18 Sturm-Liouville Eigenvalue Problems - UMBC

1 18 Sturm-Liouville Eigenvalue ProblemsUp until now all our Eigenvalue Problems have been of the formd2 dx2+ = 0,0< x < l(1)plus a mix of boundary conditions, generally being Dirichlet or Neumanntype. This is too narrow of a viewpoint, which I wish to point out througha few 1:Simple spatial variation in diffusivity:D=D0(1 +x) the problem ut=D0[(1 +x)2ux]x0< x <1, t >0u(x,0) =f(x)0< x <1u(0,t) = 0 =u(1,t)t >0(2)From the separation of variables method,u(x,t) =T(t) (x), we obtaindTdt= D0 Tandddx[(1 +x)2d dx] + = 0,with (0) = (1) = 0. Note that by carrying through the differentiation, the equation(1 +x)2d2 dx2+ 2(1 +x)d dx+ = 0(3)is aCauchy-Eulerequation (recall the Review of ODEs appendix for a dis-cussion of the equations). If we write (x) = (1 +x)r, the characteristicequation forrbecomesr(r 1) + 2r+ = 0 r={ 1 1 4 } we have to start at = 0 atx= 0, and end at = 0 atx= 1, assumewe need oscillatory solutions like we got out of (1), since 0.

2 Hence,assume >1/4, and define (for notational convenience) := 1 the roots can be written asr= 1/2 i , and since(1 +x) 1/2 i = (1 +x) 1/2(1 +x) i = (1 +x) 1/2e i ln(1+x),1a suitable fundamental set of solutions to (3) is(1 +x) 1/2cos( ln(1 +x)),(1 +x) 1/2sin( ln(1 +x)).Now (x) can be written as a linear combination of these functions, and since (0) = 0, we have (x) =B(1 +x) 1/2sin( ln(1 +x)) satisfying (3) and thisboundary condition. Now0 = (1) = 2 1/2 Bsin( ln(2)) sin( ln(2)) = 0 ln(2) =n , n , 2= 1/4 = (n /ln(2))2. Therefore, the eigenvalues and associatedeigenfunctions for this problem are{ n= 1/4 + (n ln(2))2n= 1,2,3,.. n(x) = (1 +x) 1/2sin(n ln(2)ln(1 +x))(4)This gives us the solution form for problem (2):u(x,t) = (1 +x) 1/2e D0t/4 n=1 Bne D0n2 2t/(ln(x))2sin(n ln(2)ln(1 +x)).}

3 (5)Remark:If we would not have made the above positivity assumption on in Example 1, then assume <1/4 and define :=12 1 4 > solution of the characteristic equation would ber= 1/2 , and so (x) = (1 +x) 1/2{A(1 +x) +B(1 +x) }. Now (0) = 0 =A+B, so (x) =A(1 +x) 1/2{(1 +x) (1 +x) }, while (1) = 0 =A2 1/2[2 2 ],which impliesA= 0 since >0; so 0. Therefore, there is no Eigenvalue <1/4. We ll leave it as an exercise to draw the same conclusion about = 1 :A variable density vibrating string problemDetermine the Eigenvalue problem for the following problem , and derive theeigenvalues and associated eigenfunctions: (1 +x) 2utt=c2uxx0< x < l , t >0, c >0 is a constantu(x,0) =f(x), ut(x,0) = 0 0< x < lu(0,t) = 0 =u(l,t)2 Exercise:In considering acoustic measurements in a thin tube scaled to beof unit length, letp(x,t) be acoustic pressure, andv(x,t) be volume certain assumptions, one model to consider is p x= A(x) v t, v x= A(x) c2 p t,whereA(x) is the variable cross-sectional area of the tube at locationx, is the air density in the tube, andcis the speed of sound.

4 Suppose we havescaled the problem so thatc= 1 and = 1. Assume also thatA(x) is acontinuously differentiable function, andA(x)>0 on [0,1]. First eliminatevin the above system to obtain a single equation forp(x,t). Letp(0,t) = 0andpx(1,t) = 0 for allt >0. Then separate variables,p(x,t) =T(t) (x),to obtain the EVP for thisp- problem . (The Eigenvalue equation is some-times called Webster s horn equation.) For generalA(x) satisfying the aboveconditions, if we assume real eigenvalues, then1. Show that any Eigenvalue must satisfy Show that = 0 is not an Eigenvalue of the In the special caseA(x) =eax, wherea6= 0, write out what the EVPis in this case. Then show that the eigenvalues nmust satisfy thetranscendental equation 2 a2/4/a=tan( a2/4), and hence,there is an infinite ordered set of them, with n ast.

5 Example 2:Symmetric diffusion in a diskFor multidimensional diffusion equations, special cases arise in the case ofdomains with nice geometry, for example disk and wedge shaped spatial do-mains inR2, and spherical domains inR3. In the 2D situation, the Laplacianin polar coordinates is 2=1r r(r r) +1r2 2 2(6)or in cylindrical coordinates, we have 2=1r r(r r) +1r2 2 2+ 2 z2(7)3 Figure 1: Coordinate angle definitions that will be used for spherical coordi-nates in these in spherical coordinates (see Figure 1), 2=1 2 ( 2 ) +1 2sin2 2 2+1 2 2 2+cot 2 .(8)In fact, theradial part of the Laplacian, in arbitraryndimensions (n 1),withrnotationally denoting the radial distance from the origin, is givenby 2r= 2 r2+n 1r will briefly look at some special Problems in higher dimensions later inthese Notes, but for now let us consider the diffusion equation on the diskspatial domain :={(r, ) : 0 r < a,0 <2 }, and consider thesymmetriccase (uis independent of angle ) ut=Dr(rur)rr < a , t >0, D >0 is a constantu(r,0) =f(r)r < au(a,t) = 0uremains bounded on (9)Letu(r,t) =T(t) (r), then (1/DT)dTdt=1r ddr(rd dr) = , sodT/dt= DT, as usual, andddr(rd dr) + r = 0 =rd2 dr2+d dr+ r (10)4with (a) = 0 and is bounded atr= 0.

