Introduction To Sturm-Liouville Theory
OrthogonalitySturm-Liouville problemsEigenvalues and eigenfunctionsIntroduction to Sturm-Liouville TheoryRyan C. DailedaTrinity UniversityPartial Differential EquationsApril 10, 2012DailedaSturm-Liouville Theory
OrthogonalitySturm-Liouville problemsEigenvalues and eigenfunctionsInner products with weight functionsSuppose that w (x) is a nonnegative function on [a, b]. If f (x) andg (x) are real-valued functions on [a, b] we define their innerproduct on [a, b] with respect to the weight w to behf , g i Zbf (x)g (x)w (x) dx.aWe say f and g are orthogonal on [a, b] with respect to theweight w ifhf , g i 0.Remarks:The inner product and orthogonality depend on the choice ofa, b and w .When w (x) 1, these definitions reduce to the “ordinary”ones.DailedaSturm-Liouville Theory
OrthogonalitySturm-Liouville problemsEigenvalues and eigenfunctionsExamples1The functions fn (x) sin(nx) (n 1, 2, . . .) are pairwiseorthogonal on [0, π] relative to the weight function w (x) 1.2Let Jm be the Bessel function of the first kind of order m, andlet αmn denote its nth positive zero. Then the functionsfn (x) Jm (αmn x/a) are pairwise orthogonal on [0, a] withrespect to the weight function w (x) x.3The functionsf0 (x) 1,f1 (x) 2x,f4 (x) 16x 4 12x 2 1,f2 (x) 4x 2 1,f3 (x) 8x 3 4x,f5 (x) 32x 5 32x 3 6xare pairwise orthogonalon [ 1, 1] relative to the weight function w (x) 1 x 2 . They are examples of Chebyshevpolynomials of the second kind.DailedaSturm-Liouville Theory
OrthogonalitySturm-Liouville problemsEigenvalues and eigenfunctionsSeries expansionsWe have frequently seen the need to express a given function as alinear combination of an orthogonal set of functions. Ourfundamental result generalizes to weighted inner products.TheoremSuppose that {f1 , f2 , f3 , . . .} is an orthogonal set of functions on[a, b] with respect to the weight function w . If f is a function on[a, b] and Xan fn (x),f (x) n 1then the coefficients an are given byhf , fn ian hfn , fn iRbDailedaf (x)fn (x)w (x) dx.Rb2a fn (x)w (x) dxaSturm-Liouville Theory
OrthogonalitySturm-Liouville problemsEigenvalues and eigenfunctionsRemarksThe series expansion above is called a generalized Fourierseries for f , and an are the generalized Fourier coefficients.It is natural to ask:Where do orthogonal sets of functions come from?To what extent is an orthogonal set complete, i.e. whichfunctions f have generalized Fourier series expansions?In the context of PDEs, these questions are answered bySturm-Liouville Theory.DailedaSturm-Liouville Theory
OrthogonalitySturm-Liouville problemsEigenvalues and eigenfunctionsSturm-Liouville equationsA Sturm-Liouville equation is a second order linear differentialequation that can be written in the form(p(x)y ′ )′ (q(x) λr (x))y 0.Such an equation is said to be in Sturm-Liouville form.Here p, q and r are specific functions, and λ is a parameter.Because λ is a parameter, it is frequently replaced by othervariables or expressions.Many “familiar” ODEs that occur during separation ofvariables can be put in Sturm-Liouville form.ExampleShow that y ′′ λy 0 is a Sturm-Liouville equation.We simply take p(x) r (x) 1 and q(x) 0.DailedaSturm-Liouville Theory
OrthogonalitySturm-Liouville problemsEigenvalues and eigenfunctionsExamplePut the parametric Bessel equationx 2 y ′′ xy ′ (λ2 x 2 m2 )y 0in Sturm-Liouville form.First we divide by x to get m2xy y λ x {z }x′′′2 y 0.(xy ′ )′This is in Sturm-Liouville form withp(x) x,q(x) m2,xr (x) x,provided we write the parameter as λ2 .DailedaSturm-Liouville Theory
