Hyperbolic Heat Equation as Mathematical Model for Steel Quenching of L-shape and T-shape Samples, Direct and Inverse Problems

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SEAS RANSACIONS o HEA MASS RANSER Bos M Be As Bs Hpeo He Eo s Me Moe o See Qe o L-spe -spe Spes De Iese Poes ABIA BOBINSKA o Pss Mes es o L Ze See 8 L R LAIA e@o MARARIA BIKE ANDRIS BIKIS Ise o Mes Cope See es o L R B 9 L459 R LAIA e@e s@e p://3/sess/s As: - I s ppe e eeop e oe o D 3D peo e eo os soo o ese poe o L-spe -spe spes Soos o o e ese poes e oe ose o s ee e eo s eo e eo espe o spe ooe oe e eo espe o e Ke-os: - See e Hpeo e eo L-spe spe -spe spe De poe Iese poe E soo Ie eo Ioo s s oo o o eso [6] I soe o o peos ppes e e osee poes o e oo o oe eee o e sse e I e ppes []-[5] [7] e e ese o se-se poes se poes Soe es o ese ppes e se ee o so e poe o L-o spe ee e e e eo e e eo s sse o ss oe s ee e e opeo s ee 3D s s eee o e ooe spe s e eoes e epee e o e osee e o P eo o e o oseo o ee e s o eos ρ ee s spe e p ρ s es o e o eoes e soe o oe e eeo e oe es e eo oe s e oo eo ρ 3 s po eo o se sos pes e e spee o popo Oe o s ps sos e oe s s o oe s o epe Iese Qe IQ e es e ee p o e e e es o s e [] So oe s e e s o oe se o ese e popo I ese sos oe s e epe > o s ppoo see e [4] 4 s e e oe oe s eo o Ceo s eo Hee s eo e ρ e ee epee o o e e B e see e s sse ese e oss Co eo 4 e o peo e oo eo s e spee o popo o o ρ ρ 5 ISSN: 79-544 63 Isse 3 oe 5 J

C I s so o s eepe s eo o pe e eo Hee C s e spee o e popo C ρ s e s oee Me oo o 3D Poe I s seo e e o o se eo 5 o ese IQ poess e e see o 3D poe ee Me See o 3D Poe o L-spe Spe Le s e L-spe spe e p o o e os e oe o e se e e e p e se A opes e o { } B e oo p ee { } < e e oo e spe s ee e ooe p IQ poess se e e o e As eoe eoe o es ese see e o e spe e so se peo e eo es o o e e spee o e popo Se e spe osss o o ps e e e o ee IQ poess o e p sepe o se e o-e poe e ee o ee e eos o e os es IQ poess e spe e ess o oos Le s sse eoes e epee so e se eeoe e epee so e oo e ese e eos o e oo eeoe e e oo o Ψ 6 ] ; Ψ 7 ] ee Ψ ρ Ψ ρ e e e eeo s se Ψ Ψ e es ρ e e se e s e pee seo O e se o e spe e ee es pe e so e e epee o s so e e 3 pe o oos e o oos o e se e e o 8 9 3 ee s e-se oee e o oos pose o e oe ses o e oo e o e se pe 4 5 6 7 8 As o e ee eee e e ses o e os e pese eess oos o SEAS RANSACIONS o HEA MASS RANSER Bos M Be As Bs ISSN: 79-544 64 Isse 3 oe 5 J

