SOLUTION OF PARABOLA EQUATION BY USING REGULAR,BOUNDARY AND CORNER FUNCTIONS

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1 SOLUTION OF PAABOLA EQUATION BY USING EGULA,BOUNDAY AND CONE FUNCTIONS Dr. Hayder Jabbar Abood, Dr. Ifchar Mdhar Talb Deparme of Mahemacs, College of Edcao, Babylo Uversy. Absrac:- we solve coverge seqece by sg he parabola eqao whch have a small posve parameer ξ ad fd a qe solo for a gve coverge seqece. -Irodco : - Dffereal eqaos a very mpora ool for solvg may pheomeas.there are may ahers whch sded he appled mahemacs as physcal mahemacs.levesham [4] sded he ordary dffereaol eqaos of he frs order ad he frs degree.dedoe [] sded he ordary eqao of he secod degree wh al ad bodary codos.techaoff ad Samarck [5] sded oly he paral dffereaol eqaos of he frs order ad he frs degree wh bodary codos.levesham [3] sded he parabola eqao (paral dffereal eqao of he secod order ad he frs degree ).They p al ad bodary codos o solve hs eqao.they sed coverge seres from form fco wh wo bodary fcos.they proved he esece ad qess of he solo. Oe of he mahemacal brach whch ake care by sdyg pheomea's whch comes from he he evrome we lve. Several mahemacal models are formed for may researchers o solve hese pheomea's ad really mos of hese models have a hgh ably o sdy see (Kreysg [] ad Smh [6]). Some effecs o he accracy of he solo may be small ad some researchers do ake care o sdy ad abo he mporace of hese effec may be have a effec o he resl of he pheomea's ad versely, o he oher sde some researchers are emphacs ha hese effecs ms be sded ad oe of hem s (Dedoe J.[]), we also lke hm. I hs work,we devlope he reserch of [5],we sdy he coverge seres whch s form fco,wo bodary fcos ad wo corer fcos ( see seco 4 eqao (6)).We prove he coverge fco s qe solo of parabola eqao. -Ma problem I hs research, we sdy he followg problem :- ١

2 ξ ( ) = ξ f ), << l, << () =,ξ ) = ϕ (), (, ξ ) = ( l, ξ ) () where s a arbrary vecor ad ξ a very small posve parameer.we wll dscss he problem he rego = { l } { Τ} We wll prove ha he aalyc coverge seres (,, ξ ) = = ξ (, ) s a solo for he problem () wh bodary codos ad al codo ().I order o prove he esece of a solo for () ad ()., we fd a esmao for he coverge seres. I oher meag, for cosrc lke hs covergece ad for more accrae hs coas wo fcos: -The frs s he fco, o he bodary of whe =, =, = l. -The corer fco a he pos (,), (, l ) he rego. l T (,) (l,) 3-Some Specal codos o Solve The Ma Problem We ca fd he solo of or ma problem (),ad by sg he followg codos :- = Ι The fcos f,ξ ) = ξ ) ad ϕ () whch are (+) f dffereable o cosrc a coverge seres of order ad sasfy () a (, ), ( l, ) ϕ ( ) = ϕ( l) =. From () ad whe ξ = we ge he followg eqao ٢

