1. On the line, i.e., on R, i.e., 0 x L, in general, a x b. Here, Laplace s equation assumes the simple form. dx2 u(x) = C 1 x + C 2.

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1 Lecture 16 Lplce s eqution - finl comments To summrize, we hve investigted Lplce s eqution, 2 = 0, for few simple cses, nmely, 1. On the line, i.e., on R, i.e., 0 x L, in generl, x b. Here, Lplce s eqution ssumes the simple form with generl solution d 2 u = 0, (1) dx2 u(x) = C 1 x + C 2. (2) Depending on the boundry conditions imposed t the endpoints x = nd x = b, there could be unique solution, fmily of solutions, or no solution t ll. 2. In the plne, i.e., in R 2, for two specil cses: () Rectngulr region, 0 x L, 0 y H, with prescribed vlues of the function (e.g., temperture) long the boundries. Crtesin coordintes were used. (b) Circulr regions both including s well s excluding the singulr point (0,0), for which plnr polr coordintes (r,θ) re more convenient. Lplce s eqution is of gret importnce in higher dimensions, e.g., 3. In R 3, where it is most convenient to use the coordinte system tht is dpted to the symmetry of the problem being considered. Of course, this implies tht we shll hve to express the Lplcin opertor in such coordinte system. Exmples include: () Rectngulr symmetry use Crtesin coordintes (x, y, z). (b) Sphericl symmetry use sphericl polr coordintes (r,θ,φ). (c) Cylindricl symmetry use cylindricl coordintes (r,θ,z). Recll tht in ll cses, we re solving for time-independent functions u : R n R. In the pplictions considered to dte, these functions could represent stedy-stte distributions, e.g., 1. the stedy-stte or equilibrium temperture distribution u eq for system, s determined by boundry conditions, 100

2 2. the electrosttic potentil V in region, s produced by prescribed distribution of electric potentil. We lso emphsize tht in these cses, there re no sources, i.e., no sources of het or electric chrge. In the cse tht sources re present, the equilibrium/stedy-stte function provided it exists will hve to obey Poisson s eqution, e.g., 2 u = Q (3) K 0 in the cse of the het eqution. It my be possible to solve such problems nlyticlly, using n eigenfunction expnsion pproch (Question No. 8 of Problem Set No. 3 lso see Sections 8.3 nd 8.4 of text). In prctice, however, one normlly resorts to numericl methods. It is not overstting the cse to mention once gin tht the Lplce nd Poisson equtions re of gret importnce in science nd engineering. Becuse of their pplicbility to problems in electromgnetism, het nd fluid mechnics, these equtions nd methods for their solution received gret del of ttention in the 1800 s nd erly 1900 s. And from this work rose the field of Potentil Theory, the mthemticl nlysis of solutions of Lplce s eqution (hrmonic functions). We now turn our ttention to nother mjor re of mthemtics tht ws developed during the sme time period, lso due to its importnce in the understnding of solutions to the Lplce nd Poisson equtions. An introductory look t Sturm-Liouville theory Relevnt sections of textbook by Hndelsmn: Sections In ll of the PDEs exmined so fr het eqution nd wve eqution with vrious homogeneous boundry conditions the boundry vlue problem tht determined the eigenvlues, i.e., the discrete vlues λ n of the seprtion constnts, hd the simple form with homogeneous boundry conditions, such s the following: d 2 φ + λφ = 0, (4) dx2 φ(0) = 0, φ(l) = 0, Result: Fourier sine series 101

3 φ (0) = 0, φ (L) = 0, Result: Fourier cosine series φ(0) = 0, φ (L) = 0, Result: shifted Fourier sine series φ( π) = φ(π), Result: Fourier series (sine nd cosine functions) (5) In ech cse, we found tht the ssocited eigenfunctions φ n (x) formed n orthogonl set on [0,L]. We lso sw tht in ech cse, the orthogonl set ws complete, i.e., function f(x) could be expnded in terms of this set. Of course, the PDEs tht led to this boundry vlue problem were quite simplified, rising from the ssumption of constnt coefficients in other words, homogeneous medi. We would like to now generlize the bove results to the cse of PDEs with nonconstnt coefficients. In these cses, seprtion of vribles will yield ODEs with nonconstnt coefficients. Generlly speking, the resulting boundry vlue problems re no longer solvble in terms of elementry functions such s sines nd cosines. We ll show, however, tht even in these cses, the eigenvlues λ n re discrete, nd the corresponding eigenfunctions φ n (x) re orthogonl to ech other. As motivting exmple, let us consider the het eqution for nonuniform rod, without sources: c(x)ρ(x) u t = ( K 0 (x) u ). (6) x x We ll lso ssume some kind of homogeneous boundry conditions, sy, u(0,t) = 0, u(l,t) = 0, (7) so tht the seprtion of vribles method my be pplied. As before, we ssume solution of the form u(x,t) = φ(x)g(t). (8) Substitution into (6) yields which cn be seprted s follows cρφ dg dt = d ( K 0 (x) dφ ) G, (9) dx dx G G = 1 cρφ d dx ( ) dφ K 0 = λ. (10) dx The resulting boundry vlue problem for φ(x) becomes ( d K 0 (x) dφ ) + λc(x)ρ(x)φ(x) = 0 dx dx φ(0) = φ(l) = 0. (11) 102