OrthogonalitySturm-Liouville problemsEigenvalues and eigenfunctionsExamplePut Legendre’s differential equationy ′′ 2xµy′ y 021 x1 x2in Sturm-Liouville form.First we multiply by 1 x 2 to get(1 x 2 )y ′′ 2xy ′ µy 0. {z}((1 x 2 )y ′ )′This is in Sturm-Liouville form withp(x) 1 x 2 ,q(x) 0,r (x) 1,provided we write the parameter as µ.DailedaSturm-Liouville Theory
OrthogonalitySturm-Liouville problemsEigenvalues and eigenfunctionsExamplePut Chebyshev’s differential equation(1 x 2 )y ′′ xy ′ n2 y 0in Sturm-Liouville form. First we divide by 1 x 2 to getn2x1 x 2 y ′′ y′ y 0.1 x21 x2{z} p ( 1 x 2 y ′ )′This is in Sturm-Liouville form withpp(x) 1 x 2 , q(x) 0,1r (x) ,1 x2provided we write the parameter as n2 .DailedaSturm-Liouville Theory
OrthogonalitySturm-Liouville problemsEigenvalues and eigenfunctionsSturm-Liouville problemsA Sturm-Liouville problem consists ofA Sturm-Liouville equation on an interval:(p(x)y ′ )′ (q(x) λr (x))y 0,a x b,(1)together withBoundary conditions, i.e. specified behavior of y at x aand x b.We will assume that p, p ′ , q and r are continuous and p 0 on(at least) the open interval a x b.According to the general theory of second order linear ODEs, thisguarantees that solutions to (1) exist.DailedaSturm-Liouville Theory
OrthogonalitySturm-Liouville problemsEigenvalues and eigenfunctionsRegularity conditionsA regular Sturm-Liouville problem has the form(p(x)y ′ )′ (q(x) λr (x))y 0,a x b,′(2)′(3)c1 y (a) c2 y (a) 0,d1 y (b) d2 y (b) 0,where:(c1 , c2 ) 6 (0, 0) and (d1 , d2 ) 6 (0, 0);p, p ′ , q and r are continuous on [a, b];p and r are positive on [a, b].The boundary conditions (2) and (3) are called separatedboundary conditions.DailedaSturm-Liouville Theory
OrthogonalitySturm-Liouville problemsEigenvalues and eigenfunctionsExampleThe boundary value problemy ′′ λy 0,0 x L,y (0) y (L) 0,is a regular Sturm-Liouville problem (recall that p(x) r (x) 1and q(x) 0).ExampleThe boundary value problem((x 2 1)y ′ )′ (x λ)y 0,′ 1 x 1,y ( 1) y (1) 0,is a regular Sturm-Liouville problem (here p(x) x 2 1, q(x) xand r (x) 1).DailedaSturm-Liouville Theory
OrthogonalitySturm-Liouville problemsEigenvalues and eigenfunctionsExampleThe boundary value problemx 2 y ′′ xy ′ (λ2 x 2 m2 )y 0,0 x a,y (a) 0,is not a regular Sturm-Liouville problem.Why not? Recall that when put in Sturm-Liouville form we hadp(x) r (x) x and q(x) m2 /x. There are several problems:p and r are not positive when x 0.q is not continuous when x 0.The boundary condition at x 0 is missing.This is an example of a singular Sturm-Liouville problem.DailedaSturm-Liouville Theory
OrthogonalitySturm-Liouville problemsEigenvalues and eigenfunctionsEigenvalues and eigenfunctionsA nonzero function y that solves the Sturm-Liouville problem(p(x)y ′ )′ (q(x) λr (x))y 0,a x b,(plus boundary conditions),is called an eigenfunction, and the corresponding value of λ iscalled its eigenvalue.The eigenvalues of a Sturm-Liouville problem are the valuesof λ for which nonzero solutions exist.We can talk about eigenvalues and eigenfunctions for regularor singular problems.DailedaSturm-Liouville Theory
OrthogonalitySturm-Liouville problemsEigenvalues and eigenfunctionsExampleFind the eigenvalues of the regular Sturm-Liouville problemy ′′ λy 0,0 x L,y (0) y (L) 0,This problem first arose when separated variables in the 1-D waveequation. We already know that nonzero solutions occur only whenλ λn n2 π 2L2(eigenvalues)y yn sinnπxL(eigenfunctions)andfor n 1, 2, 3, . . .DailedaSturm-Liouville Theory