3 f(,, ) =. (3) f ΙΙ Eqao(3) a he rego has a arbrary solo assme = ΙΙΙ W he s a parameer ( l) he:- db = f, ) + B,), > dτ τ.. (4) Ιν By sg of he codos (), he solo of eqao (4) becomes B ) = ϕ ( ) ).... (5) 4- The coverge seres whch s aalyc wh respec o ξ o solve () ad () gve as follow:- ξ ) = ξ ) + B(, τ, ξ ) + Q ( ζ,ξ ) + Q ( ζ, ξ ) + P( ζ, τ, ξ ) + p ( ζ, τ, ξ ). (6) l Sc h ha τ =, ζ =, ζ = ξ ξ ξ Where s he reglar fco Q, Q, B bodary fcos ad p, p he corer fcos whch are seres rased o powers wh respec oξ, for eample ξ ) = ξ =. To fd he coeffces of hese seres,we p (6) (), () ad a he fco f(,,ξ ) as follows:-, ξ ) ξ f + B τ ) = + Q ( ζ, + Q ( ζ, + P ( ζ, τ ) + P ( ζ, τ ) = we ca wre as below f = f + Bf + Qf + Q f + Pf + P f sch ha f ξ ) = f ξ ), ξ ); Bf τ, ξ ) = f ξ τ, ξ τ, ξ ) + B τ, ξ ), ξ )_ f ξ τ, ξ ) ; Qf ( ζ, ξ ) = f ( ξζ, ( ξζ, ξ ) + Q( ζ, ξ ), ξ ) f ( ξζ, ξ ); Pf ( ζ, ξ ) = f ( ξζ, ξ τ, ( ξζ, ξ τ, ξ ) + B( ξζ, τ, ξ ) + Q( ζ, ξ τ, ξ ) + P( ζ, τ, ξ ), ξ ) Bf ( ξζ, τ, ξ ) Qf ( ζ, ξ τ, ξ ) + f ( ζ, ξ τ, ξ ) ; By he same way, we ca defeq f, P f. By he sadard procedres for he aalyss of seres wh respec o ξ powers ad by pg he coeffces for he eqal powers wh respec Ο oξ,we ge a eqao for ay coverge seres, for eample whe ξ : ٣

4 (a) he fco has The followg eqao:- f,) = Whch ca be fod by he correspodece wh he eqao (3),hs we ge = ad for he fcos,. We have he followg lear eqaos f = f where f represeed by j, j< ad from hs = f ( f., (b) Eqao of he fco B τ ) s B = B f f,,, ) + B ( τ ) f (,),)),hs correspod wh he eqao (4) added o he bodary codo (5) from he codos (I IV), he fco B τ ) have he followg epooal esmao (see [] page 57). B τ ) C ep( b ), l, τ > (7) Where b>, c> are arbrary cosas.the fcos B τ ) whe s lear wh he formla B = f (, ) + B τ ), ) B + π (, τ ), j We defe B ) = ) ad π τ ) represeed by B j τ ) where j j <. The solo for hese lear eqaos whch possesses he epooal esmao s as (7). 5- The Ma esals We fd he fc o Q ( ζ,,where s as a parameer from he followg eqao ϕ = Q f f (, (, + Q ( ζ,,) f (, (,,) From he eqao () for Q ( ζ,,we ge he wo codos ϕ ϕ (, =, ( ), =,. (8) The bodary codo () gves Q ( ξ, whe ζ, so Q ( ζ, T ) whle he fcos Q ( ζ,, are defe d from he lear eqaos,( ζ cosa coeffce ) τ Q - = f (, Q + q ( ζ,.. (9) ٤

5 We ca fd q ( ζ, hrogh he fcos Q j ( ζ,, j < The solos of (8) ad (9) ad from codo III wh he bodary codo,we ge he sgle vales whch have he followg epooal esmao ( ζ, C ep( bζ ), ζ, T.() Q The fco P ( ζ, τ, ξ ) has a role o make he fco B acve wh he bodary codos ad he fco Q wh s al codo also, he lear dffereal eqao for P ( ζ,τ ) s p p = p f f τ ζ >, τ > (,, (,) + B (, τ ) + P ( ζ, τ ),) f (,, ) + B (, ),), By sbso (6) he codos of () ad for he fco P ( ζ,τ ) we ge p p ( ζ, ) =, (, τ ) () p = ζ p B ζ,, (, τ ) = ( ζ,) = Q ( ) (,, () From he codos () ad (), he fco p s bodary fco for boh varables,ha s p ( ζ, τ ) whe ζ + τ. (3) A p ( ζ,τ ),whle he fcos p ( ζ,τ ) ca be defed from he p p lear parabola eqaos A( τ ) p ( ζ, τ H ) (4) where A( τ ) = P (,, (,) + B (, τ ),). The fco H ( ζ,τ ) represes he fcos Q, B j, j ad p j, > j. The solos of () ad (4) ca be defed by (Techaoff, Samarck [4]) τ ( ζ, τ ) g( ζ, τ ) + dτ G( ζ, τ, ζ, τ ) h( ζ, τ ) dζ p = ; here g ( ζ,τ ) s a arbrary fco whch s dffereable (smooh) sasfy he codos () ad (3) g g h( ζ, τ ) = H ( ζ, τ ) + + A( τ )g, G ζ, τ, ζ τ ) _ (Greea fco), (, ( ζ ζ ) a( τ τ ) ( ζ + ζ ) ( ) a τ τ G ( ζ, τ, ζ, τ ) = φ( τ ) φ ( τ ) ep + ep πa( τ τ ) 4 4, where φ ( τ ) s fdameal mar have he followg epooal esmao []: ٥