4 This is n exmple of Sturm-Liouville eigenvlue problem. Subject to some conditions on the functions K 0 (x) (positive, piecewise C 1 ) nd c(x),ρ(x) (piecewise continuous), the existence of n infinite set of discrete (positive) eigenvlues λ n nd ssocited orthogonl eigenfunctions φ n (x) is gurnteed. Associted with these eigenvlues/eigenfunctions will be the time-dependent functions G n (t) stisfying the ODE, G n (t) + λ ng n (t) = 0, n = 1,2,. (12) Of course, we my write down the solutions immeditely: Up to constnt, G n (t) = e λnt. (13) Our seprtion of vribles method hs produced set of solutions to the nonuniform het problem in Eq. (6) of the form u n (x,t) = φ n (x)g n (t) = φ n (x)e λnt. (14) And becuse the BCs were ssumed to be homogeneous, ny liner combintion of these solutions is lso solution. If the solutions φ n (x) to the BVP in Eq. (11) cn be shown to form complete bsis set on [0,L], then ll solutions to this het problem cn be expressed in the form u(x,t) = n=1 n φ n (x)e λnt. (15) In this prticulr cse, i.e., the het eqution, the eigenvlues λ n will determine the rte of decy of the modes, i.e., the sptil functions φ n, in time. In the cse of the wve eqution tht models vibrting string, the eigenvlues λ n will correspond to the nturl frequencies of the string. The Min Result The Regulr Sturm-Liouville Eigenvlue Problem In wht follows, we dopt the nottion of the textbook see pges A regulr Sturm-Liouville eigenvlue problem consists of the Sturm-Liouville differentil eqution ( d p(x) dφ ) + q(x)φ + λσ(x)φ = 0, < x < b, (16) dx dx subject to the generl homogeneous boundry conditions of the form φ β 1 φ() + β 2 () = 0, dx φ β 3 φ(b) + β 4 (b) = 0, (17) dx 103

5 where the β i re rel. Note tht boundry conditions listed erlier in this lecture, with the exception of the periodicity condition, re specil cses of this generl form. In order for the Sturm-Liouville problem to be regulr, the coefficients p(x), q(x) nd σ(x) must stisfy some conditions over [,b], including: 1. p(x) is piecewise C 1, q(x) nd σ(x) re piecewise continuous, 2. p(x) > 0 nd σ(x) > 0. These, in prticulr the ltter, my seem to be quite restrictive, but they pply to most physicl situtions. Here, we stte the min results regrding the eigenvlue problem in Eq. (16). We ll discuss them in more detil lter. (These re listed on p. 163 of the textbook.) 1. All eigenvlues λ re rel. 2. There exists n infinite number of eigenvlues, λ 1 < λ 2 < < λ n < λ n+1 < (18) () There is smllest eigenvlue, usully denoted s λ 1. (b) There is no lrgest eigenvlue, nd λ n s n. 3. Corresponding to ech eigenvlue λ n, there is n eigenfunction, denoted s φ n (x), which is unique to within n rbitrry multiplictive constnt. The function φ n (x) hs exctly n 1 zeros in the open intervl (,b). 4. The eigenfunctions φ n (x) form complete set or bsis for the spce of functions L 2 [,b]. More on this lter. 5. Eigenfunctions belonging to different eigenvlues re orthogonl reltive to the weight function σ(x), i.e., φ n (x)φ m (x)σ(x) dx = 0 if m n. (19) 104

6 Why do we cll it n eigenvlue problem? Becuse the φ n re eigenfunctions of liner opertor, in this cse second-order liner differentil opertor. One normlly writes the Sturm-Liouville problem in the generl form Lφ + λφ = 0, (20) where the liner opertor L is given by As result, the Sturm-Liouville eqution cn be written s L = d ( p(x) d ) + q(x), (21) dx dx Lφ = λφ. (22) In other words, the ction of the opertor L on φ is to produce sclr multiple of φ: Here, the sclr is λ, the eigenvlue. In postscript to n erlier lecture, I discussed this ide for the prticulr cse L = d2 dx2. We re used to seeing eigenvlue problems expressed in the form of Eq. (22), in which cse we would sy tht φ is n eigenfunction of the opertor L. In the differentil equtions literture, however, one normlly sttes tht Eq. (20) implies tht φ is n eigenfunction of the opertor L. 105

7 Lecture 17 A Lightning Tour of Sturm-Liouville theory The following discussion is not intended to be complete. We shll briefly cover the min ides tht re responsible for the results stted in the previous lecture, i.e., the existence of discrete set of eigenvlues λ n nd ssocited eigenfunctions φ n (x). Sturm-Liouville theory is sometimes discussed in detil in AMATH 351. (Unfortuntely, however, it depends upon the instructor). An excellent reference for SL-theory is, Differentil Equtions nd Applictions with Historicl Notes, Second Edition, by G.F. Simmons, McGrw-Hill (1991), Chpter 4, Qulittive Properties of Solutions. We first clim tht it is sufficient to study the following simplified Sturm-Liouville eigenvlue/boundry vlue problem, u + λσ(x)u = 0, u() = 0, u(b) = 0, (23) where we ssume tht λ > 0 nd tht the function σ(x) > 0 for ll x. We ll lso ssume tht σ(x) is sufficiently nice, i.e., continuous nd bounded. Note: In cse you re worried bout the neglect of the p(x) nd q(x) functions in the originl SL eqution of the previous lecture, it is possible to find function α(x) > 0 so tht the following substitution, u(x) = α(x)φ(x), (24) produces function u(x) tht stisfies Eq. (23) (with different σ(x) function but it s still positive). As such, the qulittive properties of the two functions, most notbly their oscilltory behviour, re the sme in fct, they hve the sme zeros. Solutions re oscilltory In wht follows, we shll lso keep in mind the specil cse σ(x) = 1 in Eq. (23), i.e., u + λu = 0, λ > 0. (25) ignoring the boundry conditions for the moment. Of course, the generl solution to this eqution is u λ (x) = C 1 cos( λx) + C 2 sin( λx). (26) 106

