Hyperbolic Heat Equation in Bar and Finite Difference Schemes of Exact Spectrum

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1 atest reds o heoretical ad Applied Mechaics Fluid Mechaics ad Heat & Mass raser Hyperbolic Heat Equatio i Bar ad Fiite Dierece Schemes o Exact Spectrum A BUIKIS H KAIS Abstract he solutios o correspodig -D iitial-boudary value problem ad iverse problem or hyperbolic heat coductio equatio are obtaied umerically usig or the approach dieretial equatios the discretizatio i space applyig the iite dierece method ad the best scheme with exact spectrum (BSES) Numerical solutios i the time are obtaied by the MAAB solver usig the method o cojugate operators ad the method o superpositio For iite dierece approximatio with cetral diereces strog umerical oscillatios are preseted whe the iitial ad boudary coditios are discotiuous or the problem he method o BSES is without oscillatios ad this is eective way or umerical solutios Keywords Exact solutio Fiite dierece Hyperbolic heat equatio Itesive quechig I INRODUCION he hyperbolic heat coductio problems are used or modellig o itesive steel quechig []-[6] [9] [] he solutios o correspodig -D iitial-boudary value problem ad iverse problem are obtaied umerically usig the iite dierece methods ad best scheme with exact spectrum (BSES) he iite dierece methods o secod order approximatio i the uiorm grid are used or the approximatio o the dieretial operator o secod order derivatives i the space he solutio i the time is obtaied aalytically ad umerically with cotiuous ad discreet Fourier methods New trascedetal equatio ad algorithms or obtaiig the last eigevalues ad eigevectors o iite dierece scheme are obtaied by the usage o the spectral method We deie the best scheme with exact spectrum (BSES) where the iite dierece matrix A is represeted i the orm A PDP ( P D are the matrixes o iite dierece eigevectors ad correspodetly eigevalues) ad D ad the elemets o diagoal matrix are replaced with the irst eigevalues rom the dieretial operator Numerical Mauscript received Jue 3 his wor was supported partly by the Grat ZS 957 ad ESF Project No: 9/38/DP//9/IPIA/VIAA/4 Adris Buiis Proessor at Uiversity o atvia Faculty o Physics ad Mathematics Head o aboratory o Mathematical echologies Istitute o Mathematics ad Computer Sciece Uiversity o atvia Raia bulv 9 Riga V 459 atvia phoe ax: e- mail: buiis@latetlv Dr Habil Math Full Member o atvia Academy o Sciece Harijs Kalis Proessor at Uiversity o atvia Faculty o Physics ad Mathematics Researcher o aboratory o Mathematical echologies Istitute o Mathematics ad Computer Sciece Uiversity o atvia Raia bulv 9 Riga V 459 atvia phoe ax: alis@laetlv Dr Habil Math solutios i the time or iverse problem are obtaied by the MAAB usig the methods o the cojugate operators ad o the superpositio he paper is orgaized as ollows I Sectio II we cosider the two mathematical models o hyperbolic heat coductio problems with boudary coditios o irst ad third id (the direct ad iverse problems) I Sectio III IV is the method o lies Fourier series matrix o derivatives ad averaged methods [7]-[9] or solvig the used direct problem We deie the best scheme with exact spectrum (BSES) or solvig the problem I Sectio V the method o lies or solvig the iverse problem is give We cosider umerical methods or solvig the iverse problem: the method o cojugate operators ad method o superpositio Sectio VI presets some examples o umerical solutios o two typical problems related to hyperbolic heat coductio equatios obtaied by dieret methods he results are represeted with 8 igures Ad ially we summarize our coclusios i Sectio VII II HYPERBOIC HEA RANSFER MODE For umerical experimets we cosider the hyperbolic heat coductio (temperature ( x t ) ) problem i bar [5] [6] [9]: τ x ( ) ( ) ( ) + x t t α ( l ) + α ( r ) x x ( x ) ( x) V ( x) x ( ) where / is the heat diusivity - heat coductivity cρ coeiciet c - is speciic heat capacity ρ - desity ial time τ is the relaxatio time ( ) () t is the τ < l r are give temperatures α α are the heat traser coeiciets or boudary coditios From the practical poit o view the secod iitial coditio is uow [6] [7] [9] I this case we ca use additioal coditio: ( ) ( ) x t x () Here is the give ial temperature ISSN: ISBN:

2 atest reds o heoretical ad Applied Mechaics Fluid Mechaics ad Heat & Mass raser I the temperatures are costat values the α α l r or we have the problem with homogeous boudary coditios o irst id: V V V τ + x ( ) t ( t ) ( ) ( ) ( ) V t V t t t ( ) (3) V x V ( x ) v ( x) V ( x) x ( ) xr ( x) v ( x) ( x) h ( x) h ( x) Assumig α α α r we have mixed boudary problem: V V V τ + x ( ) t ( t ) V ( t) V + αv t ( t ) (4) ( ) x V x V ( x ) ( x) V ( x) x ( ) III MEHOD OF INES FOURIER SERIES MARIX OF DERIVAIVES FOR HE DIREC PROBEM We cosider uiorm grid i the space x h N Nh Usig the iite diereces o secod order approximatio or partial derivatives o secod order respect to x [3] we obtai rom (3) the iitial value problem or system o ordiary dieretial equatios (ODE) o secod order i the ollowig matrix orm: τuɺɺ ( t) + Uɺ ( t) AU ( t) (5) U ( ) U Uɺ ( ) V Here A is the stadard 3-diagoal matrix o N order with 4 elemets { ; ; } he colum-vectors h U ( t) U ɺ ( t) U ɺɺ ( t) U V o N order with elemets u ( t) V ( x t) V ( x ) t V ( x t) uɺ ( t) uɺɺ ( t) u () v ( x ) v () V ( x ) N he matrix A has eigevalues 4 π µ si h N he system o ODE (5) ca be rewritte i a ormal orm uɺ Bu u() u (6) Here u uɺ u are the colum-vectors o N order i ollowig orm( U; Uɺ ) ( Uɺ ; Uɺɺ ) ( U ; V ) B is the matrix o N order i the orm E B τ A τ E he uit matrix E is rom the order N is the symbol o traspositio For the best scheme with exact spectrum (BSES) the matrix A is represeted i the orm A PDP where symmetrical orthogoal matrix with elemets: h πij si P P is the pi j i j N N N he colum o the matrix P ad the diagoal matrix D cotais the irst N orthoormal eigevectors h π x j p ( x j ) si x j jh j N ad eigevalues π d N From the dieretial operator correspods ( AP PD) he spectral method is more stable as the method o iite dierece approximatio with cetral dierece because the eigevalues are larger the modulus 4 π d > µ i d si the we have i the h N method o iite dierece approximatio with 3-diagoal matrix A We ca cosider the aalytical solutios o (5) usig the spectral represetatio o matrix A i the orm A PDP From the trasormatio W PU ollows the separate system o ODE: τ Wɺɺ ( t) + Wɺ ( t) DW ( t) (7) W () PU Wɺ () PV Here W ( t) Wɺ ( t) Wɺɺ ( t) W Wɺ () are the colum-vectors o N order with elemets w ( t) wɺ ( t) wɺɺ ( t) w () wɺ () N he solutio o this system (7) is the uctio ISSN: ISBN:

3 atest reds o heoretical ad Applied Mechaics Fluid Mechaics ad Heat & Mass raser t w ( t) exp τ ( t) sih w () w () cosh ( t) w () ɺ + + τ d where 4τ τ I 4 d τ > the the hyperbolic uctios are replaced with the trigoometrically uctios ad the parameter is: d τ For the case 4τ the t w () w ( t) exp t w () + + w () τ ɺ τ We ca cosider the aalytical solutios usig the Fourier method i ollowig orm V ( x t) w ( t) P ( x) (8) where P ( x) si ( π x) are the orthoormal eigevectors ( P P ) P ( x) P ( x) dx δ m m m o the dieretial operator with homogeous boudary coditios ( δ m - the symbol o Kroecer) w ( t) is the solutio (8) with w ( v P ) wɺ ( V P ) () () For the approximatios o the derivatives with matrix o derivatives [4] we ca cosider o-uiorm grid with the grid poits o the roots o the Chebyshev