Notes 17 Sturm-Liouville Theory

Transcription

1 ECE 638 Fll 017 Dvid R. Jckso Notes 17 Sturm-Liouville Theory Notes re from D. R. Wilto, Dept. of ECE 1

2 Secod-Order Lier Differetil Equtios (SOLDE) A SOLDE hs the form d y dy 0 1 p ( x) + p ( x) + p ( x) y = f( x) dx dx If f( x) = 0, the equtio is sid to e "homogeeous." The ihomogeeous equtio c e solved oce we kow the solutio to the homogeeous equtio usig the method of Gree's fuctios (discussed lter). Boudry coditios (BC) re usully of the form y ( ) = y ( ) = 0 ( Dirichlet) y () = y () = 0( Neum)

3 Sturm-Liouville Form If we multiply the geerl differetil equtio p( xy ) + p( xy ) + p( xy ) = f( x) p0 e y the itegrtig fctor wx ( ) = we hve : p ( x) x p1() t x p1() t dt dt p () t p () t p ( x) + p ( x) e y e y p x w 0 x p () t dt () t 0 ( ) ( xy ) = wx ( ) f( x) x p1() t dt d p0 () t e y + p ( xwxy ) ( ) = wx ( ) f ( x ) dx Dividig this result y wx ( ) yields 1 d dy( x) Px ( ) Qxyx ( ) ( ) f( x) w( x) dx + = dx where p () t x 1 dt p 0() t ( ), ( ) ( ) Px e Qx p x 3

4 Sturm-Liouville Opertor This is clled the Sturm-Liouville or self-djoit form of the differetil equtio: 1 d dy( x) Px ( ) Qxyx ( ) ( ) f( x) w( x) dx + = dx or (usig u isted of y): = u f Where is the (self-djoit) Sturm-Liouville opertor: 1 d d Px ( ) + Qx ( ) w( x) dx dx Note: The opertor is ssumed to e rel here. The solutio u does ot hve to e rel (ecuse f is llowed to e complex). 4

5 Ier Product Defiitio A ier product etwee two fuctios is defied: We defie ier product s, ( ) ( ) ( ) < uv> ux v x wx where wx ( ) is clled weight fuctio. dx Although the weight fuctio is ritrry, we will choose it to e the sme s the fuctio w( x) i the Sturm - Liouville equtio. This will give us the ice "self - djoit" properties s we will see. 5

6 The Adjoit Prolem The djoit opertor is defied from < uv, >= < u, v> < uv, >= < u, v > = For Sturm - Liouville opertor so. (proof give ext) Hece, the Sturm-Liouville opertor is sid to e self-djoit: 1 d d = = Px ( ) + Qx ( ) w( x) dx dx Note: Self-djoit opertors hve ice properties for eigevlue prolems, which is discussed little lter. 6

7 Proof of Self-Adjoit Property Cosider the ier product etwee the two fuctios u d v:, ( ) ( ) ( ) < u v > = v x w x u x dx = v ( x) wx ( ) 1 wx ( ) d P( x) + Q( x) u( x) dx dx d d v x P x Q x u x dx = ( ) ( ) + ( ) ( ) dx dx d dx The first term isdie the squre rckets is first itegrted y prts, twice : d du( x) v x P x dx = v x P du( x) du( x) dv ( x) ( ) ( ) ( ) dx dx ( x) P( x) dx dx + dx dx du( x) dv ( x) du( x) v xpx Px dx = ( ) ( ) ( ) dx + dx dx du( x) dv ( x) d dv ( x) v xpx Pxux ux Px dx = ( ) ( ) + ( ) ( ) ( ) ( ) dx dx dx dx 7

8 Proof of Self-Adjoit Property (cot.) The ier product c ow e writte s du x dv x < uv> = v xpx + Pxux dx dx ( ) ( ), ( ) ( ) ( ) ( ) 1 d d + u( x) w( x) P( x) + Q( x) v ( x) dx w( x) dx dx This result c e writte s < uv, >= J( uv, ) + < u, v> du( x) dv ( x) where J ( uv, ) Px ( ) v( x) + ux ( ). dx dx Note : ( v) = v ( = rel opertor) From oudry coditios, we hve J( uv, ) = 0 < uv, >= < u, v> ( proof complete) 8

9 Eigevlue Prolems We ofte ecouter eigevlue prolem of the form u = λu (The opertor c e the Sturm-Liouville opertor, or y other opertor here.) The eigevlue prolem is usully stisfied oly for specific λ = λ, = 1,,. eigevlues For ech distict eigevlue, there correspods eigefuctio u = u stisfyig the eigevlue equtio. 9

10 Property of Eigevlues The eigevlues correspodig to self-djoit opertor equtio re rel. Proof: u = λu ( u) u wdx λ = uu, = λ uu, u, u = λ uu, u, u = λ uu, u u wdx u = λu ( u) ( ) = λ u u u wdx = λ u u wdx,, u u = λ uu Hece: λ = λ 10

11 Orthogolity of Eigefuctios The eigefuctios correspodig to self-djoit opertor equtio re orthogol if the eigevlues re distict. Cosider oe solutio of the eigevlue prolem : u u = λ u m m m m = λ u,, ( ) ( 1) Next, cosider other solutio of the eigevlue prolem with differet idex d distict eigevlue λ λ We multiply ( 1) y u, the cojugte of ( ) y u, sutrct the secod from the first,weight the result y wx ( ), d itegrte o x (, ) : m m m m ( ( ) ( ) ) = ( λ λ ) u u u u wdx u u wdx Next, we cosider the LHS of the ove equtio. m m : ( 3) 11

12 Orthogolity of Eigefuctios (cot.) The LHS is: ( ( ) ( ) ) u u u u w dx = < u, u > < u, u > m m m m =< u, u > < u, u > ( from the defiitio of djoit) m m = < u, u > < u, u > ( from the self - djoit property) m m = 0 Hece, for the RHS we hve ( λm λ ) umu wdx = 0 umu wdx = 0 ( orthogolity) sice λ = λ, λ λ m 1

13 Summry of Properties Assume eigevlue prolem with self-djoit opertor: u = λu, = The we hve: The eigevlues re rel. The eigefuctios correspodig to distict eigevlues re orthogol. Tht is, λ m = λ m, m = m = 0, m u u u u wdx λ λ 13

14 Exmple Recll : if Orthogolity of Bessel fuctios m ( ) ( ) ( ) u x u x w x dx 0 ( ( ) ( ) ( ) ( ) u = u = u = u = m m = for Sturm - Liouville prolem) 0 ( Dirchlet oudry coditios) Cosider : th ( ) = ( ), = root of ( ) u x J p x p m J x m m m Choose : = 0, = 1 ( ) ( ) J p 0 = 0, J p 1 = 0 m m (The first equtio is true for 0. The cse = 0 c e cosidered s limitig cse.) Wht is w(x)? We eed to idetify the pproprite DE tht u(x) stisfies i Sturm-Liouville form. 14

15 Exmple (cot.) Bessel equtio : ( ) 0, ( ) t y + ty + t y = y = J t Use t = p x, dt = p dx m 1 = t p x m m ( ) y = J p x m ( m ) + + = x y xy p x y 0 1 y + y + pm y = x x 0 15

16 Exmple (cot.) Rerrge to put ito Sturm-Liouville form: 1 y + y + pm y = x x 0 1 y y + y p = m y x x 1 y y + y λy, y J = = ( p x), λ = p x x m m 1 d d x y λy, y J ( p x), λ p x dx + = = = dx x m m 16

17 Orthogolity of Eigefuctios (cot.) Hece, we hve: 1 d d x y λy, y J ( p x), λ p x dx + = = = dx x m m Compre with our stdrd Sturm-Liouville form: 1 d d Px ( ) + Qx ( ) y= λ y w( x) dx dx We c ow see tht the u(x) fuctios come from Sturm-Liouville prolem, d we c idetify: ( ), ( ), ( ) P x = x w x = x Q x = x 17

18 Orthogolity of Eigefuctios (cot.) Hece we hve: ( ) ( ) ( ) u x u x w x dx = m ( ) ( ) J p x J p x xdx = 0, m m m m th ( ) 0 ( p m J ) J p = = root of Bessel fuctio where m m 18

19 Adjoit i Lier Alger For complex mtrix the djoit is give y : H t A = A A ( i.e., the cojugte of the trspose) Proof : To show this, we eed to show : H Au, v = u, A v where, = = i i i To show thi s : ( ) Au, v = A u v i j = u Av ij j i j ij i i j H i ( ij j ) i ji j j ij i ( relelig id j ) H u, A v = u A v = uav = u Av ) i j i j i j 19

20 Adjoit i Lier Alger (cot.) For complex mtrix we hve estlished tht H A = A Therefore, if complex mtrix is self-djoit, this mes tht the mtrix is Hermeti: Defiitio of Hermeti mtrix : H [ A] = A ( Aij = Aji ) Note: For rel mtrix, self-djoit mes tht the mtrix is symmetric. 0

21 Orthogolity i Lier Alger Becuse Hermeti mtrix is self-djoit, we hve the followig properties: The eigevlues of Hermeti mtrix re rel. The eigevectors of Hermeti mtrix correspodig to distict eigevlues re orthogol. The eigevectors of Hermeti mtrix correspodig to the sme eigevlue c e chose to e orthogol (proof omitted). 1

22 Digolizig Mtrix If the eigevectors of N N mtrix [A] re lierly idepedet, the it c e digolized s follows: [ A] = [ e][ D] [ e] 1 (proof o ext slide) [ D] λ λ λn = e e e e = = e e [ e] [ e ] [ e ] [ ] [ e ] N 1 N 1 N e31 e3 e 3N e e e N [ ] th e = eigevector ( colum vector) correspodig to eigevlue λ

23 Digolizig Mtrix (cot.) Proof [ e][ D] λ [ e ] λ [ e ] λ [ e ] ( from properties of digol mtrix) = 1 1 N N [ A][ e] λ [ e ] λ [ e ] λ [ e ] ( from properties of eigevectors) = 1 1 N N Hece, we hve [ A][ e] = [ e][ D] so tht [ A] = [ e][ D][ e] 1 Note: The iverse will exist sice the colums of the mtrix [e] re lierly idepedet y ssumptio. 3

24 Digolizig Mtrix (cot.) If the mtrix [A] is Hermeti, the we c show tht: [ e] 1 = e H ( the eigevlue mtrix is uitry) This follows from the orthogolity property of the eigevectors: [ ] [ ] ( ) H e e = I idetity mtrix Note: For the digol elemets, the eigevectors c lwys e scled so tht e e = 1 Therefore, for Hermeti (self-djoit) mtrix we hve: = [ ] [ ][ ] H A e D e A Hermeti mtrix is lwys digolizle! 4