8 As we ll know, ll of these solutions re oscilltory, thnks to the fct tht λ > 0. Wht we now wnt to show is tht ll solutions to the generl eqution Eq. (23) re lso oscilltory, by virtue of the fct tht the term λσ(x) is positive. The beutiful thing is tht this cn be done by mens of qulittive nlysis of Eq. (23), i.e., determining the qulittive properties of solutions to n eqution without hving to solve the eqution. Some qulittive nlysis of Eq. (23) First of ll, let s rewrite the DE in (23) s follows, u (x) = λσ(x)u(x). (27) Now ssume tht u(x 0 ) > 0 t some point x 0. By continuity u(x) > 0 on some intervl I = (,b) contining x 0. But from Eq. (25), u (x) < 0 on I, which mens tht the grph of u is concve down on I. In fct, s x increses wy from x 0, the grph of u(x) must cross the x-xis. You might sk, Why does it hve to cross the xis? Why cn t it pproch the x-xis symptoticlly, s sketched below? u (x 0 ) < 0 u (c) = 0 y = u(x) x 0 c x I 0 The reson is tht for the grph to pproch the x-xis, there would hve to be point of inflection c > x 0, fter which the grph is concve up. But from Eq. (27), the only points of inflection occur t zeros of u(x), i.e., where the grph of u(x) crosses the x-xis. There s nother question nd it ws sked in clss: Why couldn t the grph of u(x) be incresing towrds horizontl symptote s x, s sketched, for exmple, in the figure below. The nswer is tht s u(x) pproches the symptote, its second derivtive u (x) would hve to pproch zero. From Eq. (23), this implies tht u(x) zero, etc.. 107

9 v(x) = C u (x) < 0 y = u(x) x Using the sme resoning s bove, we cn conclude tht ny prt of the grph of u(x) tht lies beneth the x-xis must be concve upwrd. Consequently, it would hve to intersect the x-xis. The net result is tht u(x) must oscillte bout the x-xis. Note tht this does not imply tht u(x) is periodic there is difference between function being oscilltory nd it being periodic. (Tht being sid, in the specil cse σ(x) = 1, without even knowing tht the solutions re sine nd cosine functions, one cn, with little extr work, show tht the solutions re periodic. This is done in the book by Simmons mentioned bove.) We now come to the importnt fct: Since the solutions u(x) re oscilltory, they will hve zeros, i.e., vlues x i for which u(x i ) = 0. We shll now chrcterize the rte of the oscilltions of u(x) in terms of the distnce between its consecutive zeros. As the prmeter λ increses, the distnces between consecutive zeros of the oscilltory solutions decreses Let us now return to the specil cse σ(x) = 1 in Eq. (25), i.e., with generl solution, u + λu = 0, λ > 0, (28) u λ (x) = C 1 cos( λx) + C 2 sin( λx). (29) Obviously, u λ (x) is periodic, with period T = 2π λ. This implies tht the distnces between consecutive zeros of ny prticulr solution re constnt nd given by d λ = T 2 = π λ. (30) As expected, d λ decreses s λ increses. After ll, λ is relted to the frequency of oscilltion, nd d λ is relted to the wvelength. 108

10 We now show how continuous increse of λ will produce discrete set of eigenvlues λ n nd ssocited eigenfunctions u n (x) tht stisfy the boundry condition u() = u(b) = 0. Consider the following initil vlue problem ssocited with this DE, u + λu = 0, u() = 0, u () = 1. (31) The first condition is cler. The second condition is kind of normliztion condition tht is imposed in order to isolte prticulr solution u(x). We cn replce 1 by nything we wnt. The solution to this IVP is u λ (x) = 1 [ ] sin λ(x ). (32) λ Let λ be very smll, i.e., close to zero, nd let x 1 > denote the first zero of u λ (x) fter x =. It is esy to determine: x 1 (λ) = + π. (33) λ Since λ is very smll, x 1 is very lrge, in prticulr, lrger thn b. y = u λ (x) b x 1 (λ) x Now let λ increse continuously. As result, x 1 (λ) will move continuously leftwrd, i.e., it decreses. There is prticulr vlue of λ, cll it λ 1, for which x 1 (λ) = b: + π λ1 = b λ 1 = π 2 (b ) 2. (34) This produces our first solution to the BVP u() = u(b) = 0, the eigenfunction u 1 (x). y = u λ1 (x) b x 1 (λ 1 ) x 2 (λ 1 ) x Now increse λ further. The first zero x 1 (λ) will decrese towrd nd the second zero, x 2 (λ) = + 2 π λ, (35) 109

