he uco () represeg he emperaure o he rgh boudary have o be calculaed order o oba he measured values he po T( x ) T ( ) x x : x l. (5) Such measuremes may coa radom errors bu all he oher quaes appearg problem are cosdered o be kow wh suce degree o accuracy. We assume ha o ormao s avalable regardg he ucoal orm o he ukow () excep ha belogs o he space o square egrable ucos he doma. 3. CONJUGATE GRADIENT METHOD The rs preseed mehod s called Alaov s erave regularzao mehod ha s a e dmesoal aalog o he Cojugae Grade mehod [ ]. I he mplemeao o Alaov s mehod oe has o solve he drec problem he sesvy problem he adjo problem ad o use a opmzao algorhm. To avod he coamao o he esmaed uco by measureme errors oe ca use a soppg crero or he erao procedure ha akes o accou he level o ose expermeal daa. We assume ha he ukow uco () equao () belogs o he Hlber space o square-egrable ucos he me doma: ( )d. (6) The soluo o he verse problem s he uco or whch he ucoal ( ) ( ( ; ) ( )) d I T x T (7) s mmzed uder he cosra mposed by he drec problem. I erave grade ( mehods he mmzg sequece k ) k K ca be cosruced by he rule: where ( k) ( ) ( ) d ( ) (8) d s he desced dreco ad s he desced parameer. ( k The desced dreco ) ( ) d () depeds o grade o he ucoal (7) I ( k ): ( k) ( k) ( k) d I d ( ) ( ) ( ). (9) Cosderg he ukow uco he space o square egrable ucos he varao o he ucoal (7) I ( ) ca be expressed as: I( ) Id () INCAS BULLETIN Volume 6 Issue 4/ 4

4 Sera DANAILA Ala-Ioaa CHIRA 6 ( ) where () s a small varao o. To evaluae he grade I ( k ) he chose mehod amely he adjo problem wll be preseed laer he paper. (k ) The cojugao coece s esmaed usg he Flecher-Reeves mehod []: The sep sze (7) wh respec o I I ( ) d ( k ) ( ) d () k.... () (k ) s deermed by mmzg he ucoal I( ) gve by equao (k ) ha s ( k) ( k) m ( ) m ( ; ) d ( k) ( k) ( I T x T m T ( x ; d ) T d k ). () The Taylor seres expaso equao or ( k) ( k) d () s: Deog: T ( x ( y) d ( )) cosderg T T( x ; ( ) d ( )) T( x ; ( )) he equao (3) becomes: T T ( x ; ) ( k) ( k). (3) (4) T( x ; ( ) d ( )) T( x ; ( )) T ( ). (5) Replacg () resuls: ( k) m I( ) m T( x ; ) T ( ) T d= ( k) ( k) m T ( x ; ) T T T ( T( x ; ) T ) d. ( ) For small varaos T k dereae wh respec o resulg: (6) s egleced ad o mmze equao (6) we (k ) ad se he resulg expresso equal o zero: ( k ) I( ) T T ( T( x ; ) T ) d= (7) T ( d )( T ( x ; ) T )d T ( k) ( k) ( d ) d (8) INCAS BULLETIN Volume 6 Issue 4/ 4

5 7 Mahemacal ad umercal modelg o verse hea coduco problem ( ) ( ) where ( k k T d ) T ( ) d s he small varao o emperaure dsrbuo T produced by small varao o ukow boudary codo () d. The uco T( ) s esmaed as he soluo o he sesvy problem. I cocluso he cojugae grade mehod supposes he soluo o wo auxlary problems: he sesvy problem ad he adjo problem respecvely. 3. Sesvy problem I sesvy aalyss s assumed ha whe () udergoes a creme () he emperaure T( x ) chages by a amou T( x ). Thereore we replace T( x ) by T( x ) T( x ) ad () by ( ) ( ) he drec problem () ad subrac rom he orgal problem () order o oba he sesvy problem: T T x l (9) wh he ollowg boudary ad al codos: T ( l ) ( ) () T ( ) () T( x) x l. () The problem (9)-() s well-posed he creme () s gve. 3. Adjo problem ad grade equao To oba he adjo problem we mulply equao () by Lagrage mulpler ( x ) ad egrae he resulg expresso over he spaal doma rom x = o x = ad he over he me doma rom o. The expresso obaed hs maer s added o he ucoal (7): l T T ( ) ( ( ; ) ( )) d ( ) d d. (3) I T x T x x Perurbg () by () ad T( x ) by T( x ) resuls: where l ( ) ( ( ; ) ( )) ( )d d I T x T x x x l l T T ( T ) ( T ) ( x ) dxd ( x ) dxd x x ( x x ) s he Drac uco. The rs erm (4) reads: l l ( ( ; ) ( )) ( )d d T x T x x x ( T( x ) T ) T ( T( x ) T ) T ( x x )ddx. (4) (5) INCAS BULLETIN Volume 6 Issue 4/ 4

