18 Sturm-Liouville Eigenvalue Problems

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1 18 Sturm-Liouville Eigenvlue Problems Up until now ll our eigenvlue problems hve been of the form d 2 φ + λφ = 0, 0 < x < l (1) dx2 plus mix of boundry conditions, generlly being Dirichlet or Neumnn type. This is too nrrow of viewpoint, which I wish to point out through few exmples. Exmple 1: Simple sptil vrition in diffusivity: D = D 0 (1 + x) 2. Consider the problem u t = D 0 [(1 + x) 2 u x ] x 0 < x < 1, t > 0 u(x, 0) = f(x) 0 < x < 1 u(0, t) = 0 = u(1, t) t > 0 (2) From the seprtion of vribles method, u(x, t) = T (t)φ(x), we obtin dt dt = λd 0T nd d dφ [(1 + x)2 dx dx ] + λφ = 0, with φ(0) = φ(1) = 0. Note tht by crrying through the differentition, the φ eqution (1 + x) 2 d2 φ + 2(1 + x)dφ + λφ = 0 (3) dx2 dx is Cuchy-Euler eqution (recll the Review of ODEs ppendix for discussion of the equtions). If we write φ(x) = (1 + x) r, the chrcteristic eqution for r becomes r(r 1) + 2r + λ = 0 r = { 1 ± 1 4λ}/2. Becuse we hve to strt t φ = 0 t x = 0, nd end t φ = 0 t x = 1, ssume we need oscilltory solutions like we got out of (1), since λ 0. Hence, ssume λ > 1/4, nd define (for nottionl convenience) ω := λ 1/4. Then the roots cn be written s r = 1/2 ± iω, nd since (1 + x) 1/2±iω = (1 + x) 1/2 (1 + x) ±iω = (1 + x) 1/2 e ±iωln(1+x), 1

2 suitble fundmentl set of solutions to (3) is (1 + x) 1/2 cos(ωln(1 + x)), (1 + x) 1/2 sin(ωln(1 + x)). Now φ(x) cn be written s liner combintion of these functions, nd since φ(0) = 0, we hve φ(x) = B(1 + x) 1/2 sin(ωln(1 + x)) stisfying (3) nd this boundry condition. Now 0 = φ(1) = 2 1/2 B sin(ωln(2)) sin(ωln(2)) = 0 ωln(2) = nπ, n 1. Thus, ω 2 = λ 1/4 = (nπ/ln(2)) 2. Therefore, the eigenvlues nd ssocited eigenfunctions for this problem re { λn = 1/4 + ( nπ ln(2) )2 n = 1, 2, 3,... φ n (x) = (1 + x) 1/2 sin( nπ ln(1 + x)) (4) ln(2) This gives us the solution form for problem (2): u(x, t) = (1 + x) 1/2 e D 0t/4 n=1 ( ) B n e D 0n 2 π 2 t/(ln(x)) nπ 2 sin ln(1 + x). ln(2) (5) Remrk: If we would not hve mde the bove positivity ssumption on λ in Exmple 1, then ssume λ < 1/4 nd define α := λ > 0. Then solution of the chrcteristic eqution would be r = 1/2 ± α, nd so φ(x) = (1 + x) 1/2 {A(1 + x) α + B(1 + x) α }. Now φ(0) = 0 = A + B, so φ(x) = A(1+x) 1/2 {(1+x) α (1+x) α }, while φ(1) = 0 = A2 1/2 [2 α 2 α ], which implies A = 0 since α > 0; so φ 0. Therefore, there is no eigenvlue λ < 1/4. We ll leve it s n exercise to drw the sme conclusion bout λ = 1/4. Exercise: A vrible density vibrting string problem Determine the eigenvlue problem for the following problem, nd derive the eigenvlues nd ssocited eigenfunctions: (1 + x) 2 u tt = c 2 u xx 0 < x < l, t > 0, c > 0 is constnt u(x, 0) = f(x), u t (x, 0) = 0 0 < x < l u(0, t) = 0 = u(l, t) 2

3 Exercise: In considering coustic mesurements in thin tube scled to be of unit length, let p(x, t) be coustic pressure, nd v(x, t) be volume velocity. Given certin ssumptions, one model to consider is p x = ρ v A(x) t, v x = A(x) p ρc 2 t, where A(x) is the vrible cross-sectionl re of the tube t loction x, ρ is the ir density in the tube, nd c is the speed of sound. Suppose we hve scled the problem so tht c = 1 nd ρ = 1. Assume lso tht A(x) is continuously differentible function, nd A(x) > 0 on [0, 1]. First eliminte v in the bove system to obtin single eqution for p(x, t). Let p(0, t) = 0 nd p x (1, t) = 0 for ll t > 0. Then seprte vribles, p(x, t) = T (t)φ(x), to obtin the EVP for this p-problem. (The eigenvlue eqution is sometimes clled Webster s horn eqution.) For generl A(x) stisfying the bove conditions, if we ssume rel eigenvlues, then 1. Show tht ny eigenvlue λ must stisfy λ Show tht λ = 0 is not n eigenvlue of the problem. 3. In the specil cse A(x) = e x, where 0, write out wht the EVP is in this cse. Then show tht the eigenvlues λ n must stisfy the trnscendentl eqution 2 λ 2 /4/ = tn( λ 2 /4), nd hence, there is n infinite ordered set of them, with λ n s t. Exmple 2: Symmetric diffusion in disk For multidimensionl diffusion equtions, specil cses rise in the cse of domins with nice geometry, for exmple disk nd wedge shped sptil domins in R 2, nd sphericl domins in R 3. In the 2D sitution, the Lplcin in polr coordintes is 2 = 1 r or in cylindricl coordintes, we hve 2 = 1 r r (r r ) + 1 r 2 2 θ 2 (6) r (r r ) + 1 r 2 2 θ z 2 (7) 3

4 Figure 1: Coordinte ngle definitions tht will be used for sphericl coordintes in these Notes. nd in sphericl coordintes (see Figure 1), 2 = 1 ρ 2 ρ (ρ2 ρ ) + 1 ρ 2 sin 2 φ 2 θ ρ 2 φ + cotφ 2 ρ 2 φ. (8) In fct, the rdil prt of the Lplcin, in rbitrry n dimensions (n 1),with r nottionlly denoting the rdil distnce from the origin, is given by 2 r = 2 r + n 1 2 r r. We will briefly look t some specil problems in higher dimensions lter in these Notes, but for now let us consider the diffusion eqution on the disk sptil domin Ω := {(r, θ) : 0 r <, 0 θ < 2π}, nd consider the symmetric cse (u is independent of ngle θ) u t = D(ru r r) r r <, t > 0, D > 0 is constnt u(r, 0) = f(r) r < (9) u(, t) = 0 u remins bounded on Ω Let u(r, t) = T (t)φ(r), then (1/DT ) dt λdt, s usul, nd = 1 d dt rφ dr dφ (r ) = λ, so dt/dt = dr d dφ (r dr dr ) + λrφ = 0 = r d2 φ dr + dφ + λrφ (10) 2 dr 4

5 with φ() = 0 nd φ is bounded t r = 0. Note tht (10) is not Cuchy- Euler eqution, becuse of the r dependence ssocited with the λ term. But it is well-studied eqution becuse it rises so much in prctice; (10) is Bessel s eqution of order 0, nd we ll study this vrible coefficient EVP lter. However, n introduction to Bessel s eqution nd Bessel functions is given in Appendix F. The point here in introducing these exmples is to motivte us to briefly study more generl clss of EVPs clled regulr Sturm-Liouville Eigenvlue problems. They hve the form d dx dφ (p(x) ) q(x)φ + λσ(x)φ = 0 dx αφ() + β dφ dx () = 0 γφ(b) + δ dφ dx (b) = 0 < x < b The functions nd prmeters in (11) must meet the following conditions: (11) p(x) is continuous on [, b], continuously differentible on (, b), p(x) > 0 on [, b]. q(x), σ(x) re continuous on [, b], σ(x) > 0, q(x) 0 on [, b]. α, β, γ, δ re rel constnts. Remrk: The sign convention on the q term in the eqution is not universl. We use the negtive sign in the eqution so ll the inequlities in the bove conditions re either > or. Exmple 3: In our usul exmple φ + λφ = 0, 0 < x < l, φ(0) = 0 = φ(l), p(x) 1, q(x) 0, σ(x) 1, nd β = δ = 0. Then we obtin λ = λ n = n 2 π 2 /l 2, φ = φ n (x) = sin(nπx/l), n = 1, 2, 3,.... Given tht we hve explicit representtions for the eigenvlues nd eigenfunctions we cn mke some strightforwrd observtions: 1. The eigenvlues re rel nd ordered; tht is, λ 1 < λ 2 < λ 3 <..., with λ n s n. 2. Corresponding to ech λ n is n eigenfunction, φ n = sin(nπx/l), tht hs n 1 zeros in the intervl (0, l) (see Figure 2). 5

6 Figure 2: Note the splicing of zeroes of successive eigenfunctions for Exmple The eigenfunctions {sin(nπx/l)} n 1 form n orthogonl set of functions on (0, l); tht is, l 0 l < φ n, φ m >:= φ n (x)φ m (x)dx = 0 { 0 if n m sin(nπx/l) sin(mπx/l)dx = l/2 if n = m. 4. The eigenfunctions re complete with respect to the set of piecewise smooth functions f on (0, l); tht is, we cn, for such function f, write f(x) n=1 nφ n (x), where the infinite sum converges for ll x (0, l), to [f(x+) + f(x )]/2, if the coefficients re chosen to be the Fourier coefficients of f. Tht is, n =< f, φ n > / < φ n, φ n >. The gol here is to present the cse tht problems of the form (11) with coefficients stisfying the bulleted items hve the sme properties s our prototypicl EVP we hve been working with. (So our prototypicl EVP is regulr Sturm-Liouville Eigenvlue problem.) 6

7 Figure 3: This shows the first 5 eigenfunctions ssocited with the eigenvlue problem (1 + x) 2 φ + λφ = 0, φ(0) = φ(1) = 0. Exmple 1, gin: Returning to exmple 1, p(x) = (1 + x) 2, q(x) 0, nd β = δ = 0, so this exmple leds to regulr Sturm-Liouville EVP. Exmple 2, gin: From (10), p(r) = r, q(r) 0, nd σ(r) = r, nd on the intervl [0, ], α = 0, δ = 0. Now the smoothness conditions in the bulleted conditions is stisfied by this problem, but p(r) nd σ(r) re not strictly positive on the closed intervl [0, ]. However, p, σ re zero only t the boundry point r = 0, otherwise the conditions re met. So exmple 2 is n exmple of singulr Sturm-Liouville EVP, but it is close enough to the regulr cse tht wht properties re brought up below for the regulr Sturm-Liouville EVP will lso hold the singulr Sturm-Liouville EVP too. Exercise: For the exercise on pge 2, wht is the p, q, σ for the derived EVP? Sturm-Liouville Theorem: The regulr Sturm-Liouville EVP defined by (11) nd the bulleted points below (11) stisfies 7

8 1. There exists n infinite number of discrete eigenvlues, λ n, n = 1, 2,..., tht re rel, positive, ordered, nd λ n s n. 2. The eigenfunctions corresponding to different eigenvlues re orthogonl on [, b] with respect to σ; tht is, for the eigenvlue-eigenfunction pirs {λ i, φ i }, {λ j, φ j }, λ i λ j, < φ i, φ j >= φ i(x)φ j (x)σ(x)dx = Eigenfunctions of the sme eigenvlue re unique up to multiplictive constnt. 4. The nth eigenfunction φ n (x) ssocited with the nth eigenvlue λ n hs exctly n 1 zeros in (, b). (For n exmple, see Figure 3.) 5. {φ n } n 1 re complete with respect to piecewise smooth functions f on [, b]. Thus, {f(x) N 1 nφ n (x)} 2 σ(x)dx 0 s N. Our intention is not go through full proof of this theorem here, but to go through some prts of it to illustrte the rguments. Consult more dvnced tretment of Sturm-Liouville EVPs to get the full story. For purposes here let the boundry conditions for (11) be the specil Dirichlet conditions: φ() = 0 = φ(b). Clim 1: Any eigenvlue of (11) is rel Let λ be ny eigenvlue of (11), with ssocite eigenfunction φ(x). If λ is complex, then λ = λ r + iλ i nd its complex conjugte is λ = λ r iλ i, with ssocited eigenfunction ψ(x) = φ(x). Since {λ, φ} stisfies d dx (pdφ ) qφ + λσφ = 0 on < x < b (12) dx φ() = 0 = φ(b) then, by tking the complex conjugtion of the eqution (12), nd noting tht p, q nd σ re rel functions, { λ, ψ} stisfies d dx (pdψ dx ) qψ + λσψ = 0 on < x < b (13) ψ() = 0 = ψ(b) 8

9 So, multiply eqution (12) by ψ nd multiply (13) by φ, then subtrct the two resulting equtions. This gives us Now integrte: By integrtion-by-prts, then ψ(pφ ) φ(pψ ) + (λ λ)σψφ = 0. [ψ(pφ ) φ(pψ ) ]dx + (λ λ) [ψ(pφ ) φ(pψ ) ]dx = ψpφ b (λ λ) σφψdx = (λ λ) σφψdx = 0. pψ φ dx {φpψ b φ 2 σdx = 0. Since φ 2 = φψ = φ φ > 0, then λ = λ, which implies λ is rel. pφ ψ dx} = 0 Clim 2: λ > 0 Let {λ, φ} be ny eigenvlue-eigenfunction pir, then by (12), φ(pφ ) qφ 2 + λσφ 2 = 0 on intervl (, b). So, 0 = φ(pφ ) dx qφ 2 dx + λ σφ 2 dx. By integrtion-by-prts, the first integrl, fter pplying the boundry conditions, is p(φ ) 2 dx. Thus, λ = {p(φ ) 2 + qφ 2 }dx 0. (14) φ2 σdx Becuse of the positivity conditions we imposed on p, q, σ, nd the fct the φ is non-zero function, the nomintor in (14) is positive, so λ > 0. Remrk: The right side quotient of (14) cn be considered functionl of φ, so write λ = R[φ]. 9

10 R[ ] is clled the Ryleigh quotient, nd plys big prt in chrcterizing the eigenvlues in Sturm-Liouville EVPs. An outline of the use of the Ryleigh quotient in chrcterizing eigenvlues through minimiztion principle is presented in Appendix E. Clim 3: Eigenfunctions corresponding to different eigenvlues re orthogonl with respect to σ(x). Let {λ, φ}, {µ, ψ} be two rbitrry eigenvlue-eigenfunction pirs s solutions to (12), with λ µ. Thus, (pφ ) qφ + λσφ = 0, φ() = φ(b) = 0 (pψ ) qψ + µσψ = 0, ψ() = ψ(b) = 0 Multiply the first eqution by ψ, the second eqution by φ, subtrct nd integrte: [ψ(pφ ) φ(pψ ) ]dx + (λ µ) σφψdx = 0. The first integrl is 0 vi integrtion-by-prts nd boundry conditions. Since λ µ, then σφψdx = 0, which ws to be proved. Clim 4: Eigenfunctions of the sme eigenvlue re unique up to multiplictive constnt. Let φ, ψ be two eigenfunctions ssocited with the sme eigenvlue λ. Then 0 = φ[(pψ ) qψ + λσψ] ψ[(pφ ) qφ + λσφ] = φ(pψ ) ψ(pφ ) = [p(φψ ψφ )] which implies p(φψ ψφ ) = constnt = C. Applying the boundry conditions leds to C = 0, so φψ ψφ = 0. (15) This sttement should be recognizble s the Wronskin of ψ nd φ. Assume neither φ or ψ vnish in the intervl, then we cn write this expression s ψ /ψ = φ /φ, or (ln ψ) = (ln φ), or ln ψ ln φ = constnt ln(ψ/φ) = 10

11 constnt ψ/φ = constnt; tht is, ψ = kφ for some constnt k. Now if there is n x 0 where φ(x 0 ) = 0, for exmple, then from (15), ψ(x 0 )φ (x 0 ) = 0. But if φ (x 0 ) = 0, then φ(x) 0 becuse φ(x) is the solution to homogeneous second-order liner ode with zero initil conditions. Since φ is n eigenfunction, this cn not be the cse, so ψ(x 0 ) = 0, which mens ψ = kφ holds utomticlly t tht point. We will not pursue proving further conclusions of the Sturm-Liouville theorem, but you cn see the pttern of resoning behind it. A point here is tht in most cses involving vrible coefficient EVPs, we do not hve much hope of obtining n explicit formuls for the eigenvlues nd eigenfunctions, but the generl Sturm-Liouville problems behve qulittively exctly like our simpler, constnt coefficient EVP. Remrk: About n infinite number of eigenvlues going off to infinity: consider the Sturm-Liouville problem (11) gin, nd write (p(x)φ ) q(x)φ = F (x) < x < b φ() = 0 = φ(b) (16) where now we forget for moment tht F (x) = λσ(x)φ(x). It turns out, s we discuss lter, tht there exists function G(x, ξ), clled the Green s function for the problem (16), such tht the solution to the problem cn be written s φ(x) = G(x, ξ)f (ξ)dξ. For our eigenvlue problem, F is in terms of the solution (nd its eigenvlue), so this sttement gives the integrl eqution φ(x) = λ G(x, ξ)σ(ξ)φ(ξ)dξ. (17) Tht is, given λ, its ssocited eigenfunction φ stisfies (17), Fredholm integrl eqution of the first kind. The study of integrl equtions ws very intense in the erly prt of the twentieth century, nd hs been vluble wy to obtin properties of solutions to ordinry (nd prtil) differentil equtions. One of the consequences, when λ = λ n nd φ = φ n (x) is 11

12 Bessel s inequlity which gives n=1 λ 2 n φ 2 n σφ2 ndx G(x, ξ) 2 σ(ξ)dξ, 1 λ 2 n=1 n = σ(x) n=1 λ 2 n φ 2 n(x) σφ2 ndξ dx G(x, ξ) 2 σ(ξ)σ(x)dξdx <, so 1 n=1 is convergent series. This implies 1/λ 2 λ 2 n 0 s n ; tht is, n λ n s n. Summry: Mke sure you know the definition of regulr Sturm-Liouville EVP, nd in our limited discussion wht distinguishes it from singulr Sturm-Liouville EVP. You should know the sttement of the Sturm-Liouville Theorem, tht is, the properties of the solutions {λ n, φ n }. Finlly, be ble to recll the Ryleigh quotient for given problem. Exercises: Consider the eigenvlue problem d 2 φ ν dφ + λφ = 0 0 < x < π dx 2 dx dφ dφ (0) = (π) = 0 dx dx ν > 0 is constnt 1. Put the eqution into Sturm-Liouville form. Wht functions correspond to p(x), q(x), σ(x)? From this wht do you know bout the eigenvlues nd eigenfunctions without trying to compute them? 2. Derive the set of eigenvlues nd ssocited eigenfunctions for this EVP. 3. In the next section we will mention PDE eigenvlue problems, but non-stndrd one is the Stekloff problem 2 u = 0 u = λu ν in Ω on Ω 12

13 where the eigenvlue ppers in the boundry condition 1. Consider 1D problem with Ω = (0, 1), so the eqution becomes u = 0 in (0, 1). Determine the set of eigenvlues for this problem. 1 ν is the unit vector defined on the boundry of Ω, so u/ ν is the flux of u out of the domin. 13