11 will pproch b from the right. It will coincide with b t the prticulr vlue λ 2 given by + 2 π = b λ 2 = 4π2 λ2 (b ) 2. (36) This produces the second solution to the BVP u() = u(b) = 0, the eigenfunction u 2 (x). y = u λ2 (x) x 1 (λ 2 ) b x 2 (λ 2 ) x 3 (λ 2 ) x Get the picture? The nth zero x n (λ) will coincide with b t λ = λ n given by x n (λ n ) = + n π = b λ n = n2 π 2 λn (b ) 2. (37) This produces the eigenfunction u n (x). Clerly, we cn continue this procedure the result is the discrete set of eigenvlues, λ 1 < λ 2 < < λ n <, (38) with ssocited eigenfunctions u n (x) tht stisfy the boundry condition u() = u(b) = 0. Moreover, by construction, ech function u n (x) hs n 1 zeros in the open intervl (,b) (i.e., we re excluding the endpoints). We now return to the more generl eigenvlue problem of Eq. (23), i.e., u + λσ(x)u = 0, u() = 0, u(b) = 0. (39) Our gol is to show tht procedure nlogous to wht ws done for the specil cse σ(x) = 1 cn be performed for this eqution: As we increse λ, the zeros of the oscilltory solution u λ (x) move towrd x =. The result is discrete set of λ-vlues, λ n, t which u λ (b) = 0. But in order to ensure tht, in fct, ll of the zeros will move inwrd, we ll need nother theoreticl result. The Sturm Comprison Theorem We now consider the following differentil eqution u + q(x)u = 0, (40) 110

12 where it is ssumed tht q(x) > 0 for ll x R. Clerly, q(x) = λσ(x) in Eq. (23). We shll use the results involving distnces between consecutive zeros of sine/cosine functions to provide upper nd lower bounds of the distnces between consecutive zeros of u(x). Lemm: Let u(x) be solution of (40) nd I = [c,d] n intervl tht contins t lest two zeros of u(x). Suppose further tht Then if x k 1 nd x k re successive zeros of u(x) in I, 0 < m < q(x) < M, x [c,d]. (41) π < x k x k 1 < π. (42) M m The result follows from comprison of u(x) nd the solutions of u 1 + mu 1 = 0, distnce between consecutive zeros is u 2 + Mu 2 = 0, distnce between consecutive zeros is π m π. (43) M The consequences of this result re significnt. If we cn control the behviour of q(x) over the intervl [c, d], we cn control the spcing of its zeros. For exmple, by incresing m, lower bound to q(x), we cn ensure tht given number of consecutive zeros of u(x) lie in the intervl (c,d). Returning to our SL-eigenvlue problem of Eq. (23), we mke the correspondence q(x) = λσ(x). (44) The intervl [c,d] will be our BVP intervl [,b]. As we increse λ, the Sturm Comprison Theorem tells us tht the distnces between ll pirs of consecutive zeros, i.e., d k,λ = x k (λ) x k 1 (λ), k = 1,2,, (45) will get smller nd smller. As result, we shll hve more nd more zeros entering the intervl [,b] from the right. In other words, the procedure tht we employed for the sine nd cosine functions bove will pply to the generl problem in Eq. (23). Just to recll how we would proceed, first consider the following initil vlue problem, u + λσ(x)u = 0, u() = 0, u () = 1. (46) 111

13 y = u λ (x) b x 1 (λ) x For λ sufficiently smll, the first zero x 1 (λ) of u λ (x) will be greter thn b. (This is gurnteed by the Sturm Comprison Theorem - Exercise.) This is sketched bove, with the sme figure s used erlier. As λ is incresed continuously, x 1 (λ) will decrese continuously. At some vlue λ 1 > 0, x 1 (λ 1 ) = b. The solution u λ1 (x) is the eigenfunction tht corresponds to eigenvlue λ 1. It hs no zeros in (,b). y = u λ1 (x) b x 1 (λ 1 ) x 2 (λ 1 ) x As λ is incresed from λ 1, the first zero x 1 (λ) moves towrd nd second zero x 2 (λ) moves leftwrd towrd b. At some vlue λ 2 > λ 1, u 2 (λ 2 ) = b. We now hve the second eigenfunction u λ2 (x), with one zero in (,b). y = u λ2 (x) x 1 (λ 2 ) b x 2 (λ 2 ) x 3 (λ 2 ) x Clerly, we cn continue the process to produce solutions to (23) tht correspond to the discrete vlues λ 1 < λ 2 < < λ n < (47) The solution u λn, which we shll simply cll u n (x), will hve n 1 zeros in (,b). The u n (x) correspond to the eigenfunctions φ n (x) of the Sturm-Liouville eigenvlue problem. 112

