BE.430 Tutorial: Linear Operator Theory and Eigenfunction Expansion

Transcription

1 BE.43 Tuorial: Liear Operaor Theory ad Eigefucio Expasio (adaped fro Douglas Lauffeburger) 9//4 Moivaig proble I class, we ecouered parial differeial equaios describig rasie syses wih cheical diffusio. We leared abou he ehod of separaio of variables o solve he PDE. However, ay probles ivolve hoogeeous reacios i he syse or coplicaed coordiae syses, akig he goverig PDE ore coplicaed ad aybe requirig a ore sophisicaed ehod o solve i, such as his oe: c c c r α c wih B.C.s ad c( r ) r r r r r () This is a ypical diffusio-reacio proble i spherical coordiaes wih firs-order cosupio (recogize i?). Solvig his PDE wih separaio of variables ca be soewha cofusig ad cubersoe. Here, we iroduce he liear operaor heory ad he eigefucio expasio ehod, which build he basis of all ehods of solvig liear PDEs, icludig separaio of variables ad fiie fourier rasfors ad which eeds full udersadig o properly apply he. You ca read abou Fiie fourier rasfor ehods i Dee chapers 4. o 4.. afer readig his uorial. Suary of he Liear Operaor Theory Here, we prese he aheaical basis of he eigefucio expasio ehod of solvig PDEs. This igh o ake sese o you a firs; bu wih he iroducio of he Sur-Liouville operaors, you will see why he followig properies are ipora for solvig PDEs.. Liear Vecor Spaces ad Liear Operaors Vecor spaces are defied as a space i which a se of vecors exis ad i which he followig operaios are allowed: Vecor addiio u+ v w Scalar uliplicaio of vecors Au z Liear operaors, L, defie ad describe such liear vecor spaces. For exaple, a liear operaor ca be d L, dx he he vecor, u i he vecor space is described by du Lu dx The liear operaor has he followig lieariy propery:

2 du dv L( Au+ Bv) ALu+ BLv A + B dx dx Why do we care? We are goig o hik of vecors as fucios or variables fro ow o, so u ad v are fucios i he followig discussio. We will use vecor space ad fucio space ierchageably. We will herefore drop he vecor oaio.. Ier Producs Aoher liear operaor is he ier produc (also called do produc for vecors). I a liear vecor space, a ier produc exiss such ha uv, where α is a scalar, ad α (ier produc of u wih respec o v) Au + Bv, z A u, z + B v, z The ier produc of a vecor or fucio wih iself is defied as he square of he agiude of he fucio or vecor: zz, z () I he coiuous fucio (or vecor) space bouded by a ad b, he ier produc is defied as b uv, wxuxvxdx ( ) ( ) ( ) (3) a where w(x) is a weighig facor, whose iporace will be discussed laer. A liear vecor space is called Hilber space if all vecors have a fiie agiude, i.e. for all vecors or fucios z, zz, z < 3. Self Adjoi Liear Operaors Liear operaors i Hilber spaces are self-adjoi if Lu, v u, Lv for all u, v i space. (4) 4. Eigevalue Proble For self-adjoi liear operaors, here exiss a se of fucios, φ, such ha Lφ λφ (5)

3 where λ is are cosas. These fucios, φ, are called eigefucios ad λ heir correspodig eigevalues. These fucios are liearly idepede ad saisfy he boudary codiios of he fucio space. 5. Orhogoaliy of Eigefucios The eigefucios defied by he liear operaor L are orhogoal o each oher. So for all, φ, φ if φ if (6) The ier produc wih respec o iself ( ) is he squared agiude of he fucio, accordig o (). 6. Liear cobiaio of eigefucios he soluio Ay fucio i he fucio space ca be wrie as a liear cobiaio of he eigevecors of he liear operaor defiig space. u Aφ (7) where A are a se of cosas (o eigevalues!). Le s ake he ier produc of his equaio wih respec o φ. The, u, φ A φ, φ The suaio sig ca be ake ou of he ier produc sice he ier produc operaio is liear. Wih equaio (6) The, u, φ A φ, φ A A A u, φ φ φ, φ φ (we swiched wih here) 7. Soluio o he liear operaor The soluio o he liear operaor proble is he give by

4 If φ φ u u, φ (8) φ, he we call φ orhooral eigefucios. So, if φ are orhooral, he u u φ φ (9), This soluio is a powerful ool as i does oly provide he soluio o he parial differeial equaio se by he liear operaor, i also provides soluios o Algebraic (arix) equaios Ordiary differeial equaios Iegral equaios Sur-Liouville Operaors Now, why was he liear operaor heory so ipora, i.e. wha does ha have o do wih PDEs? May probles i egieerig syses ca be described by a se of operaors called Sur-Liouville operaors. d ( ) d L p x ( ) w( x + q x ) dx dx defied i he space bouded by a x b Our oivaig proble give by equaio () is described by he Sur-Liouville operaor: c Lc where ( ) ( ) ( ) w x r a p x r b q x α r The ier produc is defied as uv, b a ( ) ( ) ( ) w xu xv xdx I urs ou ha he liear fucio space described by he Sur-Liouville operaor lies i he Hilber space. Also, he Sur-Liouville operaor is self-adjoi wih Dirichle, Neua or Robi boudary codiios, i.e. dc dc c ; ; A dx dx + c where A is a cosa. So, he Sur-Liouville operaor is self-adjoi oly wih boudary codiios (you ca prove i yourself wih equaio (4)). Aoher propery of his operaor is ha all of is eigevalues are real ad egaive, so

5 λ α Sice he liear operaor describig our oivaig proble lies i he Hilber space, is self-adjoi, here exis eigevalues ad eigefucios, such ha he soluio is give by c c, φ φ if he boudary codiios are hoogeeous. Exaples: Makig he Boudary Codiios Hoogeeous Exaple Recall ha he boudary codiios o our oivaig proble was give by c r r ( ) ; c r The secod boudary codiio is o-hoogeeous. A his sage, we cao ake advaage of he liear operaor heory for a soluio of his proble. To ge aroud his proble, we defie ( ) ( ) c' r, c r, so ha he boudary codios becoe c ' r r ( r ) ; c' Where did he o-hoogeeiy i he boudary codiio disappear o? Le s chage our goverig equaio accordigly: c' c' r α c' α r r r The o-hoogeeiy appears i he goverig equaio as α. The goverig equaio wih he Sur-Liouville operaor becoes c ' Lc' α Exaple Le s ur o a sipler exaple, siilar o he proble discussed i class: c c D x wih I.C. c( x,) ad B.C.s c(, ) c ad ( ) c L,

6 To avoid cofusio wih uis, i is wise o o-diesioalize he variables before aepig o solve he equaio. Le c x η L The he proble is rewrie o γ γ τ η γ c wih I.C. (,) D τ L γ η ad B.C.s γ (, τ ) ad ( ) γ, τ The boudary codiios are o-hoogeeous. This ca be fixed by dividig he soluio io a seady-sae soluio ad a rasie soluio: (, ) ( ) + (, ) γ ητ γ η γ ητ ss The seady-sae soluio of he proble is obaied by solvig η γ ss wih he boudary codiios give above; his gives γ ss η The rasie soluio of he proble is obaied by solvig γ γ τ η () Wih he help of he seady-sae soluio, we ca rewrie our boudary ad iiial codiios of he rasie par of he proble o γ (, τ ) ; (, ) γ τ ad ( ) γ η, η I his case, he o-hoogeeiy ow appears i he iiial codiio. The boudary codiios are ow hoogeeous, ad he proble ca be solved by equaio (8). Exaple: Solvig PDEs Usig Eigefucio Expasio Mehod We ll ake he sipler proble (he proble discussed i class) for our exaple. Our proble ow is γ Lc τ where L η wih w( η ) ; p ( η ) ; q ( η ) The ier produc is defied i his space as

7 ( ) ( ) uv, uη vη dη Therefore, he liear operaor L is self-adjoi. There exiss a eigevalue proble, such ha Lφ λφ wih φ ( ) φ ad ( ) Sice all eigevalues are real ad egaive, we ca wrie d φ j α φ dη The soluio o his eigevalue proble (ordiary differeial equaio) is ( ) si ( ) + cos( α η ) φ η A αη B Applyig boudary codiios, we obai ( ) A si ( ) φ η πη B ad α π λ α wih j,,3. We are o solve A ca be deeried by he fac ha φ should be orhooral accordig o (9). Therefore, by he defiiio of he ier produc, φ φ, φ A si A A ( πη) dη ( x ) si( πη ) φ The soluio so far accordig o equaio (9) is γ φ Wha is, γ γ, φ si( πη) ()? Followig is a crucial sep, which akes his eigefucio expasio ehod possible for solvig PDEs. Le s plug i equaio () io he PDE goverig γ () (leavig φ as is) γ, φ φ Lγ, φ φ τ Sice L is self-adjoi,

8 γ, φ φ γ, Lφ φ τ γ, λφ φ λ γ, φ φ The secod equaliy is derived fro he properies of he eigevalue proble. The las equaliy is valid sice λ is a cosa ad he ier produc is a liear operaio. If we ow ake he ier produc of boh sides wih respec o φ, suaio sigs are eliiaed due o orhogoaliy properies of eigefucios: τ γ, φ λ γ, φ γ φ which is a liear firs-order ordiary differeial equaio wih, idepede variable. Solvig he equaio, as he depede variable ad η as he oly, The iiial codiio is give by Applyig he iiial codiio, γ φ ce λ τ where c is a cosa γ( η, ), φ ( η ) si( πη) π γ φ π, e The eigevalues were foud above as λ τ λ α π π, e γ φ The fial soluio of γ is he π τ γ ( η, τ) γ, φ φ si( πη) e π The fial, fial soluio of γ is he π τ

9 γ ( η, τ) η si( πη) e π π τ