6 Sera DANAILA Ala-Ioaa CHIRA 8 Replacg(5) (4) ad subracg rom (3) resuls: l l ( T) ( T) ( ) ( ( ) ) ( )d d ( ) d d (6) I T T x T x x x x x where he hgh order erms were egleced. The secod erm (6) ca be egraed by pars he mposg codos ()-() he equao (6) becomes: where l ( ) ( ( ) ) ( )d d I T T x T x x x l l d d T x T d ( T ) d T d x xl x xl x x. The ollowg adjo problem ca be ormulaed: The alˮ codo: ( T( x ) T ) ( x x ) ( xx ) (7) x l (8) or x x. (9) or x x or (3) ad he boudary codos: ( l ) ( ) (3) wll complee he adjo problem ormulao. Noe ha he adjo problem (8)-(3) s well-posed bu he egrao me s vered. Takg o accou he adjo problem rom equao (7) he ucoal varao I ( ) resuls: I( ) d. x l Cosderg () leas square egrable o he doma ucoal I ( ) ca be expressed as: I( ) Id. (3) he varao o he Comparg (3) ad (33) resuls: I ( ). (34) x x l 3.3 Soppg crero The soppg crero or he erave sequece (8) s based o he dscrepacy prcple [3]: (33) INCAS BULLETIN Volume 6 Issue 4/ 4

7 9 Mahemacal ad umercal modelg o verse hea coduco problem ( ) ( ( ; ) ( )) d I T x T. (35) The olerace s chose so ha smooh soluos are obaed wh measuremes coag radom errors. I s assumed ha he soluo s sucely accurae whe: T( x ; ) T ( ) (36) where s he sadard devao o measureme errors. From equao (36) resuls: (37) For cases volvg errorless measuremes ca be speced a pror as a sucely small umber. 3.4 Numercal dscrezao I hs paper a e derece dscrezao s cosdered. For a uorm grd x x x ad a mplc orward-meceered-space dscrezao o he drec problem ()-(4) reads []: T T T T T o ( ) wh boudary codos: ad al codo: T T T x ( ) x (38) () (39) T x. (4) Cosequely he mplc scheme yelds o he ollowg r-dagoal algebrac sysem o lear equaos: where: ad: at bt ct d (4) T a b c d ( ) ( ) ( ) x (4) a b c d a b c d ( ). x x x x The r-dagoal sysem s solved by Thomas algorhm [4]. Smlarly he orward-me ceered-space e derece represeao o he sesvy problem resuls: (43) a T bt c T d (44) where a b c are gve by expressos (4)-(43) ad: INCAS BULLETIN Volume 6 Issue 4/ 4

8 Sera DANAILA Ala-Ioaa CHIRA 3 T d x d d ( ). x For he adjo problem (8)-(3) applyg a backward-me ceered space dscrezao oe obas he sysem: (45) a b c d (46) where he marx elemes a b c have he decal expressos as or he drec ad sesvy problems (4)-(43) ad: d T x T x x The al codo a s: 3.5 Compuaoal algorhm ( ( ) ) ( ) d d. x x (47). (48) () Assumg a al guess or he ukow uco () ad akg k he ma seps o he compuaoal algorhm are:. Solve he drec problem (4)-(43).The resul s he emperaure eld T( x ) or he ( boudary codo o he rgh boudary k ) () ;. Check he soppg crero (35). I s o vered coue; 3. Solve he adjo problem (46)-(48). ( ) 4. Calculae he grade I ( k ) usg equao (34); 5. Calculae he parameer 6. Solve he sesvy problem or wh () ad he desce dreco x () d. The resul s he eld ( k d ) () wh (9); ( ) T( x ; d k ). (k ) 7. Calculae he desce parameer applyg equao (8); 8. Calculae he ew esmao o he ukow boudary codo wh equao (8) ad reur o sep. 4. TIKHONOV REGULARIZATION METHOD Aoher echque or solvg he verse problem s roduced by Tkhoov [ ]. As already meoed problems codoally well-posed accordg o Tkhoov we have o do o jus wh a soluo bu wh a soluo ha belogs o some arrower class o soluos. To solve he verse problem ()-(5) he dereal equao o he emperaure eld () s approached by a well-posed hyperbolc boudary problem. Le's cosder a dereal equao: D( T) (49) where. D s a rs kd dereal operaor. Le us suppose he case whch he rgh-had sde o (49) s gve accurae o some where: INCAS BULLETIN Volume 6 Issue 4/ 4