14 A note on numericl shooting methods to pproximte eigenvlues/eigenfunctions The method outlined bove my pper somewht rtificil or contrived, nd perhps of limited use. (I think I used the word hokey in clss.) However, it ctully provides the bsis of numericl shooting methods tht provide pproximtions to boundry vlue/eigenvlue problems. One essentilly strts with n estimte of the eigenvlue λ 1, nd then numericlly solves the initil vlue problem in Eq. (46), integrting outwrd from x = nd checking the vlues of u(x) until they chnge sign, i.e., the pproximtion to u(x) crosses the x-xis. If this occurs t n x-vlue greter/less thn b, then the estimte of λ 1 is incresed/decresed. This is done in some kind of systemtic wy tht llows the routine to zero in on n estimte of λ 1 to desired ccurcy. Such method cn lso be used to provide estimtes of higher eigenvlues λ n, with the condition tht x = b is the nth zero to the right of x =. One would lso hve to keep trck of how mny zeros there re in the intervl (,b). Orthogonlity of eigenfunctions We now prove n orthogonlity result for eigenfunction solutions of the BVP problem u + λσ(x)u = 0, u() = u(b) = 0. (48) Eigenfunctions u n (x) corresponding to different eigenvlues re orthogonl with respect to the weight function σ(x): If λ m λ n, then u m (x)u n (x)σ(x) dx = 0. (49) To prove this result, consider two distinct eigenvlues of the bove problem, λ m λ n, with ssocited eigenfunctions u m nd u n, i.e., u m + λ m σ(x)u m = 0 () u n + λ n σ(x)u n = 0 (b). (50) Now multiply () by u n nd (b) by u m nd subtrct the ltter from the former to obtin u n u m u nu m + (λ m λ n )σ(x)u m u n = 0. (51) With n eye to the desired finl result, it look like we should integrte both sides of the eqution over [,b], i.e., integrte over [,b], then we rrive t the result (Exercise): [u n u m u n u m] dx + (λ m λ n ) 113 σ(x)u m u n = 0. (52)

15 But the integrnd hppens to be the derivtive of the Wronskin W ssocited with u n nd u m, i.e., u W(u n,u m ;x) = n u m = u nu m u nu m. (53) To see this, we tke derivtives, u n u m dw dx = u nu m + u n u m u nu m u nu m = u n u m u nu m. (54) Therefore, the first integrl in Eq. (52) becomes dw dx dx = W(b) W() (55) = [u n (b)u m(b) u n(b)u m (b)] [u n ()u m(b) u n()u m ()] = 0, since u() = u(b) = 0. Eq. (52) then becomes the desired result, u m (x)u n (x)σ(x) dx = 0. (56) Of course, we knew this in the specil cse σ(x) = 1, i.e., sine nd/or cosine functions tht stisfy homogeneous boundry conditions. But we now hve n orthogonlity result for the generl cse of σ(x). 114

16 Lecture 18 A Lightning Tour of Sturm-Liouville theory (conclusion) In the previous lecture, we focussed on the eigenvlue-bvp problem u + λσ(x)u = 0, u() = u(b) = 0, (57) where λ > 0 nd σ(x) > 0. We showed the existence of discrete, infinite set of eigenvlues, λ n, n = 1,2,, such tht λ 1 < λ 2 <. (58) Associted with ech eigenvlue λ n is n eigenfunction u ( x) tht stisfies the bove BVP for λ = λ n. Furthermore, u ( x) hs n 1 zeros in the open intervl (,b). (This mens, of course, tht u n (x) hs n + 1 zeros on [,b] for the bove BVP.) There is one more result tht cn be obtined from qulittive nlysis of this problem, nmely, tht the eigenvlues λ n re rel nd positive. Techniclly, we should hve derived this result first, before proceeding on to the shooting method of constructing the discrete set of eigenvlues λ n nd ssocited eigenfunctions u n (x). When the shooting method ws introduced, we took the liberty of ssuming tht λ ws rel nd positive. Eigenvlues λ n re rel nd positive For given n 1, we strt with the eigenvlue eqution, u n + λ nσ(x)u n = 0, u n () = u n (b) = 0. (59) Now multiply both sides by u n nd integrte from x = to x = b: u n (x)u n (x) dx + λ n We integrte the first integrl by prts to obtin u n (x)u n(x) dx = u n (b)u n(b) u n ()u n() Substitution into the previous eqution yields, λ n [u n (x)] 2 σ(x) dx = 115 u n (x) 2 σ(x) dx = 0. (60) [u n(x)] 2 dx. (61) [u n (x)] 2 dx. (62)

17 Note tht (i) u n (x) is not identicly zero on [,b] nd (ii) u n (x) is not constnt (or even piecewise constnt) on [, b]. Therefore the two integrls in the bove eqution re positive. This implies tht the λ n re positive. Exmple: We consider the following eigenvlue problem u + λ(1 + x)u = 0, u(0) = u(π) = 0. (63) Here σ(x) = 1 + x. In the figure below, the eigenfunctions u 1 (x), u 4 (x) nd u 10 (x) re plotted. (The eigenfunctions nd ssocited eigenvlues were computed using softwre routine tht is bsed on the shooting method.) Also plotted for comprison re the corresponding eigenfunctions v n (x) = sin nx of the eqution v + λv = 0, v(0) = v(π) = 0. (64) In this cse the eigenvlues re esily found to be ν n = n 2. Since the function σ(x) = 1 + x increses over the intervl [0,π], one would expect tht solutions to (63) would oscillte more quickly s x increses from 0 to π. This is somewht evident in the plot of u 4 (x) but quite evident for u 10 (x) the spcing between consecutive zeros of u 10 decreses s we move to the right. For the sin eigenfunction v 10 (x) = sin 10x, the spcing of consecutive zeros is constnt. Tht being sid, one might wonder why the spcings between the first few pirs of zeros of u 10 (x) re less thn their counterprts for v 10 (x) = sin 10x. After ll, isn t the function σ(x) = 1 + x greter thn the function 1 in Eq. (64) for v? The reson lies in the fct tht the eigenvlues λ n corresponding to u n re less thn the eigenvlues ν n corresponding to v n : n λ n ν n Becuse σ(x) = 1 + x > 1, smller vlue of λ n is required to bring the nth zero to the endpoint b = 1. As result, we expect the spcings of the u n to be less thn those of the v n. But s x increses, σ(x) does s well, nd the spcings of u n decrese. In fct, they hve to eventully be less thn the spcings of the v n, becuse the grph hs to mke the sme number of crossings of the x-xis before it reches the point (π,0). 116