SEAS RANSACIONS o HEA MASS RANSER Bos M Be As Bs ese e o o e epees e e es e e o [ ] [ ] [ ] [ ] 9 e soos o eos 6 7 s so ss e oos ee e e os o epee po o e oos e es ese e e eses e o e o B e s e e o e opso e eses o pe e se oes see e D IQ s ee o e es o e ess e e s e es o ee o see [] o eeo o e eses e sse e epee sos e so o e es e e e e o e poess 3 4 ee Me See o 3D Poe o -spe Spe I s se e e spe o -o I osss o o L-spe spes e e oe s o e Oe o e s e spe ese e pee sseo ese o e oe o e se e o-e poes o ee e epee o s spe e e e se o s e pee sseo eep e o oo e pe Ao s pe e so se oo s e ppe [ ] 5 I s so o L-spe spes e e se epee so 3 De Poe o L-spe Spe Hee e soo o e poe e ep e e o e o e e o ee s o eo o o oseo ee s o o so e e oo o e e s o epese e o o s e o o o sos ppope o oos o e es pes oe o eos I e se o L-spe spe e e e e o o ees e os 3 De Poe o 3D L-spe Spe e so eos 6 7 o oe oo o o e e o ssos ep 6 ep 7 P ese epessos o eos 6 7 e oos s e e e e eos o s e ees ese e e Ke-oo eos 4 ep Ψ 8 ; ] 4 ep Ψ 9 ] ee e e o oos o ese eos e e s eoe 8 8 s os se o ep ISSN: 79-544 65 Isse 3 oe 5 J

o e se o ep o e oo se o B e o oos 9 so o e o sos so e epee e e e es e oe se ep 3 ep 3 e e oespo oos o e e poe 3 33 o o oos oe o e ese poe ep 34 ep ep 35 ep e ose e ese poe e e soos o e o 3D poes 6 7 so e poes o eos 8 9 s soos 6 7 Se soo o 3D e poe s e s s D se e ose o e e 3 De Poe o L-spe Spe I s se e e spe s e eo << << << o o e oo o D poe o e poe osee eoe e oe ee es e eo o e os se eoe o se 36 37 B s ese ppoos pp e o oos e oe o 3 7 8 o D ee eos soe es e oe 38 ] ; 39 ] ee 4 ep 4 ep I s posse o se oe e o ppoo I se e e epessos o oees os e o ose e oe e se pe o o oos s 3D se e se ee 4 4 4 SEAS RANSACIONS o HEA MASS RANSER Bos M Be As Bs ISSN: 79-544 66 Isse 3 oe 5 J

43 44 45 46 ee ep ep e oos sse e oo o 47 48 ee A e ses o ese oos e oe pp e e poee o e oos o 3D poe I oe o sp soos o ese D poes e s oe osees o oe e se eo e o o ps o e spe eeoe oees se 38 39 e e e o oos o e epees e e es e o eee e se e oo e o sse ] [ 49 ] [ 5 s o oos 49 5 e o oo o e se oe o e se e e e o ] 5 B e o oo o e e- se oe o e oo s e o 5 No e e o-e poes e se e soo o e p o e spe s ee s o eo 3 Soo o e Bse I s e o o e ee e eo 38 ss e o oos 4 43 5 e oos 47 s soo see e [3] 53 e ee s o s o see [3] s p SEAS RANSACIONS o HEA MASS RANSER Bos M Be As Bs ISSN: 79-544 67 Isse 3 oe 5 J

s p 54 e e p ppes o 54 e oe o e ees < o p > o p e eeos e e oo epessos s s e e e eees e oos o ese see eos o o I e e -spe spe e oo 4 s o A oos o As o soo 53 s o e epe eo s se 3 Soo o e oo e soo o e seo ee s s o 53 55 B s ee s o s o s o 54 s s 56 e e s e e oo ees < o > o e eees e eeos o e ee s o 56 e e ese epessos o o ] s[ s e e 33 Jo o Bo Soos I e sse 53 o o 5 e e SEAS RANSACIONS o HEA MASS RANSER Bos M Be As Bs ISSN: 79-544 68 Isse 3 oe 5 J

57 I s e o 55 5 e o epeseo o e oo o e soo o e oo s ee e oe eee o ps 58 Hee No es o p epesso 57 o 58 o o o-ooeeos oe-eo e eo o e o e oe eee o ps o e L-spe spe Φ K 59 ee Φ K e ee e oos e-e os K ee B e o-ooeeos e s e o Φ e o s ese o e oo esos I e o s oe e soo e se e oe e e o ee e soo e seo p o e spe oo 33 Soo o 3D Poe Soo o 3D poe e so e se s o D As es oe o e eo s o 59 oe e e espe o e sp ooe Φ SEAS RANSACIONS o HEA MASS RANSER Bos M Be As Bs ISSN: 79-544 69 Isse 3 oe 5 J

