Newton-Product Integration for a Stefan Problem with Kinetics

Transcription

1 Joural of Sceces Islamc Republc of Ira (): 6 () versy of ehra ISS 64 hp://scecesuacr ewoproduc Iegrao for a Sefa Problem wh Kecs B BabayarRazlgh K Ivaz ad MR Mokharzadeh 3 Deparme of Mahemacs versy of abrz abrz Islamc Republc of Ira Research Ceer of Idusral Mahemacs of versy of abrz abrz Islamc Republc of Ira 3 Isue for Sudes heorecal Physcs ad Mahemacs ehra Islamc Republc of Ira Receved: Augus 9 / Revsed: December / Acceped: 8 February Absrac Sefa problem wh kecs s reduced o a sysem of olear Volerra egral equaos of secod kd ad ewo's mehod s appled o learze Produc egrao soluo of he lear form s foud ad suffce codos for covergece of he umercal mehod are gve A eample s provded o llusraed he applcably of he mehod Keywords: Sefa problem; Kec fuco; ewo's mehod; Produc egrao; Weakly sgular Volerra egral equaos Iroduco Cosder he followg modfed oephase Sefa problem oe spaal Varable u = u gu < < s ( ) () u =V ( ) = s () g( u ) = V ( ) = s () () = (3) u( ) u ( ) u( ) s he emperaure ad g ³ he dampg erm s due o volumerc hea losses he wo boudary codos deerme he problem ad make possble o fd he free boudary wh poso s () ad velocy V() = s &() Furher assume ha g () s mooocally decreasg dffereable fuco o [ ) wh g ' C ad sasfyg V g() v for some v V > (4) he free boudary problem () (3) arses aurally as a mahemacal model of a varey of eohermc phase raso ype processes such as codesed phase combuso [6] also kow as selfsusaed hghemperaure syhess or SHS [7] soldfcao wh udercoolg [] laser duced evaporao [4] rapd crysallzao h flms [9] ec hese processes are characerzed by produco of hea a he erface ad her dyamcs s deermed by he feedback mechasm bewee he hea release due o he kecs gu ( = ) ad he hea dsspao by he medum s () he frs boudary codo () (he Sefa boudary codo) epresses he balace bewee he hea Correspodg auhor el: 98(4) Fa: 98(4) Emal: vaz@abrzuacr

2 Vol o Wer BabayarRazlgh e al J Sc I R Ira produced a he free boudary ad he hea dffuso hrough he adace medum Problem () (3) descrbes propagao of he phase raso fro he secod boudary codo () s a mafesao of he oequlbrum aure of he raso; ad s aalog for he classcal Sefa problem s us u = s () = I he coe of codesed phase combuso kec boudary codo epresses depedece of propagao velocy o he flame fro emperaure he res of he paper s orgazed as follows I Seco a local esece codo s obaed I Seco he Sefa problem wh kecs reduced o a sysem of olear Volerra egral equaos of he secod kd ad ewo's mehod s appled o learze A covergece aalyss of ewo s mehod for he problem s provded he subsecos of Seco Produc egrao soluo of he lear form s obaed seco 3 Covergece of produc egrao mehod s gve Subseco 3 Fally seco 4 umercal resuls of es problem solved by he proposed mehod s repored Esece of Local Classcal Soluo I order o o cluer formulas wh facors of he ype e g from ow o ul seco 4 we se he dampg coeffce g = he modfcaos o he g > case are rval A shorme soluo of he free boudary problem ()(3) wll be sough he form of a superposo of hea poeals u( ) = G( () s )() d G u d ( ) ( ) () G s he fudameal soluo of he hea equao é ( ) G( ) = epê 4 p( ) ë 4( ) [ ] (6) he desy of he sgle layer poeal ad he fro poso s () are o be deermed Frakel ad Royburd Ref [3] show ha he sgle layer poeal s couous up o he boudary ad s dervave possesses he sadard ump propery lm G ( s ( ) ) ( ) d = s( ) () G ( ( ) ( ) ) ( ) s s d (7) hs resul s of course wellkow f s couous I urs ou however ha by he aure of he free boudary problem a had mus have a sgulary a = hus a usfcao of (7) wll requre a era effor If he ump propery (7) holds he for he soluo represeed by () he boudary codo () yeld he followg equaos u s g V ( ( ) ) = ( ( )) = G ( s ( ) s ( ) ) ( ) d G( s( ) ) u ( ) d (8) () u(()) s = V () = G ( s ( ) s ( ) ) ( ) d G( s( ) ) u ( ) d (9) sasfes he balace codo u () lm () = p We ca rewre he egral equaos (8)(9) erms of ad V [3]: V = K ( V ) () = K ( V ) K ( V ) () he olear operaors K ad K are defed as follows { K ( V ) = g G( s( ) s( ) ) ( ) d } G( s( ) ) u ( ) d () { K ( V ) = G ( s( ) s( ) ) ( ) d } G( s( ) ) u ( ) d (3) here as usual s( ) = V ( ) d (4) he equaos are supplemeed by he al codos