6 Note that (10) isnota Cauchy-Euler equation, because of therdependence associated with the it is a well-studied equation because it arises so much in practice; (10) isBessel s equation of order 0, and we ll study this variable coefficient EVPlater. However, an introduction to Bessel s equation and Bessel functions isgiven in Appendix point here in introducing these examples is to motivate us to brieflystudy a more general class of EVPs calledregular Sturm-Liouville Eigen-value Problems . They have the form ddx(p(x)d dx) q(x) + (x) = 0a < x < b (a) + d dx(a) = 0 (b) + d dx(b) = 0(11)The functions and parameters in (11) must meet the following conditions: p(x) is continuous on [a,b], continuously differentiable on (a,b),p(x)>0 on [a,b]. q(x), (x) are continuous on [a,b], (x)>0, q(x) 0 on [a,b].

7 , , , are real :The sign convention on theqterm in the equation is not use the negative sign in the equation so all the inequalities in the aboveconditions are either > or .Example 3:In our usual example + = 0, 0< x < l, (0) = 0 = (l),p(x) 1,q(x) 0, (x) 1, and = = 0. Then we obtain = n=n2 2/l2, = n(x) = sin(n x/l),n= 1,2,3,.. Given that wehave explicit representations for the eigenvalues and eigenfunctions we canmake some straightforward observations:1. The eigenvalues are real and ordered; that is, 1< 2< 3< .., with n asn .2. Corresponding to each nis an eigenfunction, n= sin(n x/l), thathasn 1 zeros in the interval (0,l) (see Figure 2).5 Figure 2: Note the splicing of zeroes of successive eigenfunctions for The eigenfunctions{sin(n x/l)}n 1form an orthogonal set of functionson (0,l); that is,< n, m>:= l0 n(x) m(x)dx= l0sin(n x/l) sin(m x/l)dx={0ifn6=ml/2 ifn=m.}

8 4. The eigenfunctions arecompletewith respect to the set of piecewisesmooth functionsfon (0,l); that is, we can, for such a functionf,writef(x) n=1an n(x), where the infinite sum converges for allx (0,l), to [f(x+) +f(x )]/2, if the coefficients are chosen to be theFourier coefficients off. That is,an=< f, n> / < n, n>.The goal here is to present the case that Problems of the form (11) withcoefficients satisfying the bulleted items have the same properties as our pro-totypical EVP we have been working with. (So our prototypical EVP is aregular Sturm-Liouville Eigenvalue problem .)6 Figure 3: This shows the first 5 eigenfunctions associated with the eigenvalueproblem (1 +x)2 + = 0, (0) = (1) = 1, again:Returning to example 1,p(x) = (1 +x)2,q(x) 0,and = = 0, so this example leads to a regular Sturm-Liouville 2, again:From (10),p(r) =r,q(r) 0, and (r) =r, and onthe interval [0,a], = 0, = 0.

9 Now the smoothness conditions in thebulleted conditions is satisfied by this problem , butp(r) and (r) are notstrictly positive on the closed interval [0,a]. However,p, are zero only atthe boundary pointr= 0, otherwise the conditions are met. So example 2is an example of asingularSturm- liouville EVP, but it is close enough tothe regular case that what properties are brought up below for the regularSturm- liouville EVP will also hold the singular Sturm-Liouville EVP :For the exercise on page 2, what is thep,q, for the derived EVP? Sturm-Liouville Theorem:The regular Sturm-Liouville EVP defined by(11) and the bulleted points below (11) satisfies71. There exists an infinite number of discrete eigenvalues, n,n= 1,2,..,that are real, positive, ordered, and n asn .2. The eigenfunctions corresponding to different eigenvalues are orthogo-nal on [a,b]with respect to ; that is, for the Eigenvalue -eigenfunctionpairs{ i, i},{ j, j}, i6= j,< i, j>= ba i(x) j(x) (x)dx= Eigenfunctions of the same Eigenvalue are unique up to The nth eigenfunction n(x) associated with the nth Eigenvalue nhasexactlyn 1 zeros in (a,b).

10 (For an example, see Figure 3.)5.{ n}n 1are complete with respect to piecewise smooth functionsfon[a,b]. Thus, ba{f(x) N1an n(x)}2 (x)dx 0 asN .Our intention is not go through a full proof of this theorem here, butto go through some parts of it to illustrate the arguments. Consult a moreadvanced treatment of Sturm-Liouville EVPs to get the full purposes here let the boundary conditions for (11) be the specialDirichlet conditions: (a) = 0 = (b).Claim 1: Any Eigenvalue of (11) is realLet be any Eigenvalue of (11), with associate eigenfunction (x). If iscomplex, then = r+i iand its complex conjugate is = r i i, withassociated eigenfunction (x) = (x). Since{ , }satisfiesddx(pd dx) q + = 0 ona < x < b(12) (a) = 0 = (b)then, by taking the complex conjugation of the equation (12), and notingthatp,qand are real functions,{ , }satisfiesddx(pd dx) q + = 0 ona < x < b(13) (a) = 0 = (b)8So, multiply equation (12) by and multiply (13) by , then subtract thetwo resulting equations.