GENERALIZED METHOD OF LIE-ALGEBRAIC DISCRETE APPROXIMATIONS FOR SOLVING CAUCHY PROBLEMS WITH EVOLUTION EQUATION

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1 Joural of Appled Maemacs ad ompuaoal Mecacs 24 3( GENERALIZED METHOD OF LIE-ALGEBRAI DISRETE APPROXIMATIONS FOR SOLVING AUHY PROBLEMS WITH EVOLUTION EQUATION Arkad Kdybaluk Iva Frako Naoal Uversy of Lvv Lvv Ukrae Absrac We cosder solvg e aucy problem w a absrac lear evoluo euao by meas of e Geeralzed Meod of Le-algebrac dscree appromaos Dscrezao of e euao s performed by all varables euao ad leads o a facoral rae of covergece f Lagrage erpolao s used for buldg uas represeao of dffereal operaor Te rak of a fe dmesoal operaor ad appromao properes ave bee deermed Error esmaos ad e facoral rae of covergece ave bee proved Keywords: geeralzed meod of Le-algebrac dscree appromaos dyamcal sysems evoluo euao fe dmesoal uas represeao appromaos Lagrage s polyomal facoral covergece Iroduco A wde varey of mpora evromeal processes problems of ecues pyscs are descrbed by ordary dffereal euaos (ODE ad paral dffereal euaos (PDE Regardless of powerful maemacal ools very few of em ca be solved eacly Terefore ere s a eed for applcao of eer e umercal meods or e aalycal-umerc meods Oe of ese approaces s e meod of Le-algebrac dscree appromaos [] Ts meod was used by alogero 983 for solvg e egevalue problem cocerg dffereal euaos for specral problem [2] Le algebras ad er fe dmesoal uas represeaos appeared o be very useful e meod devsed Eeso of s meod for solvg PDE was proposed 988 ad was amed as Le-algebrac dscree appromao [] Some sory of developme of Le-algebrac meod ca be foud [3 4] Te ma problem aalyzed ese arcles s e aucy problem for lear evoluo euao [3]:

2 52 A Kdybaluk fd fuco u = u( 2 suc a u = K( ; u + f ( Ω R > u = ϕ( B = ( were B deoes some fucoal Baac space Smlarly as alogero s meod e Heseberg-Weyl algebra G = / as bee used as a basc algebrac ool for cosrucg e j= { } j j correspodg dscree appromaos j X Z I R ( ( ( j j Usg -dmesoal Lagrage erpolao sceme problem ( s reduced o e aucy problem e followg form: j= fd fuco u( = u( ( suc a du( = K( u( + f( > d u( = ϕ ( B = ( (2 were K ( deoes fe dmesoal uas represeao of dffereal operaork ad B ( deoes fe dmesoal space of appromaos Sysem (2 s solved by meas of Euler s or Ruge-Kua s meod [3] Sce reduced problem (2 s solved makg use of some umercal algorm e rae of me covergece s cosraed by e covergece rae of e meod based o Le-algebrac dscree appromaos for spaal varables rae of covergece s facoral [-3] By meas of e Geeralzed Meod of Le-algebrac dscree appromaos proposed s paper covergece rae for e me varable becomes facoral Moreover we cosruc a umercal sceme ad prove e covergece of e proposed geeralzao We frs llusrae e meod avg appled o oe dmesoal case ad laer we cosruc s geeralzao for e muldmesoal case aucy problem ad operaor euao problem formulao QT We cosder a bouded doma Ω : = ( a b R me lm T < + ad cylder = Ω ( T ] We assume a lear dffereal operaor K s a formal polyomal of elemes from e Le algebra { / } ad ca be represeed as

3 Geeralzed meod of Le-algebrac dscree appromaos for solvg aucy problems 53 K = a + a + + a k p k k p p were a R for all = p ad a k ad k We ake e Baac spaces e aucy problem V = Q Q = Q ad formulae p ( T ( T ( T fd fuco u = u( V suc a u = Ku + f ( QT u = ϕ V = (3 were ϕ = ϕ( V deoes al codos ad f = f ( represes eral sources Accordg o [5] we roduce subsuo u( = v( + ϕ( o (3 wc leads o cosderg e aulary aucy problem w omogeeous al codo fd fuco v = v( V suc a v = Kv + Kϕ + f ( QT v = = (4 Te soluo of problem (4 we seek e subspace of suc fucos wc are B = v V : v = = Deog e srucure elemes (4 by omogeeous a e al mome of me: { } A : = / K f ɶ = Kϕ + f ( Q T (5 we oba a problem for operaor euao: for gve operaor A : B ad eleme fɶ fd eleme v B suc a Av = f ɶ (6 Te aucy problem as bee reduced o e problem for e operaor euao Ts operaor euao we ed o solve by meas of e Geeralzed Meod of Le-algebrac dscree appromaos ad prove s covergece