6 φ [ τ τ ( τ ) φ ( τ ) C ep b( )], hs he epooal esmao for he fco p becomes :- p ( ζ, τ ) C ep[ C( ζ + τ )], ζ, τ. (5) To fd he esmao of he fcos P, Q by same mehod sed for P ad Q ad havg he same epooal esmao as () ad (5) represe o he coverge seres of order for he seres(6) U U = ξ = [ + B τ ) + Q ( ζ, + Q ( ζ, + P ( ζ, + P ( ζ, τ )] Therefore he followg saeme s vald. Theorem (5-): If he codos (I IV) are vald he ξ ), ( ξ small parameer) have a qe solo o he problems (),() ad for he form coverge seres coverges o he esmao Ο ξ +.e. U ( ) ma U = Ο( ξ + ) Proof:- Le w= U ; + U = U + ξ ( Q+ + Q+ + P + + P + ), where U he paral coverge seres defed (6). Ths we have he followg eqao:- w w ξ f ξ ) w h( w,, ξ = ), (7) f ξ = f, + B, / ζ,,,, where ( ) ( ( ) ( ) ), h( w,, ξ ) = f ( U + w,, ξ ) ξ f ξ )w w,, ξ The fco h ( ) have he followg wo properes:- - whe w =, s clear ha he eqao h + (,,, ) (,,, ) U U ξ = f ξ ξ = Ο( ξ ), ξ c ξ, =, he here es c >, ξ sch ha - If ( ) w < ξ ξ sasfy he followg eqaly sp h, sp > ( w,, ) h( w,, ξ ) C w w ξ. By sg (Greea mar), eqao () wh codos (),(3) becomes w ( ξ ), = G(,,, ζ ) h( w(,, ζ ),,, ζ ) d d l L( w,, ζ ) (8) ٦

7 From codo III, Greea mar have he esmao (Techaoff, Samarck [4]): c ( ) // (,,,, )// ep[ ]ep[ G ξ c c ]. ξ ξ By applyg he heorem of he covergece of seqeces for he eqao (8),we ge w =, w+ = L( w,, ξ ), =,,... From above, we cocled ha w (,, ξ ) s a qe solo for () whe ξ s a small parameer ad have he esmao ma ma + w (,, ξ ) = Ο ( ξ ), hs we ge + U = Ο ( ξ ). efereces:- [].Dedoe J.(984). Fodaos of Moder Aalyss. Naok.Moscow. [].Kreysg E. (988).Irodcory Fcoal Aalyss wh Applcaos.Joh Wley ad Sos. New York. [3].Levesham V.B.,Abood H.D.(3). Asympoc Iegrao of The Problem o Hea Dsrbo a Th od wh apdly Varyg Sorces of Hea //coemporary mahemacs ad s applcaos. vol [4].Levesham V.B. Asympoc Iegrao for Dffereal Eqaos wh Qck Oscllaory Composed.Mah. Cb. 3.T.8 N :357 [5]. Techaoff A.N. Samarck A.A.(988). Mahemacal-Physcs Eqao Naok Moscow. [6].Smh D.. (985).Sglar l'errba Theory.a Irodco wh Applcao. Cambrdge Uv.press,Cambrdge, الخلاصة:- ف ي ه ذا البح ث نح ل متتابع ة متقارب ة باس تخدام معادل ة قط ع مك افي والت ي تمتل ك معلم ة صغيرة وإيجاد حل وحيد لتلك المتتابعة المتقاربة. ٧