3 ewoproduc Iegrao for a Sefa Problem wh Kecs u () V () = g( u ())lm ( ) = () p he proof of he followg heorem s gve [3] heorem Le g < be couously dffereable moooe decreasg fuco u ÎC( ] u > he he problem ()() has a uque soluo V such ha V ad () are couous o [ s ] for some s > s depeds oly o Supu he soluo o he free boudary problem s deermed by V va he represeao () wh s( ) = V ( ) d Applcao of he ewo's Mehod ow we apply a ewo's mehod o learze of he problem ()() For hs purpose we ake V : = ìï í é : V () C[ s] ê Î ï ý ïîë ïþ wh he orm = sup ( ) s ê ë s { PVPC[ s] P P s} = ma We kow ha s a Baach space wh above orm Iroducg a operaor : hrough he formula éf ( V ) ; ê = ê Î f ( V ) ê ë ë ë (6) f ( V ) = V K ( V ) (7) f ( V ) = K ( V ) K ( V ) (8) he problem ()() ca be wre he form é ê = ê ë ë We suppose ha ( ) (9) V s he eac soluo of (9) he by aylor epaso of wo varables fucos f ad ˆ V ˆ suffcely close o ( ) V we have f a ( ) = f( V ) ˆ ˆ = f ˆ ˆ ˆ ( V ) ( V )( V V ) ˆ ( ˆ)( ˆ) ( ˆ V V )! ë ( ˆ ) f ˆ ˆ ( V qh qk) () Where h = V Vˆ k = ˆ ad q Î () Ad f V = ( ) ˆ ˆ = f ˆ ˆ ˆ ( V ) ( V )( V V ) ˆ ( ˆ)( ˆ) ( ˆ V V )! ë ( ˆ) f ˆ ˆ ( V qh qk) q Î () () We appromae above equaos by elmag O( h k ): s æ ˆ ( ˆ) ( ˆ ö ç V V ˆ) V h æf ˆ ˆ ( V ) ö ç æ ö ç f ˆ ˆ f ˆ ˆ ç k ; () ˆ ˆ ( V ) ( V ) è ø ç f ( V ) ç è ø è ø Hece he ewo's mehod for fdg roo of (9) æ ö ( V ) ( V ) ç V d ç æ ö ç ç g ( V ) ( V ) è ø ç è ø æf ( V ) ö ç èf ( V ) ø = = (3) d : = V V g : = (4) For sarg umercal process s suffce o evaluae elemes of coeffce mar = 3

4 Vol o Wer BabayarRazlgh e al J Sc I R Ira ( ) lm ( ) h V u = h f V hu ] [ f ( V ) = lm h V hu h K ( V hu ) V K ( V ) K = u V u ( ) K V u h [ K V hu ( ) = lm ( ) h K( V ) = lm h gé Gs (() êë [ ] h { hs( ) s( ) hs( ) ) ( ) d G( s() hs() ) u ( ) d g é G( s( ) s( ) ) ( ) d êë G( s( ) ) u ( ) d ' Da() = g ( a( ))lm h h { [ ' = g ( a( )) G ( s( ) s( ) ) ( ) { s( ) G( s( ) s( ) s ) ( ) ( ) d () (() ) ( ) s G s u d ' g G s s = ( a( )) ( ( ) ( ) ) [ ] s( ) s( ) ( ) d () G(() s ) u ( ) d s s( ) = u( ) d a() = G(()() s s )() d G s u d ( ( ) ) ( ) } ] } } D a() = G(() s hs() s() hs () u d a ) () d G(() s hs() ) ( ) ( ) ad oe ha ( G G )( ) = () Smlarly we oba K ( V ) u = ( V ) u ( ) K V u = ( V ) u K ( V ) u K ( V ) u = u ( V ) u K ( V ) u K V u g G s s u d ' ( ) = ( a ( )) ( ( ) ( ) ) ( ) K ( V ) u = G ( s ( ) s ( ) ) ( ) d ( s() s()) d s() G ( s() ) u ( ) d K V u = G s s u d ( ) ( ( ) ( ) ) ( ) Subsuo above resuls (3) gves [ ] d ( ) g ( ) d ( ) d d ( ) g ( ) d [ ] g ( ) g ( ) d = r ( ) [ ] [ ] 3 [ ] 4 g () g () d () d d ( ) g ( ) d [ ] g ( ) g ( ) d = r ( ) [ ] [ ] 6 (6) (7) 4