OrthogonalitySturm-Liouville problemsEigenvalues and eigenfunctionsExampleFind the eigenvalues of the regular Sturm-Liouville problemy ′′ λy 0,y (0) 0,0 x L,y (L) y ′ (L) 0,This problem arose when we separated variables in the 1-D heatequation with Robin conditions. We already know that nonzerosolutions occur only whenλ λn µ2n ,where µn is the nth positive solution totan µL µ,andy yn sin(µn x)for n 1, 2, 3, . . .DailedaSturm-Liouville Theory
OrthogonalitySturm-Liouville problemsEigenvalues and eigenfunctionsExampleIf m 0, find the eigenvalues of the singular Sturm-Liouvilleproblemx 2 y ′′ xy ′ (λ2 x 2 m2 )yy (0) is finite, 0,0 x a,y (a) 0.This problem arose when we separated variables in the vibratingcircular membrane problem. We know that nonzero solutions occuronly whenαmn,λ λn awhere αmn is the nth positive zero of the Bessel function Jm , andy yn Jm (λn x)for n 1, 2, 3, . . . (technically, the eigenvalues are λ2n α2mn /a2 .)DailedaSturm-Liouville Theory
OrthogonalitySturm-Liouville problemsEigenvalues and eigenfunctionsThe previous examples demonstrate the following generalproperties of a regular Sturm-Liouville problem(p(x)y ′ )′ (q(x) λr (x))y 0,′c1 y (a) c2 y (a) 0,a x b,d1 y (b) d2 y ′ (b) 0.TheoremThe eigenvalues form an increasing sequence of real numbersλ1 λ2 λ3 · · ·withlim λn .n Moreover, the eigenfunction yn corresponding to λn is unique (upto a scalar multiple), and has exactly n 1 zeros in the intervala x b.DailedaSturm-Liouville Theory
OrthogonalitySturm-Liouville problemsEigenvalues and eigenfunctionsAnother general property is the following.TheoremSuppose that yj and yk are eigenfunctions corresponding todistinct eigenvalues λj and λk . Then yj and yk are orthogonal on[a, b] with respect to the weight function w (x) r (x). That ishyj , yk i Zbyj (x)yk (x)r (x) dx 0.aThis theorem actually holds for certain non-regularSturm-Liouville problems, such as those involving Bessel’sequation.Applying this result in the examples above we immediatelyrecover familiar orthogonality statements.This result explains why orthogonality figures so prominentlyin all of our work.DailedaSturm-Liouville Theory
OrthogonalitySturm-Liouville problemsEigenvalues and eigenfunctionsExamplesExampleWrite down the conclusion of the orthogonality theorem fory ′′ λy 0, 0 x L,y (0) y (L) 0.Since the eigenfunctions of this regular Sturm-Liouville problem areyn sin(nπx/L), and since r (x) 1, we immediately deduce thatZ0Lsin mπx Lsin nπx Ldx 0for m 6 n.DailedaSturm-Liouville Theory
OrthogonalitySturm-Liouville problemsEigenvalues and eigenfunctionsExampleIf m 0, write down the conclusion of the orthogonality theoremforx 2 y ′′ xy ′ (λ2 x 2 m2 )y 0, 0 x a,y (0) is finite,y (a) 0.Since the eigenfunctions of this regular Sturm-Liouville problem areyn Jm (αmn x/a), and since r (x) x, we immediately deducethatZ a α αmℓmkx Jmx x dx 0Jmaa0for k 6 ℓ.DailedaSturm-Liouville Theory
Sturm-Liouville equations A Sturm-Liouville equation is a second order linear differential equation that can be written in the form (p(x)y . According to the general theory of second order linear ODEs, this guarantees that solutions to (1) exist. Daileda Sturm-Liouville Theory.
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110 partial differential equations In the Sturm Liouville operator the derivative terms are gathered together into one perfect derivative, Dp(x)D. This is similar to what we saw in the Chapter 2 when we solved linear first order equations. In that case we sought an integrating factor. We can do the same thing here.
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