18 -1 1 lmbd(1) u(x) x 1 lmbd(4) u(x) x 1 lmbd(10) u(x) x Eigenfunctions u n (x) of the BVP u + λ(1 + x)u = 0, y(0) = y(π) = 0, for n = 1 (top), n = 4 (middle) nd n = 10 (bottom). Also plotted for comprison (s dotted curves) re the corresponding eigenfunctions v n (x) = sin nx (up to constnt fctor) of v + λv = 0. A return to the generl Sturm-Liouville eigenvlue eqution We now stte, without proof, tht the results obtined for the Sturm-Liouville eigenvlue problem, u + λσ(x)u = 0, u() = u(b) = 0, (65) 117

19 crry over to the generl Sturm-Liouville eigenvlue eqution, ( d p(x) du ) + q(x)u + λσ(x)u = 0, dx dx < x < b, (66) with homogeneous boundry conditions of the form φ β 1 φ() + β 2 () = 0, dx φ β 3 φ(b) + β 4 (b) = 0. (67) dx The min results for this generl problem were presented in Lecture 16, but we repet them below: 1. There exists discrete, infinite set of eigenvlues λ 1 < λ 2 <, with λ n. (In the generl cse, becuse of the ppernce of the term q(x)u, it is not gurnteed tht ll of the eigenvlues re positive. Nevertheless, the eigenvlues re bounded from below, implying tht only finite number of them will be negtive.) 2. Associted with ech eigenvlue λ n is n eigenfunction φ n (x) which stifies the boundry conditions nd which hs exctly n 1 zeros in the open intervl (,b). 3. The eigenfunctions φ n stisfy the following orthogonlity reltion, φ m φ n σ(x) dx = 0, m n. (68) This result is proved in Section xxx of the textbook by Hbermn. Completeness of the set of eigenfunctions We now stte the finl importnt result regrding the set of solutions φ n (x) to the eigenvlue problem in Eq. (66) with boundry conditions of the form in (67): The set of functions {φ n } n=1 forms complete orthogonl bsis in the spce L2 [,b] of (rel- or complex-vlued) squre-integrble functions on the intervl [,b], i.e., L 2 [,b] = {f : [,b] R (or C) f(x) 2 dx < }. (69) This mens tht ny function f L 2 [,b] my be expressed s unique liner combintion of the form, f(x) = n φ n (x). (70) n=1 118

20 The uniqueness is s follows: For given function L 2 [,b], there will be unique set of n coefficients. As we ll qulify below, Eq. (70) does not necessry imply tht the series converges to f(x) for ech x [,b]. This is different kind of convergence convergence with respect to integrtion of L 2 functions on [, b]. We shll describe this concept very briefly below you will see more detiled discussions in n nlysis course such s AMATH 331 (or the grdute course AMATH 731). To understnd convergence in function spces, we must consider the prtil sums of the bove series, defined s follows, N S N (x) = n φ n (x). (71) n=1 In the sme wy s ws done for infinite series of rel numbers, we must hve tht the prtil sums S N converge to the function f s N, i.e., lim N S N = f or S N f s N, (72) We hve put quotes round the mthemticl sttements becuse it is not cler t time wht these limits men. To mke sense of these limits, we return to the definition of the spce L 2 [,b] in Eq. (69). The integrl in the definition my be used to define the size or mgnitude of the function f(x). In other words, it defines the norm of f(x), i.e., [ 1/2 f = f(x) dx] 2. (73) Suffice it to sy tht the bove formul stisfies ll of the properties required of norm, including the tringle inequlity. As you my recll from liner lgebr, the norm my be used to define distnce function or metric on vector spce. Here, the distnce between two functions f,g L 2 [,b], to be denoted s d(f,g), will be defined s follows, [ 1/2 d(f,g) = f g = f(x) g(x) dx] 2. (74) The bove definition my be considered s functionl nlogue of the Eucliden metric/distnce between two vectors in R n. The convergence in Eq. (72) my now be expressed in the following wy, lim d(f,s N) = 0 or f S N 0 s N. (75) N 119