5 ewoproduc Iegrao for a Sefa Problem wh Kecs [ ] ' g ( ) = g ( a ( )) G ( s ( ) ) u ( ) d g ( ) =g ( a ( )) G ( s ( ) s ( ) ) [ ] ' ( ) d [ ] ' 3 ( ) = ( a( )) ( ( ) ( ) ) g g G s s [ r ] ( ) = f ( V ) = [ ] ' 4 = é ë g () g ( a ()) G ( s () ) G ( s( ) ) u ( ) d [ ] ' éë g ( ) = g ( a ( )) G ( s ( ) s ( ) ) G ( s ( ) s ( ) ) ( ) d [ ] ' 6 ( ) = ( a( )) ( ( ) ( ) ) g g G s s G ( s ( ) s ( ) ) Equaos (6)(7) yeld he followg lear sysem () = () [ ] [ ] F [ ] K [ ] d ( ) ( ) () [ ] éd ( ) = ê g() ë (8) : ë V V é ë = ë é () = é ( V ) ( V ) é V ê ê ê () (3) ê ê ( V ) ( V ) ë êë g ( ab ) = ê ( ) êë g ( a) b q a = a( V ; ): = G( s( ) s( ) ) ( ) d G( s( ) ) u ( ) d b = b( V V ; ): = G ( s( ) s( ) ) ( )( s ( ) s ( )) d s( ) G ( s( ) ) u ( ) d G ( s ( ) s ( ) ) ( ) d Sce G = G G = G so we ca wre q = q( V V ; ) = F K ér () ( ) = ê ë () [ ] r æg () g ( ) g ( ) ö ( ) =ç è g () g ( ) g ( ) ø [ ] [ ] [ ] [ ] 3 [ ] [ ] [ ] 4 6 Covergece of ewo s Mehod I hs Subseco covergece of ewo's mehod wll be proved We ca rewre ()() as a operaor equao o Baach space (see begg of Seco ): = ë é ë (9) s defed by (6) For a arbrary Î ë Î s ca be show ha: ad [ ] G ( s( ) s( ) ) ( )( s ( ) s ( )) d s ( ) G ( S( ) ) u ( ) d G ( s( ) s( ) ) ( ) d s ( ) = V ( ) d s ( ) = V ( ) d ow we prove ha for every ê ê Î êë êë æé V ö ç ê ç ê ç ê èë ø here es L > such ha: V V L é é ê ê ê ê êë êë êë êë (3)

6 Vol o Wer BabayarRazlgh e al J Sc I R Ira æé V ö ç ê ç ê ç ê èë ø s a eghborhood of eac soluo ê Suppose ha for every ê ê Î we defe ê êë ê ê ë ë L º ê ê êë êë For arbrary ê Î ê ë s suffce show ha ad ad L ê < L ê ê êë êë êë For [ s ] ê = ê ë Î deoe a = a( V ; ) b = b( V V ; ) q = q( V V ; ) a = a( V ) b = b( V V ; ) q = q( V V ; ) So we ca wre ég ( a ) b g ( ab ) % L ê = ê º ê ê ê g ( a) b q g ( a ) b q ê ë ë ë % We defe he orm { s } (3) (33) V% : = ma V% : Î[ ] (3) cé s êë { Î } % s : = ma % : [ s] (36) Sce g s bouded ad Lpschz couous hese meas $ C g ( ) C [ ) g > " Î g $ L > " y Î [ ) g g ( ) g ( y) L y g ad hece (37) (38) V% = g ( a ) b g ( a) b = g ( a )( b b) ( g ( a )g ( a)) b (39) C b b L a a b g g ow we evaluae some upper bouds for b b a a By he mea value heorem ad og h h ha for all h > we have e he ad h he e b b C ( ) V V (4) æ p /4 ö C( ) = e ç u è 8 ø ep (4) a a C ( ) ê ê ê ë ê ë (4) C ( ) = p u ep s e (43) ad he las erm s q q C () 3 ê ê êë êë (44) C3( ) = p s u (4) p By (39)(4) ad (4) { s } V% = ma V% : Î[ ] C[ s] C ( s) 4 ê ê êë êë C4() = C C() L C() b Smlar g g evaluao lead { } % s = ma % : Î[ s] s[ C ( s) C ( s)] 4 3 ê ê êë êë 6