9 3 Mahemacal ad umercal modelg o verse hea coduco problem. (5) Because he rgh sde erm s gve wh accuracy we wll ry o ormulae a wellposed problem or aoher operaor D wch possesses mproved properes compared o D : D ( T ) (5) where T s he approxmae soluo ad he parameer α ca be relaed wh he accuracy level he rgh-had sde.e. α = α(δ). As preseed beore varaoal mehods sead o solvg equao (5) hey mmze he dscrepacy ucoal: I ( v) D ( v). (5) For a bouded soluo Tkhoov regularzao mehod roduces a addoal sablzg ucoal v he dscrepacy ucoal: I ( v) D ( v) v (53) where he regularzao parameer α > mus be relaed o he rgh-had sde accuracy level δ. The approxmae soluo o he al problem (49) s he exremal o he ucoal: I ( T ) m D ( v) v (54) vh H beg he Hlber space where he approxmae soluo belogs. Isead o solvg he varaoal problem (54) Tkhoov cosders he relaed Euler equao: D DT T D (55) where D s he adjo operaor. As a geeral rule he raser rom he ll-posed problem (49) o he well-posed problem (55) ca be made passg o a problem wh a sel-adjo operaor D D. I order o esmae he regularzao parameer rom he dscrepacy crero we dee he uco: ad ca be oud as he soluo o equao: ( ) D ( T ) (56) ( ). (57) Ths equao ca be approxmaely solved usg varous compuaoal procedures [7]. For sace we ca use he successo ( k) k q q. (58) INCAS BULLETIN Volume 6 Issue 4/ 4

10 Sera DANAILA Ala-Ioaa CHIRA 3 To sar we made k = ad coue o a cera k = K a whch equaly (57) becomes ullled o a accepable accuracy. Wh so deed regularzao parameer oly K + calculaos o he dscrepacy (or he soluos o he Euler equaos (55)) are eeded. I he prese paper we apply wo dere approaches usg he Tkhoov regularzao: he mehod wh perurbed boudary codos ad he mehod wh perurbed al equao respecvely [7 ]. 4. Perurbed boudary codo wh local regularzao For a global regularzao he soluo s o be deermed a all mes smulaeously whereas local regularzao mehods he soluo depeds oly o he pre-hsory ad ca be deermed sequeally a separae mes. Local regularzao mehods ake o accou he specc eaure o verse problems or evoluoary problems maxmal possble measure []. Le s cosder he drec ad verse problems ()-(5). We cosder verse problem whch he boudary codo a he rgh boudary s o gve (he uco ψ() () s ukow). Isead he addoal codo a he le boudary s gve: T( ) T ( ) (59) whch s equvale o (5) x. Usg or each me value he local Tkhoov regularzao or deermg he boudary codo a he rgh boudary mples mmzao o he smoohg ucoal: ( ) ( ) ( ) ( ) I T T (6) where () s he ukow uco he verse problem. The approxmae T ( x ) ca be represeed as: T ( x ) T ( x ) T ( x ) o where T o( x ) s he soluo o he homogeeous derece boudary value problem: T T T T T o o o o ( ) T o x T o x l (6) (6) ( ) (63) wh he al codo T o x. Noe ha T s he soluo o he drec problem or a gve boudary codo ( ). For he o-homogeeous par T ( x ) cosderg he equao () resuls : T T T T ( ) x l (64) T ( ) (65) x INCAS BULLETIN Volume 6 Issue 4/ 4

11 33 Mahemacal ad umercal modelg o verse hea coduco problem ad he al codo T T o x soluo o he problem (64)-(66) ca be wre as: where qx ( ) s he soluo o he problem: Replacg (6) ad (67) (6) we oba: ( ) (66). Because he equao (64) s lear he T ( x) q( x) ( ) (67) q q x l dq () ql ( ). dx ( ) o( ) () ( ) ( ) ( ) (68) I T q T (69) The mmum codo I ( ) / leads o he equao: T T o q ( ) () ( ) ( ) q () For a gve value he ma seps o he compuaoal algorhm o advace me are:. Solve he drec problem ()-(4); resuls T x.. Solve he drec problem (6)-(63) o calculae T o x. 3. Solve drec problem (64)-(66) o calculae 4. Solve drec problem (68); resuls q (). x. T 5. Esmae he value ( ) ad reur o sep or he ex value o me. For a gve accuracy he seps -4 have o be cluded a loop or solvg he equao: (7) T ( ) T ( ) (7) ollowg he sequece (58). Noe ha or he seps -3 we eed o solve a rdagoal sysem o equaos ad he marx o he sysem s he same as prevous applcao (4). 4. Perurbed al equao I hs case he al dereal equao s replaced by a perurbed dereal equao order o ormulae a well-posed problem. Because he verse problem he ukow s he rgh boudary codo s covee o rasorm he al parabolc equao o hyperbolc equao havg he x l a ree boudary. Samarsk ad Vabshchevch [7] show ha he desred equao s: wh he al codo: T T T 3 (7) INCAS BULLETIN Volume 6 Issue 4/ 4