21 The distnce d(f,s N ) my be viewed s the error of pproximtion of the function f(x) by the prtil sum function S N (x): As N increses, the error of pproximtion decreses, pproching zero s N. In this sense, we my write f(x) = lim N S N(x) = n φ n (x). (76) n=1 Note: The fct tht the pproximtion error goes to zero for ny function f(x) is due to the completeness of the set of functions {φ n (x)}. This is very importnt property for function spces, which re exmples of infinite-dimensionl spces. In finite-dimensionl spces, e.g., R n, ny set of linerly independent vectors {v} n k=1 will provide bsis for the spce. (We cn lwys construct set of orthogonl/orthonorml vectors from this set.) But in n infinite-dimensionl spce such s L 2 [,b], not ny infinite set of functions {u n } n=1 will do: The set must be ble to rech ny element f in the spce. An importnt result from functionl nlysis sttes tht the infinite set of solutions φ n (x) of Sturm-Liouville eigenvlue eqution in (66) forms complete set in the spce L 2 [,b]. (This result is proved in AMATH 731.) We must now comment on the expnsion coefficients n in Eq. (70) ssocited with function f(x). The question is, How do we determine the n from f(x)? The nswer is, In the sme wy s we do for Fourier series. For given k 1, we multiply both sides of Eq. (70) by φ k (x)σ(x) the function σ(x) must be included becuse it ppers s the weight function in the orthogonlity reltion, Eq. (68): f(x)φ k (x)σ(x) = n φ n (x)φ k (x)σ(x). (77) n=1 Now integrte both sides over the intervl x [, b]. Omitting ll technicl detils (see note below) we rrive t the following result, f(x)φ k (x)σ(x) dx = n φ n (x)φ k (x)σ(x) dx. (78) n=1 Becuse of the orthogonlity of the φ n (x) functions in Eq. (68), the only nonzero term on the righthnd-side is for n = k, i.e., We then esily solve for k, f(x)φ k (x)σ(x) dx = k φ k (x)φ k (x)σ(x) dx. (79) k = f(x)φ k(x)σ(x) dx φ k(x)φ k (x)σ(x) dx. (80) 120

22 In the specil cse tht p(x) = 1, q(x) = 0 nd σ(x) the cses tht we hve considered erlier in the course the φ n functions will be sine or cosine functions (or combintions of them). In other words, we hve Fourier series expnsions. Over the intervl [, b] = [0, L], recll tht the integrls in the denomintor hve the vlue L/2. In closing, you my hve noticed remrkble prllel between the expnsion of function f in terms of the complete bsis set {φ n } nd the expnsion of vector v R N in terms of n orthogonl bsis {u n } tht spns R N. In the ltter cse, if N v = n u n, (81) then tking sclr products of both sides with n element u k for some 1 k n yields n=1 v,u k = k u k,u k, (82) from which follows the result, Eq. (83) my be viewed s finite-dimensionl version of Eq. (80). k = v,u k u k,u k. (83) This, of course, leds to the ide tht the bsis elements φ n (x) behve s bsis vectors in the function spce L 2 [,b]. The sclr product u,v of two vectors u,v R n ppers to hve the following nlogue in our spce L 2 [,b]: f,g = f(x)g(x)σ(x) dx. (84) This indictes tht the spce of functions L 2 [,b] is equipped with sclr product, which is usully clled n inner product. Note tht the inner product involves the weighting function σ(x) which, in turn, is connected to the orthogonl bsis functions φ n (x) vi the Sturm-Liouville eigenvlue eqution. In summry, we simply stte tht the spce of squre-integrble functions L 2 [,b] is n inner product spce. Moreover, it is complete inner product spce to put it loosely, it contins its limit points. An inner product spce tht is complete is known s Hilbert spce. You my hve herd this term in other courses, e.g., course in quntum mechnics. A finl note on the derivtion involving Eq. (78): As mentioned bove, we hve omitted ll technicl detils in the derivtion of Eq. (78). Both (1) integrtion nd (2) summtion of n infinite series involve limiting opertions. Interchnging 121

23 these limiting opertions requires some cre: The proper pproch would be to work with prtil sum pproximtions of f(x). In ech cse, the order of integrtion nd summtion my be reversed. By tking limits, we rrive t Eq. (78). Het eqution for nonuniform 1D rod s S-L eqution Relevnt sections of text: 5.2.1, 5.4 In Lecture 16, we motivted the study of the generl Sturm-Liouville eigenvlue problem by exmining the het eqution for nonuniform 1D rod. We my now use the forml results of the generl Sturm-Liouville problem to complete this problem. Recll tht the het eqution ssumes the following form, c(x)ρ(x) u t = ( K 0 (x) u ), 0 < x < L. (85) x x We ssume seprtion-of-vribles solution of the form u(x,t) = φ(x)g(t), (86) which, fter seprtion, yields the following equtions for φ nd G: ( d K 0 (x) dφ(x) ) + λc(x)ρ(x)φ(x) = 0, φ(0) = φ(l) = 0, (87) dx dx nd G + λg = 0. (88) The DE for φ(x) is Sturm-Liouville eigenvlue eqution with p(x) = K 0 (x) > 0, q(x) = 0 nd σ(x) = c(x)ρ(x) > 0. As such, there exists n infinite sequence of eigenvlues λ 1 < λ 2 < λ 3 < (89) with ssocited eigenfunctions φ n (x) such tht 1. φ n (0) = φ n (L) = 0 nd φ n (x) hs n 1 zeros in (,b), 2. L 0 φ n (x)φ m (x)σ(x) dx = 0 if m n, where σ = cρ. Associted with ech eigenvlue λ n is the solution G n of the corresponding DE G + λ n G = 0 G n (t) = e λn t. (90) 122

24 Therefore, the seprtion of vribles method yields the following product solutions, u n (x,t) = φ n (x)e λnt, n = 1,2,3. (91) Superposition of these solutions yields the generl solution u(x,t) = n φ n (x)e λnt. (92) If the following initil condition is imposed, n=1 u(x,0) = f(x), 0 x L, (93) then the expnsion coefficients A n my be obtined by using the orthogonlity of the φ n s follows. From the initil condition, we hve f(x) = n φ n (x). (94) n=1 Multiplying both sides by φ k (x)σ(x), for n integer k 1, nd integrting x from 0 to L, we obtin L 0 L f(x)φ k (x)σ(x) dx = k φ k (x) 2 σ(x) dx, (95) which is rerrnged to give L 0 k = f(x)φ k(x)σ(x) dx L 0 φ k(x) 2 σ(x) dx. (96) In generl, with even knowledge of the functionl form of the functions K 0 (x), c(x) nd ρ(x) (which will not usully be the cse in prcticl cses), the vlues of the eigenvlues λ n or the form of the eigenfunctions φ n (x) will not be known. If necessry, one could resort to numericl methods for their determintion. Tht being sid, it is probbly computtionlly cheper (i.e., less computtion is involved) to compute estimtes of the solutions u(x,t) using finite difference methods. We ll touch on these methods little lter in the course. In some cses, however, it my be useful to obtin estimtes of the eigenvlues λ n. This is prticulrly true in the cse of the wve eqution for vibrting systems, which we exmine below. 0 Wve eqution for nonuniform vibrting string We now return to the wve eqution for vibrting string tht is clmped t both ends, ssuming tht it is not necessrily homogeneous: ρ(x) 2 u t 2 = x 123 ( T(x) u ), (97) x

25 where ρ(x) is the linel mss density function nd T(x) is the tension. Once gin we ssume seprtion-of-vribles solution of the form, u(x,t) = φ(x)g(t). (98) Substitution into (97) yields ρ(x)φ(x)g (t) = d ( T(x) dφ ) G(t). (99) dx dx We then seprte vribles to obtin G (t) G(t) = 1 φρ d dx ( T dφ ) = µ = λ, λ < 0. (100) dx We hve tken the liberty of mking the seprtion constnt negtive, in light of our previous experiences with the DE/BVPs ssocited with the φ(x) function. This procedure yields the following φ-eqution, nd the ssocited G-eqution, ( d T dφ ) + λρφ = 0, φ(0) = φ(l) = 0, (101) dx dx G + λg = 0. (102) Note tht the φ-eqution hs the form of generl Sturm-Liouville eigenvlue eqution with p(x) = T(x) nd σ(x) = ρ(x). From Sturm-Liouville theory, there exists n infinite, discrete set of (positive) eigenvlues, λ 1 < λ 2 <, with ssocited eigenfunctions φ n (x). From Eq. (102), the ssocited G n (t) functions will stisfy the equtions G n + λ ng n = 0, n = 1,2,. (103) The generl solution of this DE is G n (t) = n cos( λ n t) + b n sin( λ n t). (104) Reclling the form of our seprtion of vribles solution in Eq. (98), we hve obtined set of linerly independent solutions u n (x,t) of the form, u n (x,t) = φ n (x)[ n cos(ω n t) + b n sin(ω n t)], ω n = λ n, n = 1,2,. (105) 124

26 In order to ccomodte the initil conditions (initil displcement function f(x) nd initil velocity function g(x), we must consider series solution of the form, u(x, t) = = u n (x,t) n=1 φ n (x)[ n cos(ω n t) + b n sin(ω n t)]. (106) n=1 As in the cse of the homogeneous string, where T(x) = T 0 nd ρ(x) = ρ 0, the functions u n (x,t) represent the fundmentl modes of vibrtion of the nonuniform string. The sptil profile of ech mode is determined by the eigenfunction φ n (x). Ech mode will oscillte verticlly between the grphs of φ n (x) nd φ n (x) with frequency ω n. In this problem, we see the importnce of the eigenvlues λ n of the Sturm-Liouville eigenvlue eqution for φ(x): They determine the frequencies of oscilltion, ω n = λ n, of the fundmentl modes. In pplictions, it is often helpful to hve knowledge of the first few fundmentl frequencies. Just to remind the reder: Only in some very specil cses, e.g., homogeneous strings, rods, cn we obtin the exct vlues of eigenvlues λ n nd corresponding eigenfunctions φ n (x) in closed form. In relity, this is not the cse, nd one must resort to finding estimtes of these eigenvlues/frequencies. Tht being sid, good estimtes re often sufficient in pplictions. In other pplictions, the eigenvlues of SL problems correspond to vrious other physicl properties, for exmple, the energy levels of quntum mechnicl system. There re only few quntum mechnicl systems tht cn be solved in closed form. As result, much work hs been done on obtining good estimtes of these energy eigenvlues most often the lowest energy stte (the so-clled ground stte ) nd perhps few excited sttes. The methods employed in quntum mechnics cn be trced bck to the work done by people in the lte 1800 s nd erly 1900 s most notbly Lord Ryleigh to estimte eigenvlues of Sturm-Liouville problems. This is the subject of the next section. 125