7 ewoproduc Iegrao for a Sefa Problem wh Kecs By roduce L = L ( s) = L he blow form (33) holds { 4( s) s [ 4( s) 3( s) ]} Ma C C C (46) hus hypohess of followg heorem s sasfy heorem Le X ad Y be wo Baach spaces ad operaor : X Y be Freche dffereable Assume s a roo of ( ) = such ha [ ( )] ess ad s a couous lear map from X o Y Assume furher ha ( ) s locally Lpschz couous a = ( ) ( y) L y " y Î ( ) (47) ( ) s a eghborhood of ad L > s a cosa he here ess a d > such ha f d he ewo's sequece { } s well defed ad coverges o Furhermore for some cosa M we have he followg error bouds M æ ç ) Md è M Proof see [] pages 6 3 Produc Iegrao Mehod (48) (49) I Eq (8) g 3 ad g 6 are weakly sgular kerels he followg form ( ) ( ) ( ) g 7 3 = p g () p( ) : = ì ( s ( ) s ( ) ï ) ï g ( ) : = g ( a( ) ) ep 7 í ý 4p 4( ) ( ) ( ) ( ) 6 8 () () ï ï î þ g = p g (3) g8 ( ): = [ g ( a()) 4p ì( s( ) s( )) ep í ý î 4( ) þ s( ) s( ) ] (4) ow we wa o solve he weakly sgular egral equao (8) by he mehod s descrbed [8] hs mehod allows us o overcome he dffculy caused by he poor behavor of he soluo () a he al po = Gve a relavely shor erval [ b ] we frs solve he problem () = F () K( ) ( ) d Î[ b] () by a ysrom ype mehod based upo a whole erval produc egrao rule of erpolao ype wh egraes eacly he kerel p ( ) Afer he al erval he bad behavor of he dervave of s of less sgfcace We he solve he problem wh () = () bk ( ) ( ) d Î [ b ) (6) b ( ) = F( ) k( ) ( ) d (7) by a sadard sepbysep mehod for regular soluos Sce he compuao of () depeds o he sarg Î b he wo mehods appromao of () [ ] have o be regarded as pared ow we descrbe he ysromype mehod used o solve equao () umercally We ca rewre () as: {[ ] () = F() K () K ( ) d () 3 4 p( ) K ( ) g ( ) } d [ b] K 3() = ê 4 é g () g ë () é g () K 4 () = ê g ë () é g 7() K () = ê 8 g ë () Havg chose Î (8) = dsc pos { } he erval [ b ] we collocae he equao (8) a he 7