4 54 A Kdybaluk 2 Numercal sceme ad uue of dscree soluo Le deoe e cou of odes doma Ω ad deoes cou of odes erval [T ] Se of odes we deoe Q T For every varable we cosruc a se of Lagrage polyomals wc sasfy propery lj( = δj ad lj( = δj were δ j deoes Kroecker symbol Accordg o e Weersrass appromao eorem e se of polyomals w real-valued coeffces s dese se e space of couous real-valued fucos oosg l( { l j( } j= 2 = we oba sysem of polyomals wou polyomal assocaed w al mome of me Is easy verfes a j = 2 l ( = ad l( B moreover bass fucos l( l( B are j = learly depede ece sysem of ese fucos creae bass for appromao spaces B B Tus we seek e soluo as a Lagrage erpolao e followg form ( (7 v ( = v l ( l ( = v l( l( j j j j= 2 j= were deoes e dscrezao parameer j ad j are dees of odes by correspodg varables j deoes e uue umber of e ode ( = + ad v deoes e se of values v { vj} j= + j j j = f ɶ ad furer usg of calcula- Subsuo (7 o euao (6 leads o os yelds Av ( ( = l ( l( l( K l( v = f ɶ (8 = (8 we oba a sysem of lear algebrac eua- Takg os = ad 2 ( Z I I K v = f ( = = 2 ɶ (9 Deog A = Z I I K ad f ɶ = f ɶ ( 2 = = we oba dscree formulao of operaor problem for gve operaor A : B ad eleme fɶ fd eleme v B suc a A v = fɶ (

5 Geeralzed meod of Le-algebrac dscree appromaos for solvg aucy problems 55 were marces of correspodg fe dmesoal uas represeaos ave bee bul upo ese rules Z = l ( K = ( Kl ( I = l ( I = l ( j j j j j j j j Accordg o eorem deermg e rak of fe dmesoal uas represeaos [4] we oba ( ( ( ( rak Z = rak I = rak K = k rak I = Usg propery of esor produc we verfy a ( = ( ( = ( ( rak Z I rak I K k Te rak of wole mar A remas a ope ueso ad furer lemmas gve a aswer o s ueso Z K s lpoe Lemma Mar ( Proof Sce K s fe dmesoal uas represeao of operaor k p d d d Kɶ = ak + a k + + a p as e form K a Z a Z a Z p d d d ad mar K s lpoe ece ( m N m : K = k k k p = k + + k p Te propery of esor produc [6] ( N A B = A B yelds ( ( ( m N m : Z K = Z K = ece mar ( Z K s lpoe Lemma 2 Mar ( I I Z K ( e as full rak as a verse mar ad s rak s Proof Le us rewre mar ( I I Z K as a formal seres: ( + ( + ( I I Z K = Z K = Z ( K Sce marces K ( Z K = = are lpoe (Lemma we oba a e verse mar ess because of e esece of fe epaso m m ( I I Z K = ( Z K = ( Z ( K = =

6 56 A Kdybaluk However mar as ( rows ad colums ad as verse mar erefore as full rak: rak( I I Z K ( = Usg ese lemmas we ca prove e e eorem Teorem Te rak of fe dmesoal uas represeao A of operaor A ad ere ess a uue soluo of dscree as full rak ad s rak s ( problem ( Proof Le us rewre A ( ( Z I I K Z I I I Z K However rak( Z I = ( ad due o Lemma 2 rak( I I Z K = = = = ( usg propery for wo suare marces : ( = m { rak( A rak( B } yelds rak( Z I I K ( A B rak AB = = Sce mar as full rak e a uue soluo of e problem ( ess ere 3 Appromao properes of umercal sceme Accordg o cosruco of fe dmesoal uas represeao of e Av A v = Av( M AvI( M were M = v M I operaor ca be verfed a ( ( deoes Lagrage erpola ad M ( = deoes ode from Q T Le e dmeso of fe dmesoal subspaces B be dm B= dm = N I ese spaces we ca defe e orm a smlar way as was proposed [7] amely B N v = v = v N j = Assume a a v W { v : QT R : D v L ( QT a } = wc meas a all possble dervaves ll order are bouded Te resdual of Lagrage erpolao polyomal ca be wre e followg form [8]: ω ( v( ξ ω ( v( η ω ( ω ( y v( ξ η v( v ( = + + I (! (! (! (! were ω ( = ( ω ( y = ( yy y ad ξ ξ η η y j= j= 2 j Ω ( T

