AN EXPLICIT WAVELET-BASED FINITE DIFFERENCE SCHEME FOR SOLVING ONE-DIMENSIONAL HEAT EQUATION
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1 ISSN AN EXPLICI WAVELE-BASED FINIE DIFFERENCE SCHEME FOR SOLVING ONE-DIMENSIONAL HEA EQUAION Mahmmod Azz Muhammed 1 Adh Susao 1 F. Soesao 1 Soerso 1Deparme of Elecrcal Egeerg Faculy of Egeerg Gadjah Mada Uversy Yogyakara Deparme of Mechacal Egeerg Faculy of Egeerg Gadjah Mada Uversy Yogyakara ABSRAK Meode fe dfferece ekspls adalah meode yag mudah dprogram dbadgka meode fe dfferece mplc aau meode-meode umerk laya. Sela u meode ekspls u dapa dguaka uuk meyelesaka persamaa paas hea equao) lear dalam sau dmes. Aka eap meode ekspls u mempuya sebuah kekuraga yau keerbaasa sablas dar peyelesaa umerk adalah saga kea. Oleh sebab u meode ekspls u dak lag ermasuk dafar meode-meode umerk yag hadal yag dapa dguaka uuk meyelesaka persamaa-persamaa dferesal parsal. Oleh karea u maka dusulka uuk megguaka aalss wavele Haar d dalam skema umerk dar meode ekspls uuk megaas kekuraga meode u yau keerbaasa sablas dega mejaga dskresas dar meode ekspls agar dak berubah. Kekuraga dar meode fe dfferece ekspls u sudah dapa daas secara sgfka oleh aalss Haar wavele yag dak mempegaruh logka meode asl yau meode ekspls. Kaa kuc: meode fe dfferece ekspls persamaa paas 1. INRODUCION here are may praccal suaos egeerg where he geomery or boudary codos are such ha a aalycal soluo has o bee obaed a all or f he soluo has bee developed volves such a comple seres soluo ha umercal evaluao becomes eceedgly dffcul. For such suaos he mos fruful ad ye prmve approach o he problem s oe based o fe dfferece echques. Fe dffereces are used o appromae dffereal cremes me ad space coordaes; ad he smaller we choose hese fe cremes he more closely he soluo whch s umercally appromaed. here are wo fudameal approaches fe dfferece; eplc ad mplc approaches. he eplc mehod seps ou me a fasho ha s smlar o Euler s echque for solvg ordary dffereal equaos. I has he advaage ha s smple o program bu uforuaely has he dsadvaage of a very srge sably. Sably of a umercal scheme for solvg a paral dffereal equao meas ha he error a ay sage of he compuao are o amplfed bu are EKNOIN Vol. 10 No. 1 Mare
2 aeuaed as he compuao progresses. I has bee prove [] ha f he spaal sep s halved order o mprove he appromao of he spaal secod dervave hea equao he me sep mus be quarered o maa he sably of eplc mehod oherwse he soluo wll oscllae or lead o a cosderably hgh usable scheme. Furhermore he compuao of each of hese me seps wll ake wce as log because resuls eghfold crease he umber of calculaos. hus he compuao burde may be large o aa accepable accuracy. he am of hs research s o roduce wavele aalyss he eplc fe dfferece scheme whch s appled for solvg oe-dmesoal hea equao order o allevae he sably requremes of he umercal soluo obaed from ha scheme.. BASIC HEORY he oe-dmesoal hea equao ca be epressed as follows: 1) he emperaure ) equao 1) s a fuco of space ad me ha s = ). hs fuco s epaded Haar scalg fuco ) ad Haar wavele fuco ) [3] he mulpled by Haar scalg fuco me h ) as follows I N ) h )[ ) 0 0 )] A me level +1/ he epaso of Haar scalg ad wavele fucos s I N 1/ 1/ 1/ 1/ 0 0 1/ ) h1/ )[ 1/ ) 1/ )] where h ) h k 4a) ) k 4b) ) k 4c) he oao I ad N represes he umber of grds space ad me resoluo) respecvely. Equao ) ad 3) are subsued equao 1). Equao ) represes he dervave me whle equao 3) s he spaal dervave ha s ) 3) 54 Muhammed e al. A Eplc Wavele-Based Fe Dfferece Scheme for Solvg Hea Equao
3 EKNOIN Vol. 10 No. 1 Mare / 5) he er produc of hs equao wh a weghed fuco w whch ca be epressed as follows )) ) / 1 h w s he egraed usg Galerk mehod [1]. he resulg equaos are 1/ 1/ 1/ 1/ 1/ 1/ 1/ 1/ 1/ 1/ 1 ) 6a) 1/ 1/ 1/ 1/ 1/ 1/ 1/ 1/ 1/ 1/ 1 ) 6b) emperaures appeared equaos 6a) ad 6b) are show Fgure 1. I s obvous ha he scheme s mplc me sep o +1/ ad eplc + 1/ o + 1. hs wll esure a ucodoally sable scheme eve wh fesmal me seps. For more deals abou he dscrezao of hs equao see [6] Fgure 1. Posos of emperaures appeared equaos 6a) ad 6b) 3. NUMERICAL RESULS he oe-dmesoal hea equao obaed from a wall subjeced o a emperaure of zero degrees cegrade o s boh sdes ad has a hckess of L see Fgure alog wh s boudary ad al codos ca be epressed as follows:
4 0) L) 0 ) 100 s 7) he eac soluo of hs equao 7) s ) 100e s ; 0 L 0 8) Fgure. A plae wall zero emperaure a he boudares Equao 8) represes he emperaure dsrbuo wh he wall a ay po wh he wall ad a ay me. he hea coduco rasferred hrough he wall ca be he evaluaed from Fourer s law [5]. Fgure 3 shows he plo of Equao 8) a hree me levels = ad 60 secods 56 Muhammed e al. A Eplc Wavele-Based Fe Dfferece Scheme for Solvg Hea Equao
5 Fgure 3. he emperaure dsrbuo obaed from Equao 8) alog wall hckess durg fve me levels = ad 60 secods Fgure 3 was obaed applyg he eac soluo Equao 8) usg me sep 15 secods sars a 0. he oal umber of daa pos s 50 ha s L/50) where L = 1 m ad he value of hea dffusvy = m /s). he same deals me sep = 15 sec ad specal sep = L/50 m was appled o oba he umercal soluo of Equao 7) applyg he umercal he wavele-based umercal scheme Equaos 6a) ad 6b)). he resul s show Fgure 4. Fgure 4. he soluo of Equao 7) usg wavele-based umercal scheme Equaos 6a) ad 6b) EKNOIN Vol. 10 No. 1 Mare
6 4. DISCUSSION I s oblvous ha he values of emperaures fgures 3) ad 4) are o he same. For sace a = 0 he mamum value of emperaure fgure 3) ha s he eac soluo s 100C occurs a L/ whle he mamum value of emperaure fgure 4) ha s he umercal soluo obaed from he wavelebased umercal soluo s 73C whch occurs a he same poso L/. hs dfferece emperaure values s udersood as he umercal error whch occurs as a resul of rucag he epaso of emperaure fuco wavele seres. By oher words s he dfferece bewee he dervave ad dfferece equao based o wavele coeffces. he wavele-based umercal scheme as epeced resuled sable soluo whch ca be compared o he eac soluo ye wh a accepable perceage of error. hs ucodoal sably of wavele-based scheme represes oe of he dscve feaures of wavele-based schemes based o s ably o copyg wh oleary ad large dscrezao meshes [3]. I wavele-based scheme as he me ad/or specal sep ges larger some of addoal bases wll be added o keep racg wh chage of he roue of curve ha mgh resul floag pos as he case of coveoal eplc scheme. As a resul he umercal soluo wll be uaffeced by he sudde chage of he me ad/or specal sep. 5. CONCLUSION We ow ge o hs cocluso ha he draw back of he eplc fe dfferece amely sably has bee overcome by roducg Haar wavele aalyss whch s essece dd o fluece he logcal approach of ha mehod erm of geg oe emperaure value he ew me sep from hree kow emperaure values obaed a he prevous me sep []. Raher wavele has mproved hs approach by self addg of ew emperaure values foud a he mddle po bewee he prevous ad e me seps fgure 1). he same ca be sad cocerg he specal seps. hs self addg of ew values ca be erpreed as ha wavele aalyss s bul o mul-resoluo aalyss [3] coe of epressg a fuco emperaure here wo coes levels of resoluo; he coarse level s obaed by he Haar scalg fuco ad he oher refg level s obaed by Haar wavele fuco. Wh hs he frs order of appromao by wavele epaso compared o he frs order of appromao by aylor epaso o whch he eplc fe dfferece approach s based [] wll resuls oe ukow emperaure value obaed from s kow values a shorer me sep. hose hree era kow values wll resul hgher accuracy ad sably scheme ha he aylor-based umercal scheme whch gves oe kow value of emperaure correspods o hree kow values a loger me sep. I s of course more cosumg me of compuao evaluag oal umber of emperaure values a oe me level ha half of ha umber he same level. hs s rue as log as he sably resrcos of he coveoal scheme whou waveles have bee sasfed. However whe he problem eeds more deals for he emperaure dsrbuo wh he physcal space he suao wll be dffere as hs wce as log as he me requred for 58 Muhammed e al. A Eplc Wavele-Based Fe Dfferece Scheme for Solvg Hea Equao
7 compuao wll be compesaed by he hgh sably addressed o wavelebased approach. REFERENCES [1] Amaruga K. ad Wllams J. R. 1994) Wavele-Galerk Soluos for Oe-Dmesoal Paral Dffereal Equaos Ieraoal Joural for Numercal Mehods Egeerg No [] Caraha B. Luher H. A. ad Wlkes J. O. 1969) Appled Numercal Mehods Wley New York. [3] Daubaches I. 199) e Lecures o Waveles SIAM Phladelpha Pesylvaa. [4] Duchaeau P. ad Zachma D. 1986) Paral Dffereal Equaos Schaum s Oule Seres McGraw-Hll New York [5] Holma J. P. 199) Hea rasfer 7h Edo McGraw-Hll New York. [5] Muhammed M. A. 005) A Eplc Wavele-Based Numercal Scheme for Solvg Oe ad wo-dmesoal Hea Equao Dsserao Gadjah Mada Uversy Yogyakara Idoesa. EKNOIN Vol. 10 No. 1 Mare
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