12 Sera DANAILA Ala-Ioaa CHIRA 34 T ( x) or ad he al codo: T ( x ) or ad x l The boudary codos or equao (73) are: T T ( ) ( ) or (73) ad x l. (74) x ad (75) T ( ) or x ad. (76) Because he equao (7) s hyperbolc respec o x o solve hs equao s ecessary o advace he soluo hs dreco. Le's be x a arbrary mesh po. For hyperbolc equaos sable dscrezao s obaed usg upwd schemes: T T T T o( ) x ( ) (77) T T T T o( ) T T o( ) (78) T T T x T T T T T T ( ) ( ) Replacg (7) resuls : T T T ( ) ( ) ( ) ( ) T T T T T T T ( ) ( ). 3 x. The above equao represes a rdagoal algebrac sysem wh he ukows T or cos. To close he sysem (8) he codos (73) ad (74) are aached. I he dscree orm o (73) s easy o oba: (79) (8) T (8) o mpleme codo (74) we wre he derece orm o he equao (7) he po akg o accou he equaly T T or all. The resulg equao s: INCAS BULLETIN Volume 6 Issue 4/ 4

13 35 Mahemacal ad umercal modelg o verse hea coduco problem where: ad: T T ( ) ( ) ( ) T T T T ( ) ( ) The rdagoal sysem s: T T (8) at bt ct d (83) x( ) ( ) ( ) x( ) a b c T T T T d T T ( ) ( ) a x( ) ( ) ( ) b T T T T d T T ( ) ( ) To sar or we mpose(75): ad order o mpleme he codo (76) we wre a Taylor: Because I resuls: T T (84) (85) T T ( ) (86) 3 T T ( ) o ( ). (87) T T T T T x o( ) T T 3 T T ( ) o ( ). For hs mehod he compuao algorhm s represeed oly by oe loop or solvg he equao (7). A each erao he problem (8)-(89) s solved o d T ( ). x (88) (89) 5. NUMERICAL RESULTS For he umercal applcaos uorm grds wh. ad l x were used. Also he assumed al me value was bu he me sep was calculaed mposg dere values or. For each mehod we have solved rs a drec problem cosderg INCAS BULLETIN Volume 6 Issue 4/ 4

14 Sera DANAILA Ala-Ioaa CHIRA 36 hree shapes or he rgh boudary codo T( l ) ( ) deoed he ollows wh case- case- ad case-3 respecvely: case-: ( ) 4 ( ) case-: / / ; () ( ) / /.5 / 4; (9) case-3: ( ) / 4 3 / 4;.5 / 4. The soluos o he drec problems or all hree case are represeed Fgure - Fgure 3. I order o ormulae he pu daa or he verse problem T ( ) he obaed soluo o he drec problem le sde T( ) s radom perurbed wh a accuracy level. So he exac soluo o he verse problem wll be represeed by (). Fgure. Soluo o drec problem case Fgure. Soluo o drec problem case Fgure 3. Soluo o drec problem case 3 Fgure 4. Soluo o verse problem wh cojugae grade mehod-case INCAS BULLETIN Volume 6 Issue 4/ 4

15 37 Mahemacal ad umercal modelg o verse hea coduco problem Fgure 5. Soluo o verse problem wh Tkhoov local regularzao mehod-case Fgure 6. Soluo o verse problem wh Tkhoov equao regularzao mehod-case Fgure 7. Soluo o verse problem wh cojugae grade mehod-case Fgure 8. Soluo o verse problem wh Tkhoov local regularzao mehod-case Fgure 9. Soluo o verse problem wh Tkhoov equao regularzao mehod-case Fgure. Soluo o verse problem wh cojugae grade mehod-case 3 INCAS BULLETIN Volume 6 Issue 4/ 4

