Chapter 25 Sturm-Liouville problem (II)

Transcription

1 Chpter 5 Sturm-Liouville problem (II Speer: Lug-Sheg Chie Reerece: [] Veerle Ledou, Study o Specil Algorithms or solvig Sturm-Liouville d Schrodiger Equtios. [] 王信華教授, chpter 8, lecture ote o Ordiry Dieretil equtio

2 Pruermethod Sturm-Liouville Dirichlet eigevlue problem: Scled Pruer trsormtio y = ρ si S where z = py = S ρ cos S ( ; E > : sclig uctio d dy p + q y = Ew y, y = y = Simple Pruer trsormtio y = ρ si z = py = ρ cos d S Ew q S = cos + si + si cos p S S d ρ S Ew q S = si cos ρ p S S S = d cos = + ( Ew q si p d ρ = ρ p ( Ew q si cos So r we hve show tht Sturm-Liouville Dirichlet problem hs ollowig properties Eigevlues re rel d simple, ordered s E < E < E < Eige-uctios re orthogol i L ([, ], w with ier-product φ ψ φ ψ w 3 Eige-uctios re rel d twice dieretible W Moreover we hve implemeted (Scled Pruer equtio d S Ew q S = cos + si + si cos p S S with Forwrd Euler Method (ot stble, but it c be used so r

3 Sturm s Compriso [] Theorem (Sturm s irst Compriso theorem: let (, E,(, E φ φ be eige-pir o Sturm-Liouville problem. d dy p + q y = Ew y suppose E > E, the φ is more oscilltory th φ. Precisely speig φ φ Betwee y cosecutive two zeros o φ, there is t lest oe zero o φ Theorem (Sturm s secod Compriso theorem: let (, E,(, E φ φ be solutios o Sturm-Liouville problem. d d φ p = E w q d dφ p = E w q ( φ φ ( ( suppose p p d E w q < E w q o [, b] ( Betwee y cosecutive two zeros o φ, there is t lest oe zero o φ <proo o (> d dφ p = E w q d dφ p = E w q φ φ Simple Pruer d = cos + p d = cos + p ( E w q si ( E w q si 3

4 First we cosider = = Sturm s Compriso [] d = cos + p = d ( Ew q si F (, = cos + ( E w q si G (, suppose p p d E w q < E w q o [, b] = = p p ( Ew q < Ew q p = ( ( (, <, F G cotiuity o F, G d d (, [ δ ] F, < G, + < [ + δ ], = = = < ( + ] δ, = + h =, > + h, the ( = ( p ( p ( F (, ( < G (, ( ( Ew ( q ( < ( Ew( q ( = + h ( ] < + h, + h + δ 4 Suppose

5 d = Questio: How to del with the cse < Sturm s Compriso [3] <proo o (> Betwee y cosecutive two zeros o φ, there is t lest oe zero o φ Suppose φ hs cosecutive zeros t, φ φ Without loss o geerlity, we ssume φ > o (, Moreover φ (, we my ssume φ > o [, + δ d φ =, φ ( > ( φ > o [, + δ ( Apply result o (, set =, b =, the ( = ( < ( = π ( c < y = ρ si = π φ ( φ ( = = p t φ ( φ ( = t = < γ π p ( c φ = d π > o (, + δ π < γ < i φ ( > π < γ < π i φ ( < c 5

6 Pitll [] Recll Sturm-Liouville Dirichlet eigevlue problem: d dy p + q y = Ew y, y = y = Eigevlues re rel d simple, ordered s E < E < E < Questio: How bout symptotic behvior o eigevlue, sy lim E = or lim E = α Eige-uctios re orthogol i L ([, ], w with ier-product φ ψ φ ψ w Questio: re eige-uctios complete i L ([, ], w W {( E, ψ : =,,3 } is eige-pir o d dy p q y Ew y y y 3 Eige-uctios re rel d twice dieretible ( { ψ } = { = ψ } = [ ] + =, = = cl sp cl iite combitio o L,, w? L ([, ], w lim ψ, ψ = = The more importt questio is d d Questio: is opertor L = p + q w digolizble i L ([, ], w 6

7 Pitll [] Questio: How bout symptotic behvior o eigevlue, sy lim E = or lim E = α Geerl Sturm-Liouville problem Model problem d dφ + = p q φ Ew φ pm + q Mφ = Ewmφ p M = m { p : [, ] } > = φ ( = q M = m { q : [, ] } w mi { : [, ]} d = c os + ( Ew q s i m = w > = cos + ( Ewm qm φ d p = Sturm s secod Compriso theorem ( ( ( Betwee y cosecutive two zeros o φ, there is t lest oe zero o φ d φ Ew q + φ = = > p m M,, require Ewm qm Hoo s Lw: M solutio: φ = mπ π Zeros o solutio is m = with spce = m m = s E d φ p M = si φ = si Eercise: Betwee y cosecutive two zeros o φ, there is t lest oe zero o φ shows lim E = 7