8 Vol o Wer BabayarRazlgh e al J Sc I R Ira odes { } : = = F [ K K ] ( ) ( ) { ( ) ( ) d ( ) 3 4 p( ) K ( ) g( )} d (9) = hus we use he Lagrage erpolao polyomal L ( f ; ) = å l ( ) f ( ) (6) = o appromae d ( ) K ( g ) ( ) ad oba followg mehod: or 3 4 = F( ) {[ K ( ) K ( )] å = l ( d ) p( ) å l ( ) K ( g ) } d = () () = F ( ) å ( w w ) d J = å w g = = (3) () 3 (6) w = k ( ) l ( ) d (6) () 4 w = k ( ) l ( ) d (63) (3) w = k ( ) p( ) l ( ) d (64) éd = ê ê g ë (6) () (3) o cosruc he coeffces w ad w we use a Mahemaca sofware Ad we use for () w he Gaussa egrao By solvg he lear sysem (9) we oba () as a ysrom appromao for (): = () () = F() å ( w () () (3) = w ()) d å ( w () g () 3 (66) w ( ) = k ( ) l ( ) d (67) () 4 w ( ) = k ( ) l ( ) d (68) w (3) ( ) k( ) p( ) l ( ) d = (69) ow we are ready o gve he covergece of produc egrao mehod 3 Covergece of Produc Iegrao for Solvg Sysem of Weakly Sgular Iegral Equaos I our covergece aalyss we eame he lear es equao: ( ) = F( ) p( ) ( ) d Î[ ] (7) éd() ér () () = ê F () = ê êg() êr () ë ë Ad assume ha he forcg fuco g Î C[ ] Ad p s defed by () he he es equao (7) has a uque soluo Î C[ ] C[ ] ha may be epeced o have ubouded dervaves a he ed po = If for a gve mesh { } = we apply he mehod of Seco 3 o he es equao (7) ad oba as appromae soluo () he followg ysrom erpola: = ( ) = F( ) å w ( p; ) ( ) (7) = w ( p ; ) p ( ) l ( ) d I order o eame he uform covergece of he appromae soluo () o he eac soluo () of (7) oce ha by =å = () () w ( p ;) { } ( ) ( ) ( p ) (7) Where ( p ) s he local rucae error defed = å = ( p ) p( ) ( ) d w ( p; ) ( ) (73) 8

9 ewoproduc Iegrao for a Sefa Problem wh Kecs Hece we oba ( ) I A (74) A s he lear operaor defed by A : X : = C[ ] C[ ] X A ( ) = å w ( p; ) ( ) ÎX Î[ ] (7) = Frs we vesgae he covergece properes of he uderlg produc quadraure rule Lemma Le { p } = s a sequece of orhogoal polyomals o [] wh respec wegh fuco w ( ) he { q } = s a sequece of orhogoal polyomals o [ ab ] wh respec wegh fuco w% () b a q( ) = p( [ ]) Î[ ab ] b a (76) b a w% ( ) = w( [ ]) Î[ ab ] (77) b a Proof he proof of hs lemma s easy ad refer ha he reader verfy heorem 3 Le { } = be he zeros of he ( ) sdegree member of a se of polyomals ha are orhogoal o [ ] wh respec o he wegh fuco a b w( ) = u( )( ) ( ) 3 < a b > (78) Here u () s posve ad couous [ ] ad he modulus of couy of u sasfes ( u ) d d d < Le L ( ;) deoe he vecor of d erpolag polyomal of degree ha cocdes wh he vecor fuco ( ) = ( d( ) g( )) a he odes { } = wh he for every vecor fuco s ()() Î X : = C[ ] C[ ] s > (o a eger) here holds lm ( p ) = (79) I parcular we have he bouds s { } æ ö O ( ) ç = Log ( ) è ø Proof oe ha for all Î X : = C[ ] C[ ] ( ) = ( u ( ) u ( )) { } (8) = ma u u (8) Ad apply relaos (4)() heorem of [8] he vecor case Apply Lemma for balace of erval of orhogoaly he boud (8) s a mmedae cosequece of heorem [] ow we vesgae he behavor of he frs erm ( I A) he rgh had sde of (74) heorem 4 Le he operaor ad he odes { } = A be defed as (7) chose as heorem 3 he for all suffcely large here es a cosa C > depede of such ha ( I A) C (8) Proof Codos of lemmas of [8] are sasfy ad hece by heorem of [8] he resul arrve heorem Le be he eac soluo of he equao (7) Le be he appromae soluo obaed by dscrezg he egral erm of (7) by a produc quadraure rule of erpolaory ype cosruced o a se of dsc odes { } If he odes = { } = are he zeros of he ( ) sdegree member of a se of polyomals he wegh fuco (78) wh 3 < ab < he coverges uformly o Moreover he rae of covergece of o cocdes wh he oe of he basc quadraure rule we choose o appromae he egral erm of (7) he proof follows mmedaely from he esmae (74) ogeher wh heorems 3 ad 4 he boud (8) supply a esmae of he rae of covergece 9