polyomials o the secod id: π ( ) x 5 cos N + (9) N Usig this grid poits we ca approximate the derivative i the equatios (5) with matrix D o derivatives o N + order i the orm: V D V " h e h where Vh ( V ( x t) V ( x t) V ( xn + t) ) " " ( ) " ( ) " V V x t V x t V ( x t) ( + ) are the h h h h N colum-vectors o the correspodig " values V ( x t) h V x ( t) e From the agrage iterpolatio ollows [4] that elemets o matrix the orm ( x j ) D e are i dl d j j N + () dx ω ( x) where l ( x) are the elemetary ω x x x ( )( ) agrage multipliers ω ( x x ) N + For this ouiorm grid (5) the iterpolatio error is small [4] he determiats o derivatives matrix D are equal to zero (this matrix is sigular) hereore we eed decrease the order o this matrix to N order usig the homogeeous boudary coditios ad deletig the irst ad last colums ad rows I this case i (6) we eed to replace the matrix A with D e IV HE MEHOD OF INES FOR SOVING HE DIREC PROBEM Similarly we obtai rom (4) the iitial value problem or the system o ODE (5) (6) where the colum-vectors ad uit matrix are o the N + order B is the matrix o N + order ( u ( ) ( x ) ) he tree-diagoal matrix A o N + order ca be represeted with ollowig dierece operator o secod order approximatio [3] ( y y ) / h Ay ( y+ y y ) / h N + () ( yn yn ) / h N Here y is colum-vector o N + order with elemets y N Usig two vectors y y scalar product N [ y y ] h y y + 5( y y + yn yn ) ca be proved i [3] that operator A is symmetrical ad [ Ay y] he spectral problem Ay µ y N + has correspodig solutio y C cos ( p x )cos( p x ) cos( p xn ) 4 µ si ( ph / ) () h ISSN: ISBN:

4 atest reds o heoretical ad Applied Mechaics Fluid Mechaics ad Heat & Mass raser Here p are the positive roots o the ollowig trascedetal ta p si p h / h α N + ad the equatio ( ) ( ) costats C ( ) ( ) + 5h si p / ta p h m Gives the orthoormal eigevectors y y with the scalar m product [ ] or m N For p + rom y y δ m N N p π N + pn ad µ N µ N N + we have special solutio o the problem: () ollows that + For y C cosh ( p x )cosh ( p x ) cosh ( p xn ) 4 µ cosh ( ph / ) N + (3) h Here pn + is the positive root o the ew trascedetal equatio tah ( pn + ) sih ( pn + h) / h α ad the costat C N + + 5hsih p / tah p h ( ) ( ) N + N + gives the orthoormal eigevectors or all m N + he matrix A ca be represeted i the orm A PDP where P P because o P P E he matrix P ad the diagoal matrix D cotais N + orthoormal eigevectors y ad eigevalues µ N + he solutio o the spectral problem or dieretial equatio y x λ y x x ( ) ( ) ( ) ( ) ( ) α ( ) y y ě + y is i the orm ( ) ( λ ) y x C cos x C Here + 5si / ( λ ) / λ + α ( α + λ ) λ are positive roots o the trascedetal equatio: ( ) ta λ λ α 3 I the scalar product ( ) ( ) ( ) [ y y ] y y y x y x dx is approximated m m m with trapezoidal ormula ad i the limit case i h the p λ or N V MEHOD FOR SOVING HE INVERSE PROBEM For the iverse problem the vector V i (5) is uow ad we have additioal coditio U ( t ) u where u is the vector-colum with elemets u ( x ) M M N or (3) or M N + or (4) he aalytical solutios o this problem ca be obtaied rom (7) replacig the secod iitial W PV W t Pu or the ɺ with ( ) coditio ( ) problem (4) his aalytical solutio is t sih ( t) t τ e w ( t ) (4) τ w ( ) sih ( ) t e t w ( ) cosh ( t ) + w ( ) cosh ( t) W t (the where w ( t ) are the compoets o vector ( ) ote or vectors V u (4) is valid) From the ormula (4) ollows that the secod coditio is i this orm: w ( ) sih t ( ) ( ) ( ) t w τ e w ( t ) w ( ) cosh ( t ) τ V PWɺ I replacig this expressio i (8) we obtai the solutio (4) For Fourier method we obtai the Fourier coeiciets w t u Q w ( t) wɺ ( ) rom (4) where ( ) ( ) ( ) ( ) ( ) ( ) ( ) V x wɺ Q x Q x p x or the problem (3) or Q ( x) y ( x) or the problem (4) We cosider two umerical methods or solvig the iverse problem: the method o cojugate operators ad method o superpositio [] VI SOME EXAMPES FOR MIXED PROBEM he umerical experimet or the direct problem (3) (6) with τ V is made l r or the case o the iitial ad boudary coditios he umerical results are produced by MAAB solver de5s or exp Bt with the help o calculatio the matrix-uctio ( ) usig Matlab operator exp m( B t) u t exp m B t u problem (6) is ( ) ( ) he solutios o the ISSN: ISBN:

5 atest reds o heoretical ad Applied Mechaics Fluid Mechaics ad Heat & Mass raser Fig Solutio depedig o t at x 5 Fig 4 Solutio o BSES N t Fig Solutio depedig o x at t Fig 5 Solutio depedig x at t N Here were solved several more practical itesive steel quechig problems Some iverse problems solutios are preseted o igures 6 7 ad 8 Fig3 Solutio o iite dierece method [] N t Fig 6 Solutio max ( ( )) U x t depedig o t by τ ISSN: ISBN:

6 atest reds o heoretical ad Applied Mechaics Fluid Mechaics ad Heat & Mass raser 7 Solutio max ( ( )) U x t depedig o t Fig [] Kobaso N I Itesive Steel Quechig Methods Hadboo heory ad echology o Quechig Spriger-Verlag 99 [] otte GE Bates CE ad Clito NA Hadboo o Quechats ad Quechig echology ASM Iteratioal 993 [3] Aroov MA Kobaso N Powell JA Itesive Quechig o Carburized Steel Parts IASME rasactios Issue 9 Vol November 5 p [4] Kobaso NI Sel-regulated thermal processes durig quechig o steels i liquid media Iteratioal Joural o Microstructure ad Materials Properties Vol No 5 p -5 [5] Buiis A Guseiov Sh Solutio o Reverse Hyperbolic Heat equatio or itesive carburized steel quechig Proceedigs o ICCES 5 (Advaces i Computatioal ad Experimetal Egieerig ad Scieces) December -6 5 II Madras p [6] Buie M Buiis A Approximate Solutios o Heat Coductio Problems i Multi- Dimesioal Cylider ype Domai by Coservative Averagig Method Part Proceedigs o the 5 th IASME/WSEAS It Co o Heat raser hermal Egieerig ad Eviromet Vouliagmei Athes August p 5 [7] Buiis A Coservative averagig as a approximate method or solutio o some direct ad iverse heat traser problems Advaced Computatioal Methods i Heat raser IX WI Press 6 p 3-3 [8] Vilums R Buiis A Coservative averagig method or partial dieretial equatios with discotiuous coeiciets WSEAS rasactios o Heat ad Mass raser Vol Issue 4 6 p [9] Buie M Buiis A Several Itesive Steel Quechig Models or Regular Samples his Proceedig 6 p [] Ciegis R Numerical solutio o hyperbolic heat coductio equatio Mathematical Modellig ad Aalysis 4 () 9p-4 [] Na Y Computatioal methods i egieerig boudary value problems Academic Press New Yor 979 [] Maarov V Gavrilyu I P O costructio the best et circuits with the exact spectrum Dopov Aad Nau Ur RSR Ser A 975 p77-8 (I Rusia) [3] Samarsij A A heory o iite dierece schemes Moscow Naua 989 (I Rusia) [4] Cirulis Nosaturated approximatio by meas o agrage iterpolatio Proceedig o the atvia Academy o Scieces Sectio B (5) 998 p Fig 8 Solutio U ( x t ) by τ VII CONCUSIONS he hyperbolic heat coductio problems are solved by umerical modelig usig eective iite dierece scheme with exact spectrum he solutios o correspodig -D iitial-boudary value problem ad iverse problem are obtaied umerically Numerical solutios i the time or the iverse problem are obtaied usig the methods o the cojugate operators ad o the superpositio he advatages o the BSES are demostrated via several umerical examples i compariso with some other well-ow previously used methods For iite dierece approximatio with cetral diereces strog umerical oscillatios are preseted but the dierece scheme o exact spectrum is without oscillatios he spectral method is more stable tha the method o iite dierece approximatio with cetral dierece Reereces ISSN: ISBN: