A second look at separation of variables

Transcription

1 A secod look at separatio of variables Assumed that u(x,t)x(x)t(t) Pluggig this ito the wave, diffusio, or Laplace equatio gave two ODEs The solutios to those ODEs subject to boudary ad iitial coditios ofte ivolved Fourier series

2 Geeralisatio The Sturm Liouville Equatio Commo framework for the ODEs that result from the PDEs of iterest Solutios of SL equatios are sums of orthogoal fuctios More geeral defiitio of orthogoality Two other examples of orthogoal fuctios Legedre polyomials & Laplace i spherical polars Besselfuctios & diffusio i cylidrical polars

3 Sturm Liouville Equatio I the early th cetury, these two mathematicias realised that may of the ODEs that arise from PDEs, for example by separatio of variables, have the form Example: choosig Example, the SL equatio: d U dx r ( x) du q( x) + λp( x) + + U r( x) dx r( x) d U r( x) p( x), q( x) + λu dx r d U du x) x, q( x), p( x) ( x ) x + λu dx dx ( As we will see, this arises i the solutio of Laplace s Eq i spherical polars Example, the SL equatio: d U dx x du x + + ( ) U x dx x This arises i the solutio of the diffusio equatio i cylidrical polars

4 Orthogoality You are familiar with the idea that vectors u, v are orthogoal if u v u v u v + uv + u3v3 You are also familiar with orthogoality relatios i Fourier e.g. π cos mθ cos θ π m m series : We ca defie the scalar product of two fuctios f,g ad from this talk about sets of fuctios beig orthogoal. The Fourier fuctios si θ, cos mθ are but oe case.

5 Orthogoality of fuctios We begi by defiig a scalar product of two fuctios f, g: b f, g f ( x) g( x) w( x) dx a I the most geeral case, w(x) is a weightig fuctio, though it will mostly but ot always - be equal to i this course { ( x),... } f A set of fuctios is said to be orthogoal whe f m, f, m There are lots of such sets: Walsh, Bessel, Legedre,, wavelets

6 Legedre orthogoal fuctios & SL equatio The Legedre fuctios are defied by a recurrece relatio ( + ) P + ( x) ( + ) xp ( x) P ( x),,,... The first few are: P o P ( x) ( x) ; x; P x P3 ( x) (5x ( ) ( x 3x ) ; 3)

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8 Legedre orthogoal fuctios & SL equatio The Legedre fuctios are defied by a recurrece relatio ( + ) P + ( x) ( + ) xp ( x) P ( x),,,... The first few are: P o ( ) ( ) ( ) ( ) x ; P x x; P x 3x ; P ( x) (5x 3) 3 The Legedre fuctios are orthogoal: P m ( x) P ( x) The Legedre fuctios are solutios of a SL Equatio: x dx + d u du ( x ) x + ( + ) u By superpositio, the geeral solutio of this equatio is: u( x dx ) a P dx ( x) for m for m

9 Steady state solutio of a spherical capacitor We cosider a spherical capacitor, radius a, with mid-plae isulator; the potetial o the top hemispherical surface is held at V, o the bottom surface at zero Rotatioal symmetry meas that the potetial, u u(r, θ, Φ) does ot i fact deped o θ, so we ca write u(r, Φ) Mathematically, the symmetry i θ implies: u θ Sice it is steady state, we have to solve Laplace s Equatio, which i spherical polars* reduces to: r r u r + *see Kreysig. page 636 (eq 9) u siφ siφ φ φ u V φ r Isulator u Boudary Coditios u( a, φ ) V φ < π u( a, φ ) π φ π

10 Step : separatio of variables We assume that u ( r, φ) R( r) H( φ) Proceedig i the ow familiar way, we geerate two ODEs: r cosφ H + H R + rr siφ R H k Rearragig terms: cosφ H + H + kh siφ r R + rr kr () ()

11 Step a: solve for H Substitutig w cosφ we fid that It follows that dh dφ d H dφ Substitutig k (+) dh dw dw dh siφ dφ dw d dh siφ dφ dw dh cosφ dw + si d H φ dw cosφ d H dh H + H + kh ( w ) w + kh siφ dw dw d ( w ) w + ( + ) H dw H dh dw Legedre s Equatio!!