K 6 ee s o o e se 3D se s o p s p s 6 ee e eeos e e oo epessos s s e e s B e eees e oe o ese see eos o o o 4 Iese Poes o D L-spe Spe As s eoe eoe s se e epees e e es e o o B s ee o eee e eses o e opso e eses Ise o oos e e e epee so e e eses e oe e e o e poess ep 6 ep 63 ee ep ep As s po o e e es e s oe e e e 64 B es o e oe e poes o e os e e e eos e e se o s 38 39 65 ] ; 66 ] Ess ppope oos oe soos o ese eos e o oo oos e e s e e poe e oos e e o 67 68 SEAS RANSACIONS o HEA MASS RANSER Bos M Be As Bs ISSN: 79-544 7 Isse 3 oe 5 J

es o e o e poe se o ese e oo o e soo o s o-e poe s eoe e pee seo Dee e soos espe o e e ees e oe 69 7 e e e eses o 3D se e e e o e poess Soo o e 3D poe e so e se s D se 5 Coso e e ose e soo o peo e eo o D L-spe spe ose o o ee e eo o I s eo e eo espe o e spe es oe e eo espe o e e soos o D e eese poes e ee o e poe e o eee e e eses Reeees: [] A Bs M Be Cose o-eso soo o e se peo sse Poees o e L Ae SEAS RANSACIONS o HEA MASS RANSER Bos M Be As Bs ISSN: 79-544 7 Isse 3 oe 5 J

SEAS RANSACIONS o HEA MASS RANSER Bos M Be As Bs o Sees Seo B o5 No5 998 pp 8- [] A Bs Cosee e s ppoe eo o soo o soe e ese e se poes Ae Copo Meos He se IX I Pess 6 pp 3-3 [3] M Be E Soo o 3-D Se- Se Poe o Oe Eee o e Sse Re Ne Aspes o He se e Eee Eoe P II SEAS Pess 8 pp 4-46 [4] M Be A Bs E se Soo o Sse Re Poees o e 5 SEAS I Co o HEA AND MASS RANSER HM 8 Apo Meo SEAS Pess 8 pp 5-3 [5] A Bs M Be E 3-D Soo o Sse Re P Poees o e SEAS I Co o APPLIED MAHEMAICS MAH 7-II Co Ep SEAS Pess 7 pp 64-69 [6] M Be A Bs Hpeo He Eo s Me Moe o See Qe o L-spe Spes P De Poe Poees o e 3 SEAS I Co o APPLIED MAHEMAICS MAH 8 Peo De L C C Iss Sp SEAS Pess 8 pp 98-3 [7] A Bs M Be Soe 3-D se-se soos o sses e IASME sos Isse 7 o Sepee 5 pp -9 [8] A Bs M Be Bos e Reo o e Soo o Hpeo He Eo o Ie Eo o R Coe Poees o e I Co Dee Eos e Appos DEA 9 Pees L K 9 pp 4-6 [9] A Bs S seo Iese I He Reo e Poes o Hpeo He Eo Poees o e SEAS I Co o APPLIED MAHEMAICS Ds es SA SEAS Pess 6 pp-4 [] MA Aoo N Koso JA Poe Iese Qe o Ce See Ps IASME sos Isse 9 o Noee 5 pp 84-845 [] N Koso P Kos D Yeo I C He Deses Ee o e Bss o CD Moe Epees Iese Qe Poees o e 5 IASME/SEAS I Co o HEA RANSER HERMAL ENINEERIN AND ENIRONMEN Aes eee SEAS Pess 7 pp 95-98 [] NI Koso Iese See Qe Meos eo eoo o Qe Spe-e Ne-Yo NY 99 [3] AD Po Hoo o Le P Dee Eos o Eees Sess Cp&H/CRC Pess Bo Ro [4] ZM Z No/Mose He se M-H 7 Aoees: e ese s sppoe e es o L ES poe Sppo o Doo Ses e es o L No 9/38/DP//9/IPIA/IAA/4 e se o e o s sppoe e L Co o See No 957 ISSN: 79-544 7 Isse 3 oe 5 J