7 Geeralzed meod of Le-algebrac dscree appromaos for solvg aucy problems 57 Le us deoe α e ges order of dervave operaor by varable ad a α deoes coeffce sadg w ges order of dervave by varable Teorem 2 Fe dmesoal uas represeao A of e operaor A appromaes e operaor A o eleme v B ad error esmao of appromao as e followg form: α v l( l( α k= Av A v + a k α v ( Proof Sce e orm of space s vecor orm e accordg o e cosruco of fe uas represeao A of operaor A ca verfed a were I ( ( ( ( Av A v Av M Av I M M = M = v deoes Lagrage erpola ad M ( ( = deoes ode from Q T Usg e defo of e orm space afer calculaos we oba ( Av A v Av Av = A v v I I Acg w operaor A o resdual of Lagrage polyomal we oba Esmao ω ( α ω ( ( v ω v A( v( vi ( aα +!! ( ( α ( (!l( ( (! ( l( ω α k k= α ad v W ( Q B yelds α v I l( l( α k= α v Av Av + a k ad fally ( ca be obaed (

8 58 A Kdybaluk 4 overgece ad error esmaos Accordg o e Kaorovc covergece eorem [9] of absrac appromao sceme lm v v = olds f B ere ess a uue soluo of euao Av= f ɶ 2 for all operaors appromag A operaor A es verse bouded operaors 3 operaor appromaes operaor A o eleme v B : lm Av A v = Te frs reureme ca be easly verfed ad e rd reureme as bee already sasfed Teorem 2 us we sould prove e secod reureme Teorem 3 If fe dmesoal uas represeao A of operaor A as a full rak ad s e same as fe dmesoal subspace B e e bouded verse operaor ess e: A M > A : A M < + (2 Proof Aloug e orm sasfes e aom of posvy we oba A v v D( A ad A v = A v = v s possble f de A= ad Le A v = e A v= For raka< dm B However A : raka = oly Tus v D( A \ { B } : A v v > are srcly posve for v D A { B } v B B ( \ e ere ess suc cosa µ> suc a B = dm B e A v= s possble f > Sce values A v > ad A v µ v B Accordg o e eorem of esece of bouded verse operaor [] seg M = > we oba (2 µ Teorem 4 If lm ɶ ɶ = olds ad codos of eorems 2 3 are f f sasfed e lm v v = ad B α v v v M l( + a ( l( k B α k = α v (3

9 Geeralzed meod of Le-algebrac dscree appromaos for solvg aucy problems 59 Proof Le us cosder v v : B ( ( v v = A A v v A A v v B B Sce e verse operaor s bouded (2 e Le us esmae A ( v v ( ( A A v v M A v v : ( A v v = A v Av + Av A v A v Av + Av A v Sce Av= f ɶ ad A v= f ɶ we oba ( A v v A v Av + fɶ fɶ Fally we oba v v M( A B v Av + fɶ fɶ ad v v M ( A B v Av fɶ fɶ Tus lmv v = However lm lm + lm = B as e followg form: Sce ( ad fɶ s Lagrage appromao of f ɶ error esmao of e eglecg erms w fɶ fɶ fɶ fɶ + B eds o zero faser a ad fɶ fɶ B we yelds o error esmao (3 5 Te aucy problem for evoluo euao several dmesos Te resuls obaed prevous secos ca be geeralzed aural way for a muldmesoal case Le us cosder -dmesoal bouded doma

10 6 A Kdybaluk Ω : = ( a b ( a b ( a b R me lm T < + ad cylder 2 2 QT = Ω ( T ] Le a b We assume a lear dffereal operaor K s formal polyomal of elemes from Le alge- bra { j / j } = damω deoe e leg of e rage ( [3 ] Le α deoe e ges dervave by varable ad a α deoe coeffce sadg by e ges dervave by varable α αd Le us cosder Baac spaces V = ( ( ( d QT QT = QT ad formulao of e aucy problem w lear evoluo euaos [] s gve below fd fuco u = u( V suc a u = Ku + f ( QT u = ϕ V = (4 were ϕ = ϕ( d V deoes al codos ad f = f ( represes eral sources Accordg o [5] we roduce subsuo u( = v( + ϕ( o (4 wc leads o cosderg a aulary aucy problem w omogeeous al codo: fd fuco v = v( V suc a v = Kv + Kϕ + f ( QT v = = (5 Te soluo of problem (5 we seek e subspace of suc fucos wc are omogeeous a al mome of me: B { v V : v = } = = Deog srucure elemes (4 by A : = / K f ɶ = Kϕ + f ( Q T (6 we oba e problem for operaor euao: for gve operaor A : B ad eleme fɶ fd eleme v B suc a Av = fɶ (7

