Sturm Liouville Problems

Transcription

1 Sturm Liouville Problems More generl eigenvlue problems So fr ll of our exmple PDEs hve led to seprted equtions of the form X + ω 2 X =, with stndrd Dirichlet or Neumnn boundry conditions Not surprisingly, more complicted equtions often come up in prcticl problems For exmple, if the medium in het or wve problem is sptilly inhomogeneous,* the relevnt eqution my look like X V(x)X = ω 2 X for some function V, or even (x)x +b(x)x +c(x)x = ω 2 X Also, if the boundry in problem is circle, cylinder, or sphere, the solution of the problem is simplified by converting to polr, cylindricl, or sphericl coordintes, so tht the boundry is surfce of constnt rdil coordinte This simplifiction of the boundry conditions is bought t the cost of complicting the differentil eqution itself: we gin hve to del with ODEs with nonconstnt coefficients, such s d 2 R dr r dr dr n2 r 2 R = ω2 R The good news is tht mny of the properties of Fourier series crry over to these more generl situtions As before, we cn consider the eigenvlue problem defined by such n eqution together with pproprite boundry conditions: Find ll functions tht stisfy the ODE (for ny vlue of ω) nd lso stisfy the boundry conditions And it is still true (under certin conditions) tht the set of ll eigenfunctions is complete: Any resonbly well-behved function cn be expnded s n infinite series where ech term is proportionl to one of the eigenfunctions This is wht llows rbitrry dt functions in the originl PDE to be mtched to sum of seprted solutions! Also, the eigenfunctions re orthogonl to ech other; this leds to simple formul for the coefficients in the eigenfunction expnsion, nd lso to Prsevl formul relting the norm of the function to the sum of the squres of the coefficients * Tht is, the density, etc, vry from point to point This is not the sme s nonhomogeneous in the sense of the generl theory of liner differentil equtions 1

2 Orthonorml bses Consider n intervl [, b] nd the rel-vlued (or complex-vlued) functions defined on it A sequence of functions {φ n (x)} is clled orthogonl if It is clled orthonorml if, in ddition, φ n (x)*φ m (x)dx = whenever m n φ n (x) 2 dx = 1 This normliztion condition is merely convenience; the importnt thing is the orthogonlity If we re lucky enough to hve n orthogonl set, we cn lwys convert it to n orthonorml set by dividing ech function by the squre root of its normliztion integrl: ψ n (x) φ n (x) φ n(z) 2 dz ψ n (x) 2 dx = 1 However, in certin cses this my mke the formul for ψ n more complicted, so tht the redefinition is hrdly worth the effort A prime exmple is the eigenfunctions in the Fourier sine series: therefore, φ n (x) sinnx ψ n (x) π φ n (x) 2 dx = π 2 ; 2 π sinnx re the elements of the orthonorml bsis (This is the kind of normliztion often used for the Fourier sine trnsform, s we hve seen) A good cse cn be mde, however, tht normlizing the eigenfunctions is more of nuisnce thn help in this cse; most people prefer to put the entire 2/π in one plce rther thn put hlf of it in the Fourier series nd hlf in the cofficient formul Now let f(x) be n rbitrry (nice) function on [,b] If f hs n expnsion s liner combintion of the φ s, f(x) = c n φ n (x), then φ m (x)*f(x)dx = c n φ m (x)*φ n (x)dx = c m φ m (x) 2 dx 2

3 by orthogonlity If the set is orthonorml, this just sys c m = φ m (x)*f(x)dx ( ) (In the rest of this discussion, I shll ssume tht the orthogonl set is orthonorml This gretly simplifies the formuls of the generl theory, even while possibly complicting the expressions for the eigenfunctions in ny prticulr cse) Remrk: The integrl on the right side of ( ) is clled the inner product of φ m nd f nd is often written s φ m,f It cn be thought of s the generliztion to n infinite-dimensionl nd complex vector spce of the fmilir dot product of vectors in R 3 Then the normliztion integrl φ n 2 dx = φ n,φ n φ n 2 is the nlog of the squre of the length of vector, nd the distnce between two vectors f nd g is f g, s pplied in the next two prgrphs For more on this liner-lgebr nlogy, see pp in Appendix B of the notes (pp of the pdf file) The two previous pges of tht Appendix give simple exmple of complete set of functions nd set tht fils to be complete becuse some vectors hve been left out It cn esily be shown (just s for Fourier series) tht f(x) 2 dx = c n 2 This is the Prsevl eqution ssocited to this orthonorml set Furthermore, if f is not of the form c nφ n (x), then (1) c n 2 < f(x) 2 dx (clled Bessel s inequlity), nd (2) the best pproximtion to f(x) of the form cn φ n (x) is the one where the coefficients re computed by formul ( ) These lsttwo sttementsremintruewhen {φ n }isfinite set inwhich cse, obviously, the probbility tht given f will not be exctly liner combintion of the φ s is gretly incresed The precise mening of (2) is tht the choice ( ) of the c n minimizes the integrl f(x) 2 c n φ n (x) dx Tht is, we re tlking bout lest squres pproximtion It is understood in this discussion tht f itself is squre-integrble on [, b] Recll tht the spce of such functions is clled L 2 (or, more specificlly, L 2 (,b)) 3