10 Vol o Wer BabayarRazlgh e al J Sc I R Ira v 4 umercal Eamples g Subsug v = e u u = u gu yelds v = sce g g g v = g e u e u = e ( gu u ) = v hece we ca pu g = ow cosder he followg es problem u = u < < s ( ) (83) u ( ) = s () = V () (84) u( ) = ep( a) < < a > (8) g( u ) ( ) = s( ) = V (86) g( ) = ep( )a (87) e I s o dffcul o verfy ha for a > he fucos s ( ) au ( ) = ep( a a) are a eac soluo of es problem Whou loss of geeraly we ca suppose a = For hs problem we ca wre (6) he form of lear sysem ( ) R ( ) ( ) ( ) R AX = b A = a Î X bî For a arbrary { } = a g ( ) l () d g ( ) l ( ) d = a g ( ) l () d Î we have g ( ) l ( ) d ¹ a = g ( ) 7 p( ) l ( ) d = 4 a g ( ) l () d g ( ) l ( ) d a = g ( ) 8 p( ) l ( ) d a = g ( ) 8 p( ) l ( ) d ¹ ( ) X = d d g g ( ) b = r ( ) r ( ) r ( ) r ( ) ow we evaluae he eac for he es problem correspodg o a = For hs problem we have u( ) = ep( ) s( ) = u = ( ) ep( ) Subsug hs values () eds o ì ( ) p ep( ) = ep í ý î 4( ) þ ì ( ) ( ) d ep í ýd (89) î 4 þ For = s () (89) we have ì ( ) p ep í ýd î 4 þ ì( ) = ep í ý ( ) d (9) î 4 þ If we defe æ ö m( ): = ep ç è 4 ø he we ca wre ( m )( ) = p æ erf æ öö ç ç è è øø (88) (9) (9) ow we ake Laplace rasform from (9) ad oba { } { } L ( ) = 4 s (93) s Ad hece we oba he eac for our problem /4 ì ï e æ öï ( ) = í erf p ç ý ïî è øïþ (94) 6

11 ewoproduc Iegrao for a Sefa Problem wh Kecs 4 Dscusso ad Cocluso I he es problem we se b = = 3 he odal pos are zeros of q ( ) = 7b 6b 94b b 77b 8484b b b { q } = are orhogoal wh respec w ( ) = o Î [ b] Suppose y() = () he y s couous [3] ad (94) yeld /4 ïì æ öï e y ( ) = í erf ç ý ïî è øïþ p able Crude daa V ( ) V % ( ) y( ) y% ( ) able Daa afer oe sep of ewo's mehod V ( ) V % ( ) y( ) y% ( ) (9) For hs problem he eac value of V s V ( ) = ow wh al guess y( ) = (96) p V ( ) = (97) he absolue errors of he soluos usg orgal daa a pos = = are gve able able gves he same quaes usg V % ad y% sep appromaed values of V ad y respecvely Frs of all able shows he precso of he mehod s cosderable so ca be appled o may praccal problems Secodly we show able ha furher mprovemes precso are possble by usg beer appromae values for V ad y herefore hs mehod ca be appled o wde rage of problems dffere applcaos Ackowledgmes he auhors are hakful o he referees for her valuable suggesos o mprove hs paper Refereces Akso K ad Ha W heorecal umercal aalyss: A fucoal aalyss framework Sprger () Crscuolo G Masroa G ad Moegao G Covergece properes of a class of produc formulas for weakly sgular egral equaos Mah Comp : 3 3 MR 9m:63 (99) 3 Frakel M ad Royburd V Compac aracors for a Sefa problem wh kecs EJDE : 7 () 4 Golberg S M ad rbelsk M I O laser duced evaporao of olear absorbg Meda Zh ekh Fz (Sov PhysJ ech Phys) : (98) Lager J S Lecures he heory of paer formao : Chace ad Maer J Soulee J Vameus ad R Sora eds Elsever Scece Publshers (987) 6 Makowsky B J ad Svashsky G I Propagao of a pulsag reaco fro sold fuel combuso SIAM J Appl Mah 3: 3 (978) 7 Mur Z A ad Aselmeambur Selfpropagao eohermc reacos: he syhess of hghemperaure maerals by combuso Maer Sc Rep 3: 7736 (989) 8 Ors A P Produc egrao for Volerra egral equaos of he secod kd wh weakly sgular kerels Mah Comp 6: (996) 9 Weeks J ad Va Saarloos W Surface udulaos eplosve crysallzao: a olear aalyss of a hermal sably Physca D : 7994 (984) 6