12 Step b: solve for R We have This has the geeral solutio: r R R + rr kr ( ) ( +) r A r + B r Solvig first iside the sphere, we isist that the solutio remai fiite at r, so that the costats B are all zero. The geeral solutio becomes u ( r, φ ) A r P, th ( w), where w cosφ P is the Legedre polyomial To solve for the costats A we apply the boudary coditios at ra ad use the Legedre polyomial orthogoality coditios

13 Bessel fuctios Defiitio : they are the solutios to a S-L equatio; but there is o eat recurrece relatio Bessel fuctios arise very ofte i egieerig, for example ultrasoud They arise as solutios to diffusio equatios i cylidrical polars

14 aother SL Equatio, whose solutios are Bessel fuctios The Sturm-Liouville equatio is also called Bessel s equatio: x y + y + y x x y AJ ( ) + ( ) Geeral solutio to Bessel s equatio: x BY x Mercifully, it is ofte the case that B is zero! Ufortuately, Bessel fuctios J (x) ad Y (x) are much more complicated tha Legedre fuctios, which is a shame because they are so useful. There s o eat recurrece relatio, they are usually defied i terms of ifiite series, ad usually plotted.

15 ν i are the roots of J ν ν ν ν 4 Bessel fuctios J (x) ad J (x). They are rapidly dyig ripples. Note: the axes are NOT same scale.

16 Orthogoality of Bessel fuctios The situatio is more complex ad specific tha for Fourier or Legedre. For this course, the oly orthogoality relatio that is of relevace is: dx. xj o( υx) J o( υmx) 5J ( υ ) m m Note: the weightig fuctio w(x)x x is scaled by the roots of J

17 The first two Bessel fuctios of the secod kid. For this course, the oly thig to ote is that the values are ifiite at x.

18 Cardiac Ultrasoud Adult heart (left vetricle) cm legth 7 beats/mi 3-5MHz probe 5-Hz samplig rate Multi-chael D-array, ofte with >8 chaels Piezoelectric ceramic, ofte Lead-zircoatetitaate (PZT) The pressure, show here as brightess, varies spatially accordig to a Bessel fuctio

19 Diffusio equatio i cylidrical polars We cosider trasiet Heat Flow i a Log, Uiform cylidrical rod, radius a. We assume that every poit i the rod is iitially at temperature T. At time t, we pluge the rod ito a freezig lake, at ºC (r,θ) (r,θ,z) The rod is log solutio is idepedet of z. Rod is circular, uiform solutio idep of θ I this case, diffusio equatio i cylidrical polars: u u u ( r, θ, z, t) u( r, t) c t r u + r r

20 Boudary ad iitial coditios Iitial coditio u ( r,) T, all r Boudary coditios: u( a, t), all t We assume that the solutio remais fiite as r ad t

21 Separatio of variables We assume that u ( r, t) R( r) T ( t) This leads as usual to the two equatios : c T T R + r R R K, a costat

22 Case : k T T cost R + r R R( r) T ( r, θ ) Al r + C l r B + D Sice T remais fiite as r, we have C The boudary coditio u(a,t), all t implies D

23 Case : k> t c Ce t T T c T ) ( α α Fiite solutios as t implies ), ( ) ( t r u t T C

24 There is oly case 3: k< Let k β T + β c T T ( t) Ce β c to which we retur after aalysig the secod equatio t The secod equatio is : R + R + β R r

25 The emergece of Bessel The secod equatio is : substitutig s βr,ad otig that From this we fid: dr dr dr ds ds dr R + R + β R r dr β ad ds d R dr d R β ds d R ds + s dr ds + R which is Bessel s Equatio of order The solutio is : R( r) Sice u(r,t) is fiite at r, we must have B ad so: AJ o ( s) + BY ( s) The solutio is : u( r, t) AJ ( βr) e β c t

26 Boudary coditio t c a e r a AJ t r u ) ( ), ( ν ν The solutio is : t all, ), ( t a u ) ( ), ( a J t a u β a ν β The th root of J t c e r AJ t r u ) ( ), ( β β The solutio is :

27 Now use the iitial coditio u( r,) T to determie A by orthogoality... u ν r,) T AJ ( r) a ( Use orthogoality of course! xj o ( x) J o ( υmx) dx. υ 5J ( υ ) m m Multiply both sides by xj (ν m x) where x r/a ad itegrate T xj o m a ( v x) dx A J ( v ) m So that, fially: To A a J ( v ) m a rj o vmr a dr

28 The Ed