8 Pitll [3] Questio: re eige-uctios complete i L ([, ], w Geerl Sturm-Liouville problem d dy p + q y = Ew y y = = y p =, q =, w =, = π Model problem d y = Ey y = = y ( π Questio : solutio o modl problem is ψ = si, =,,3, Is such eigespce { ψ =,,3, } complete i L [, π ] Cosider spce ([ π π ] [ π π ] π { π } L, = :, R < i { ϕ e =, 3,,,,,,3, } is orthogol i L ([ π, π ] with ier-product g = g π π ϕ ϕ i { ϕ Z} m π π i m i im = = e e = π i m ( [ π, π ] cl sp e L P is closed subspce decompositio = P + P is uique P where P cl ( sp{ ϕ Z} cl ( sp{ ϕ Z} P cl sp ϕ Z ( { } 8

9 Pitll [4] P P ( { ϕ Z} cl sp Iormlly, P = cϕ or some { } = ( { } P = P cl sp ϕ Z c = to be determied P ( { ϕ } cl sp Z P ϕ Z ϕ P = Z P = P ϕ = ϕ P Z ϕ = c ϕ ϕ Z m m m= π i ϕ ϕ m = = i = m m c = ϕ Z π Formlly speig, whe we write such tht S P S i L sese. P = c ϕ, i mthemticl sese we costruct prtil sum = P i L [ π, π ] lim P S = lim P S = L π π P S sp { ϕ =,,, } P = cϕ = S ( { ϕ > } cl sp 9

10 Pitll [5] S P P { ϕ =,,, } sp ( { ϕ > } cl sp { ϕ,,, } M sp = ( { ϕ } N cl sp > ( { ϕ Z } M cl sp ([ π, π ] C L = M N M ( P = S + S + P L ([ π, π ] M N C M ( S P S S P S S = P S + P ϕ S = ϕ = ϕ S c = ϕ π Eercise: S = cϕ, c = ϕ π is the solutio o mi :{ } = = ϕ = L c c R

11 S = cϕ, c = ϕ π Pitll [6] i = + + = = S c c e c e ( cos si ( cos si S = c + c + + c = S = c + c + c cos + c c si = i S + cos + b si = cos = where si = si m cos =, m d π = π π π = cos π π π b = si π π Theorem: trigoometric bsis is complete i L ([ π, π ] i ( { ϕ Z} = ([ π, π ] cl sp e L C ([ π, π ] = M sp{ ϕ =,,, } N cl sp{ ϕ > } L M N M S i L sese, where S + cos + b si =

12 i Eercise: we hve show { ϕ Z} where Pitll [7] { } ([ π π ] cl sp e = cl sp, cos,si : =,, = L, S b L π π + cos + si, = ([ ] π π =, cos : Fourier cosie coeiciet π = π π π π b = si : Fourier sie coeiciet π π We bbrevite s lim S = + cos + b si I uctio is eve, sy = π =, the + cos, = cos π = I uctio is odd, sy = (, the π b si, b = si π = d y = Ey Modl problem hs eige-pir ψ y = y ( π = E =, = si, =,,3, From bove eercise, or y L ([, π ], we c do odd etesio odd the. Hece ψ = b si ( { = si : =,, } = ([, π ] cl sp L Questio: How bout i we do eve etesio i > eve L i < = i > L i < = ([ π, π ] ([ π, π ]

13 Pitll [8] d Questio: is opertor d L = p + q w From Pruer trsormtio, we c show Lψ E ψ ψ ψ digolizble i L ([, ], w =, = = Eigevlues re rel d simple, ordered s E < E < E <, lim E = Eige-uctios re orthogol i L ([, ], w with ier-product φ ψ φ ψ w d W Deie domi o opertor L with Dirichlet boudry coditio s D( L L ([ ] w ( {,, : } = = = Clerly we hve cl sp{ ψ : =,, } D L,but we c ot sy L is digolizble i D( L Fiite dimesiol mtri computtio iiite dimesiol uctiol lysis Jord orm: A( u v = ( u v Au = u u : eigevector Av = v + u v : geerlized eigevector Lψ = E ψ ψ ψ = = Lφ = E φ + ψ φ = = φ ψ : eigeuctio φ : geerlized eigeuctio Questio: does such φ : geerlized eigeuctio eists? ( Ide:i we c show tht cl sp{ ψ : =,, } = D L, the eve such φ eists, φ D( L The opertor L is digolizble i D( L, why? 3

14 Scled Pruer trsormtio d dy + = y = ρ si S z = py = S ρ cos S ( ; E > Scled Pruer Trsormtio [] Time-idepedet Schrodiger equtio p q y Ew y d + V ψ = Eψ ψ = = ψ ( π d S Ew q S Ew q S = + + cos + si p S p S S = A + B cos + C si d E V E V S = S S cos si S S S ; = ( E i E V Suppose we choose S = = ( E V where E V i E V > Questio: uctio is cotiuous but ot dieretible t =. How c we obti d i = i > i < d ( = i > d ( d d = = + (, ( d hs jump discotiuity t = 4

15 Observtio: i = i > Scled Pruer Trsormtio [] i < d = i > d The udmetl Theorem o Clculus lso holds, sy d d d = + s ds < <, =, udmetl Theorem o Clculus holds, does ot eist, we igore it. d = + s ds =, < < = is cotiuous ( + ( = = d 3, < =, udmetl Theorem o Clculus holds, + ( d = + s ds = + ds =, < s Questio: lthough udmetl theorem o clculus holds or uctio, but i How c we id ( umericlly d hve better ccurcy? Reso to discussio o udmetl theorem o clculus: d is give, d E V E S S V cos S = si S S S = ( + ( s, ( s ds ds d depeds o S(, ccurcy o ( d is equivlet to ccurcy o obtiig S( i S = ( E V, = i > 5

16 Numericl itegrtio [] 3 4 ( 3 ( 4 5 = O(, = (, = 3! 4! ( ( Igore odd power sice it does ot cotribute to itegrl h h 4 3 h h = h O h O h = + + 4! 3 ( h h h ( h = + h O h 3! 4! 3 4 ( 3 ( 4 5 h h h ( h = h + O h 3! 4! 3 4 ( 3 ( h + h = + h + O h 4 3 ( h 5 ( h h = h h + h O h geerl orm ( ( b b b 3 = ( ( b ( b c + b Trpzoid rule ( 梯形法 bse = b 6

17 Numericl itegrtio [] Emple: give prtitio = < < 3 < 4 < 5 = b d grid uctio = (, =,,3, 4, = We use Trpezoid rule to id F t dt = = b F F ( = = F = F ( = ( + F Eercise : let = cos, =, b = Try umber o grids =,, 4, 8, 6, compute F d mesure mimum error F ( { } m si Plot error versus grid umber, wht is order o ccurcy? = dt 3 = F = F = b Eercise : let i < = =, b = i > I = is i the grid prtitio, wht is order o ccurcy F I = is NOTi the grid prtitio, wht is order o ccurcy = = b 7

18 Scled Pruer Trsormtio [3] Questio: c we modiy uctio slightly such tht it is cotiuously dieretible, sy i = p i < < + i + d i = p i < < + i + 3 where p ( z z z z = is polyomil o degree 3,,,, 3 re chose such tht C <sol> C is chieved by ollowig 4 coditios + = p = = =, + ( ( = = p = 3 ( ( + + = + + = p + = ( + = ( = + p 3 + = + 3 =, d 3 = + + p z z z 3 where + 3 = 3 + 8

19 Scled Pruer Trsormtio [4] i = p i < < + i + d i = p i < < + i + = d, C but d hs jump discotiuity t =, + Eercise 3: try to costruct C i = p i < < + i where p ( z z z z z z = is polyomil o degree use Symbolic toolbo to determie coeiciets,,, 3, 4, 5 d d plot,, d use Trpezoid method to compute = + t dt,wht is order o ccurcy? 3 9

20 Model problem: d ψ = ψ, ψ = ψ ( π = solutio is ψ = si, =,,3 Review Fiite Dierece Method FDM π Dhψ ( j = λψ ( j or j =,,,, h = + eige-pir: { si ( : j,,,,,, N} j ψ = = + ( h 4si / λ = h ( c um cos = um = h = O h Questio: why does error o eigevlue icrese s wve umber icreses? 3 h ( 4 4 Dh = + + O h d ψ h ( 4 = ψ Dhψ + ψ ψ Substitute ψ = si ( ψ ( 4 4 ( j = ψ ( j 4 3 h 4 4 um + um + O( h d ψ h h h 4 Eercise 4: id lytic solutio o + Vψ = Eψ, ψ = ψ ( π = V The use FDM to solve D ψ ( V ( ψ ( E ψ ( ψ ψ ( π where + =, = = h j j j um j π 3π < < = 4 4 otherwise Wht is order o ccurcy? mesure Eum E