16 Sera DANAILA Ala-Ioaa CHIRA 38 Fgure. Soluo o verse problem wh Tkhoov local regularzao mehod-case 3 INCAS BULLETIN Volume 6 Issue 4/ 4 Fgure. Soluo o verse problem wh Tkhoov equao regularzao mehod-case 3 Fgures 4 5 ad 6 plo he obaed umercal resuls or case. All mehods are very sesve o he umber o me seps. I he accuracy o umercal resuls s sasacory or he cojugae grade mehod ad or he Tkhoov equao (hyperbolc) regularzao or he local regularzao srog oscllaos o he umercal soluo are prese eve or low umber o me layers. We apprecae ha he Alaov s erave regularzao mehod predcs smoohed resuls. For he case all mehods prese smlar behavor. Fgure 7 correspodg o Alaov s erave regularzao mehod shows smoohed resuls respec o he predced by Tkhoov equao regularzao mehod (Fgure 9). However he las oe succeeds o capure more exacly he peak o he reerece daa. The Tkhoov local regularzao s aga he wors predco. For he case 3 where we check he schemes or dscouous pu daa he resuls are ploed Fgures -. The plos show ha all hree mehods are o able o capure correcly he dscouy. The sudde varaos are replaced by coguous oes. Aga he cojugae grade mehod ad he Tkhoov equao regularzao predc beer resuls. 6. CONCLUSIONS I he prese paper we ry o asses hree dere approaches solvg he verse hermal coduco problem: he Alaov s erave regularzao mehod he Tkhoov local regularzao mehod ad he Tkhoov equao regularzao mehod respecvely. For he cosdered umercal applcaos he rs ad he hrd mehod provde smlar resuls havg a sasacory accuracy respec o exac reerece daa. The local Tkhoov regularzao roduces srog oscllaos umercal predco beg he mos sesve o he umber o perormed me seps. REFERENCES [] M. N. Ozsk R. B. H. Orlade Iverse Hea Traser Taylor ad Fracs New York. [] O. M. Alaov Iverse Hea Traser Problems Sprger-Verlag New York 994. [3] A. N. Tkhoov Regularzao o Icorrecly Posed Problems Sove Mah. Dokl. 4(6) [4] J. V. Beck B. Blackwell A. Shekh-Haj Comparso o Some Iverse Hea Codo Mehods Usg Expermeal Daa I. J. Hea Mass Traser 39(7):

17 39 Mahemacal ad umercal modelg o verse hea coduco problem [5] J. V. Beck B. Blackwell C. R. S. Clar Iverse Hea Coduco Ill Posed Problems A Wley Ierscece Publcao New York 985. [6] A. N. Tkhoov V. Y. Arse Soluo o ll-posed problems Wso & Sos Washgo DC 977. [7] A. A. Samarsk P. N. Vabshchevch Numercal mehods or solvg verse problems o mahemacal physcs de Gruyer ISBN Berl Germay. [8] A. N. Tkhoov Soluo o Icorrecly Formulaed Problems ad he Regularzao Mehod Sove Mah.Dokl. 4(4) [9] A. N. Tkhoov Iverse Problems Hea Coduco J.Ehg. Phys. 9() [] A. A. Samarsk P. N. Vabshchevch Compuaoal Hea Traser Vol.. The Fe Derece Mehodology Wley Chcheser 995. [] A. N. Tkhoov A. V. Samarsk Equaos o Mahemacal Physcs Dover Publcaos 99. [] V. Isakov Iverse Problems or Paral Dereal Equaos Sprger-Verlag 997. [3] G. S. Dulkravch T. J. Mar Iverse Shape ad Boudary Codo Problems ad Opmzao Hea Coduco Chaper Advaces Numercal Hea Traser Mkowycz W. J. ad Sparrow E. M. (eds.) Taylor ad Fracs 996. [4] S. Dăălă C. Berbee Numercal mehods lud dyamcs Ed Academe Romae Buchares 3. [5] E. Buckgham Model expermes ad he orms o emprcal equaostras.asme 37: [6] V. L. Sreeer ad E. B. Wyle Flud Mechacs McGraw-Hll Book Compay New York 7h edo Chaper [7] A. D. Kraus A. Azz ad J. R. Wely Exeded Surace Hea Traser Joh Wley&Sos Ic. NewYork. [8] J. H. Lehard IV J. H. Lehard V A hea raser ex book Thrd Edo paperback-augus 3. [9] O. M. Alaov Iverse Hea Traser Problems Sprger-VerlagTelos 995. [] D. Peacema H. Rachord The umercal soluo o parabolc ad ellpc equaos SIAM Joural [] R. Flecher ad C. M. Reeves Fuco Mmzao by Cojugae Grades Compuer J INCAS BULLETIN Volume 6 Issue 4/ 4