11 Geeralzed meod of Le-algebrac dscree appromaos for solvg aucy problems 6 Te aucy problem as bee reduced o problem for operaor euao Ts operaor euao we solve by meas of Geeralzed Meod of Le algebrac dscree appromaos 6 Appromao properes ad covergece muldmesoal case Te umercal sceme for problem (7 s bul usg dmesoal Lagrage erpolao Le deoe e cou of odes by varable Dscree problem s formulaed below for gve operaor A : B ad eleme fɶ fd eleme v B suc a A v = fɶ (8 Teorem 5 Te rak of fe dmesoal uas represeao A as a full rak ad s rak s ( = A of e operaor ad ere ess a uue soluo of e dscree problem (8 Proof Usg propery a fe dmesoal uas represeao K s lpoe mar Smlarly as proof of Teorem we oba a fe dmesoal uas represeao as a full rak ad ece a uue soluo of dscree problem (8 ess ere K of operaor Teorem 6 Fe dmesoal uas represeao A appromaes operaor A o eleme ad error esmao of appromao as e followg form α v Av A v l l( + a k α = k= α v (9 Proof Usg formula of dmesoal Lagrage erpolao ad acg by operaor A o resdual yelds (9 Teorem 7 If f f lm ɶ ɶ = olds ad codos of eorems are sasfed e lmv v = ad B v v M l + a l( k B = k= α v ( ( α α v (2

12 62 A Kdybaluk Proof Accordg o e Kaorovc covergece eorem of absrac appromao sceme all reuremes are sasfed us smlarly o eorem 4 usg eualy v v M Av A B v ad eglecg erms ad = we oba (2 oclusos Tus s paper we prese a Geeralzed Meod of Le algebrac dscree appromaos for solvg e aucy problem for lear dyamcal sysems Te key fdg of s researc s e opporuy o provde a facoral rae of covergece by all varables e euao cludg me varable Te aucy problem for dffereal euaos as bee reduced o a sysem of lear algebrac euaos wc geeralze e way of solvg ODE ad PDE Moreover e case of lear problem e roduced subsuo allows for e rapd solvg of e problem we al daa or eral sources ave bee caged bu coeffces of dffereal operaor e problem remaed cosa Ts s possble by keepg memory e verse mar ad mulplyg o e vecor wc represes al daa or eral sources Refereces [] Myropolsk YuA Prykarpask AK Samojleko VHr Algebrac sceme of dscree appromaos of lear ad olear dyamcal sysems of maemacal pyscs Ukraa Maemacal Joural [2] alogero F Ierpolao dffereao ad soluo of egevalue problems more a oe dmeso Le Nuovo meo [3] Bu O Pryula M Meod of Le algebrac dscree appromaos e eory of dyamcal sysems ( Ukraa [4] Bu O Pryula M Te rak of projeco-algebrac represeaos of some dffereal operaors Maemayc Sud [5] Kdybaluk AA Pryula MM Geeralzao of sceme of e Le-algebrac meod of dscree appromaos for aucy problem XIX Ukraa oferece of oemporary Problems of Appled Maemacs ad Iformacs L vv ( Ukraa [6] Hor RA Joso R Mar Aalyss ambrdge Uversy Press ambrdge 99 [7] Treog VA Fucoal Aalyss Fzmal Moscow 22 ( Russa [8] Berez IS Zydkov NP Numerc Meods Vol Fzmagz Moscow 962 ( Russa [9] Rcmayer R Dfferece Meods of Solvg Boudary-value Problems Mr Moscow 972 ( Russa [] Luserk LA Sobolev VI Elemes of Fucoal Aalyss Nauka Moscow 965 ( Russa [] Samojleko VHr Algebrac Sceme of Dscree Appromaos of Maemacal Pyscs ad s Precso Esmaos Asympoc Meods Ma Pyscs Maemacs Isue AN USSR Kyv ( Russa