4 Now suppose tht every squre-integrble f is the limit of series c nφ n (This series is supposed to converge in the men tht is, the lest-squres integrl M f(x) 2 c n φ n (x) dx for prtil sum pproches s M ) Then {φ n } is clled complete set or n orthonorml bsis This is the nlogue of the men convergence theorem for Fourier series Under certin conditions there my lso be pointwise or uniform convergence theorems, but these depend more on the specil properties of the prticulr functions φ being considered So fr this is just definition, not theorem To gurntee tht our orthonorml functions form bsis, we hve to know where they cme from The mircle of the subject is tht the eigenfunctions tht rise from vrible-seprtion problems do form orthonorml bses: Sturm Liouville theory is Theorem: Suppose tht the ODE tht rises from some seprtion of vribles L[X] = λx on (,L), ( ) where L is n bbrevition for second-order liner differentil opertor L[X] (x)x +b(x)x +c(x)x,, b, nd c re continuous on [,L], nd (x) > on [,L] Suppose further tht ( ) L L[u](x) *v(x)dx = u(x)* ( L[v](x) ) dx ( ) for ll functions u nd v stisfying the boundry conditions of the problem (In terms of the inner product in L 2, this condition is just Lu,v = u,lv An opertor stisfying this condition is clled self-djoint or Hermitin ) Then: (1) All the eigenvlues λ re rel (but possibly negtive) (2) The eigenfunctions corresponding to different λ s re orthogonl: φ n (x)*φ m (x)dx = if n m There is technicl distinction between these two terms, but it does not mtter for regulr Sturm Liouville problems 4

5 (3) The eigenfunctions re complete (This implies tht the corresponding PDE cn be solved for rbitrry boundry dt, in precise nlogy to Fourier series problems) The proof tht given L stisfies ( ) (or doesn t stisfy it, s the cse my be) involves integrting by prts twice It turns out tht ( ) will be stisfied if L hs the form d dx p(x) d dx +q(x) (with p nd q rel-vlued nd well-behved, nd p(x) > ) nd the boundry conditions re of the type αx () βx() =, γx (L)+δX(L) = with α, etc, rel* Such n eigenvlue problem is clled regulr Sturm Liouville problem The proof of the conclusions (1) nd (2) of the theorem is quite simple nd is generliztion of the proof of the corresponding theorem for eigenvlues nd eigenvectors of symmetric mtrix (which is proved in mny physics courses nd liner lgebr courses) Prt (3) is hrder to prove, like the convergence theorems for Fourier series (which re specil cse of it) Exmple: Convective boundry condition The simplest nontrivil exmple of Sturm Liouville problem ( nontrivil in the sense tht it gives something other thn Fourier series) is the usul sptilly homogeneous het eqution u t = 2 u x 2 ( < x < L, < t < ), with boundry conditions such s nd initil dt u(,t) =, u x (L,t)+βu(L,t) = u(x,) = f(x) In relistic problem, the zeros in the BC would be replced by constnts; s usul, we would tke cre of tht compliction by subtrcting off stedy-stte solution Physiclly, the constnt vlue of u (L,t)+βu(L,t) is proportionl to the x * The reson for the minus sign in the first eqution is to mke true property (7) stted below 5

6 temperture of the ir (or other fluid medium) to which the right-hnd endpoint of the br is exposed; het is lost through tht end by convection, ccording to Newton s lw of cooling Mthemticlly, such BC is clled Robin boundry condition, s opposed to Dirichlet or Neumnn The seprtion of vribles proceeds just s in the more stndrd het problems, up to the point T(t) = e ω2t, X(x) = sinωx, ω λ To get the sine I used the boundry condition X() = The other BC is X (L)+βX(L) =, or or ω β cosωl+sinωl =, ( ) tnωl = 1 β ω ( ) Itisesytofindthepproximte loctionsoftheeigenvlues, λ n = ω n 2, bygrphing the two sides of ( ) (s functions of ω) nd picking out the points of intersection (In the drwing we ssume β > ) ω π 1 ω 2π 2 ω 3π 3 ω 4 L L L 4π L π 2L 3π 2L 5π 2L 7π 2L The nth root, ω n, is somewhere between ( n 2) 1 π L nd nπ L ; s n, ω n becomes rbitrrily close to ( n 2) 1 π L, the verticl symptote of the tngent function For smller n one could guess ω n by eye nd then improve the guess by, for exmple, Newton s method (Becuse of the violent behvior of tn ner the 6

7 symptotes, Newton s method does not work well when pplied to ( ); it is more fruitful to work with ( ) insted) To complete the solution, we write liner combintion of the seprted solutions, u(x,t) = b n sinω n xe ω 2t n, nd seek to determine the coefficients from the initil condition, f(x) = u(x,) = b n sinω n x This problem stisfies the conditions of the Sturm Liouville theorem, so the eigenfunctions ψ n sinω n x re gurnteed to be orthogonl This cn be verified by direct computtion (mking use of the fct tht ω n stisfies ( )) Thus f(x) sinω m xdx = b m sin 2 ω m xdx However, the ψ n hve not been normlized, so we hve to clculte sin 2 ω m xdx ψ m 2 nd divide by it (This number is not just 1 L, s in the Fourier cse) Alterntively, 2 we could construct orthonorml bsis functions by dividing by the squre root of this quntity: Then the coefficient formul is simply B m = φ n ψ n ψ n f(x)φ m (x)dx (where f(x) = m B mφ m, so B m = ψ m b m ) The theorem lso gurntees tht the eigenfunctions re complete, so this solution is vlid for ny resonble f (Nevertheless, if β < it is esy to overlook one of the norml modes nd end up with n incomplete set by mistke See Hbermn, Figs 582 nd 583) 7

8 More properties of Sturm Liouville eigenvlues nd eigenfunctions Continution of the theorem: For regulr Sturm Liouville problem: (4) For ech eigenvlue λ there is t most one linerly independent eigenfunction (Note: This is true only for the regulr type of boundry conditions, αx () βx() =, γx (L)+δX(L) = For periodic boundry conditions there cn be two independent eigenfunctions for the sme λ, s we know from Fourier series) (5) λ n pproches + s n (6) φ n (x) is rel nd hs exctly n 1 zeros ( nodes ) in the intervl (,L) (endpoints not counted) The bsic reson for this is tht s λ increses, φ becomes incresingly concve nd oscilltory (This property, lso, depends on regulr boundry conditions Clerly it is not true of the eigenfunctions e inx in the full Fourier series) (7) If α, β, γ, δ, p(x), nd q(x) re ll nonnegtive, then the λ n ω n 2 re ll nonnegtive (Corollry: For the het eqution, the solution u(x, t) pproches st + iflltheeigenvluesrepositive; itpproches constnt ifω = occurs) Note tht prts (1) nd (7) of the theorem mke it possible to exclude the possibilities of complex nd negtive eigenvlues without detiled study of the solutions of the ODE for those vlues of λ In first lerning bout seprtion of vribles nd Fourier series we did mke such detiled study, for the ODE X = λx, but I remrked tht the conclusion could usully be tken for grnted (Indeed, Appendix A gives the proof of (1) nd (7), specilized to X = λx) A good exercise: For regulr Sturm Liouville problem with differentil opertor L = d dx p(x) d dx +q(x), prove ( ) nd (7) long the lines previously indicted Nonstndrd weight functions Unfortuntely, not ll importnt problems fit exctly into this frmework Sometimes the ODE hs the form (insted of ( )) L[X] = λr(x)x, 8

9 where r(x) > on the intervl nd the opertor is still of the form L[X] = d dx p(x) d dx +q(x) (In prticulr, the rdil opertors tht come from seprting vribles in polr nd sphericl coordintes re of this type) From the point of view of generl liner lgebr it would mke more sense to define new opertor A[X] L[X] r(x), A[X] = λx, nd to study its eigenvlues nd eigenfunctions However, it is L, not A, tht stisfies the self-djointness property ( ) with respect to the stndrd inner product on functions in L 2 Insted, A is self-djoint with respect to new inner product, u,v r u(x) v(x)r(x)dx; Au,v r = u,av r In this cse, sttement (2) of the theorem must be revised: (2 ) The eigenfunctions corresponding to different eigenvlues re orthogonl with respect to the weight function r(x): φ n,φ m r = φ n (x)*φ m (x)r(x)dx = if n m (Everything sid previously bout orthonormlity cn be generlized to the cse of nontrivil positive weight function Thinking bout the inner product bstrctly, insted of s certin integrl, is very powerful here) In principle, weight function cn lwys be voided by mking chnge of vrible so tht dy = r(x)dx But in prctice tht my complicte the differentil opertor, mking the cure t lest s bd s the disese Singulr Sturm Liouville problems If one of the coefficient functions in the opertor L violtes condition in the definitionofregulrsturm Liouvilleproblem tnendpoint (eg, ifp() =, orif q(x) s x L), or if the intervl is infinite, then the problem is clled singulr (insted of regulr) Mny of the most importnt rel-life cses re singulr Under these conditions the foregoing theory cquires complictions, which I cn discuss only very loosely here 9

10 1 The set of eigenfunctions needed to expnd n rbitrry function my depend on λ s continuous vrible, s in the cse of the Fourier trnsform 2 The boundry conditions needed to get n orthogonl nd complete set of eigenfunctions my be of different type The criticl condition tht must be kept stisfied is( ) In prticulr, if one of the endpoints moves to infinity, then usully there is no boundry condition there of the type γx (L)+δX(L) = ; insted, one merely excludes solutions tht grow exponentilly fst t infinity If ll the remining solutions go rpidly to zero t infinity, so tht they re squre-integrble, then the eigenfunction expnsion will be series, s in the regulr problems If the remining solutions do not go to zero, then typiclly ll of them re needed to form complete set, nd one hs sitution like the Fourier trnsform Eigenfunctions, delt functions, nd Green functions Let s return to the regulr cse nd ssume tht the eigenfunctions hve been chosen orthonorml (For simplicity I lso ssume tht r(x) = 1; otherwise fctors r nd 1/r will show up in some of the formuls below) We hve n expnsion formul f(x) = c n φ n (x) ( ) nd coefficient formul c m = φ m (z)*f(z)dz ( ) Substituting ( ) into ( ) nd interchnging the order of summtion nd integrtion yields [ ] f(x) = dzf(z) φ n (x)φ n (z)* In other words, when cting on functions with domin (,b), δ(x z) = φ n (x)φ n (z)* This is clled the completeness reltion for the eigenfunctions {φ n }, since it expresses the fct tht the whole function f cn be built up from the pieces c n φ n In the specil cse of the Fourier sine series, we looked t this formul erlier 1

11 We cn lso substitute ( ) into ( ), getting [ ] c m = c n φ m (x)*φ n (x)dx This eqution is equivlent to φ m(x)*φ n (x)dx = δ mn, where { 1 if m = n δ mn if m n (This is clled the Kronecker delt symbol; it is the discrete nlogue of the Dirc delt function or, rther, Dirc s delt function is continuum generliztion of it!) This orthogonlity reltion summrizes the fct tht the φ s form n orthonorml bsis Note tht the completeness nd orthogonlity reltions re very similr in structure Bsiclly, they differ only in tht the vribles x nd n interchnge roles (long with their lter egos, z nd m) The different ntures of these vribles cuses sum to pper in one cse, n integrl in the other Finlly, consider the result of substituting ( ) into the solution of n initilvlue problem involving the functions φ n For exmple, for certin het-eqution problem we would get u(t,x) = c n φ n (x)e ω 2t n This becomes u(t,x) = [ dzf(z) Therefore, the Green function for tht problem is G(x,z;t) = ] φ n (x)φ n (z)*e ω n 2 t φ n (x)φ n (z)*e ω 2t n When t = this reduces to the completeness reltion, since limg(x,z;t) = δ(x z) t Similrly, G(x,z;λ) = φ n (x)φ n (z)* ω n2 λ 2 11

12 is the resolvent kernel, the Green function such tht u(x) = G(x,z;λ)g(z)dz solvesthenonhomogeneousodel[u]+λ 2 u = g (ifr = 1)withthegivenboundry conditions (Our first Green function exmple constructed with the id of the delt function, severl sections bck, ws resolvent kernel) It my be esier to solve for the Green functions directly thn to sum the series in these formuls In fct, such formuls re often used in the reverse direction, to obtin informtion bout the eigenfunctions nd eigenvlues from independently obtined informtion bout the Green function 12