Math 412-501Theory of Partial Differential EquationsLecture 2-7:Sturm-Liouville eigenvalue problems.Sturm-Liouville differential equation:ddxpdφdx+ qφ + λσφ = 0 (a < x < b),where p = p(x), q = q(x), σ = σ(x) are knownfunctions on [a, b] and λ is an unknown constant.The Sturm-Liouville equation is a linearhomogeneous ODE of the second order.Sturm-Liouville eigenvalue problem == Sturm-Liouville differential equation ++ linear homogeneous boundary conditionsJ. C. F. Sturm J. Liouville(1803–1855) (1809–1882)The Sturm-Liouville equation usually arises afterseparation of variables in a linear homogeneous PDEof the second order.Examples.• φ′′+ λφ = 0 (heat, wave, Laplace’s equations)• r2d2hdr2+ rdhdr= λh(Laplace’s equation in polar coordinates)standard notation: x2φ′′+ xφ′− λφ = 0canonical form: (xφ′)′− λx−1φ = 0Heat flow in a nonuniform rod:cρ∂u∂t=∂∂xK0∂u∂x+ Q,K0= K0(x), c = c(x), ρ = ρ(x), Q = Q(u, x, t).The equation is linear homogeneous if Q = α(x, t)u.We assume that α = α(x).cρ∂u∂t=∂∂xK0∂u∂x+ αuSeparation of variables: u(x, t) = φ(x)G (t).Substitute this into the heat equation:cρφdGdt=ddxK0dφdxG + αφG .Divide both sides by c(x)ρ(x)φ(x)G (t) = cρu:1GdGdt=1cρφddxK0dφdx+αcρ= −λ = const.The variables have been separated:dGdt+ λG = 0,ddxK0dφdx+ αφ + λcρφ = 0.Sturm-Liouville differential equation:ddxpdφdx+ qφ + λσφ = 0 (a < x < b).Examples of boundary conditions:• φ(a) = φ(b) = 0 (Dirichlet conditions)• φ′(a) = φ′(b) = 0 (von Neumann conditions)• φ′(a) = 2φ(a), φ′(b) = −3φ(b) (Robinconditions)• φ(a) = 0, φ′(b) = 0 (mixed conditions)• φ(a) = φ(b), φ′(a) = φ′(b) (periodicconditions)• |φ(a)| < ∞, φ(b) = 0 (singular conditions)ddxpdφdx+ qφ + λσφ = 0 (a < x < b).The equation is regular if p, q, σ are real andcontinuous on [a, b], and p, σ > 0 on [a, b].The Sturm-Liouville eigenvalue problem is regular ifthe equation is regular and boundary conditions areof the formβ1φ(a) + β2φ′(a) = 0,β3φ(b) + β4φ′(b) = 0,where βi∈ R, |β1| + |β2| 6= 0, |β3| + |β4| 6= 0.This includes Dirichlet, Neumann, and Robinconditions but excludes periodic and singular ones.Regular Sturm-Liouville eigenvalue problem:ddxpdφdx+ qφ + λσφ = 0 (a < x < b),β1φ(a) + β2φ′(a) = 0,β3φ(b) + β4φ′(b) = 0.Eigenfunction: nonzero solution φ of the boundaryvalue problem.Eigenvalue: corresponding value of λ.Eigenvalues and eigenfunctions of a regular Sturm-Liouville eigenvalue problem have six importantproperties.Property 1. All eigenvalues are real.Property 2. All eigenvalues can be arranged inthe ascending orderλ1< λ2< . . . < λn< λn+1< . . .so that λn→ ∞ as n → ∞.This means that:• there are infinitely many eigenvalues;• there is a smallest eigenvalue;• on any finite interval, there are only finitelymany eigenvalues.Remark. It is possible that λ1< 0.Property 3. Given an eigenvalue λn, thecorresponding eigenfunction φnis unique up to amultiplicative constant. The function φnhas exactlyn − 1 zeros in (a, b).We say that λnis a simple eigenvalue.Property 4. Eigenfunctions belonging to differenteigenvalues satisfy an integral identity:Zbaφn(x)φm(x)σ(x) dx = 0 if λn6= λm.We say that φnand φmare orthogonal relative tothe weight fun ction σ.Property 5. Any eigenvalue λ can be related toits eigenfunction φ as follows:λ =−pφφ′ba+Zbap(φ′)2− qφ2dxZbaφ2σ dx.The right-hand side is called the Rayleigh quotient.Property 6. Any piecewise continuous functionf : [a, b] → R is assigned a seriesf (x) ∼X∞n=1cnφn(x),wherecn=Zbaf (x)φn(x)σ(x) dxZbaφ2n(x)σ(x) dx.If f is piecewise smooth then the series convergesfor any a < x < b. The sum is equal to f (x) if f iscontinuous at x. Otherwise the series converges to12(f (x+) + f (x−)).We say that th e set of eigenfunctions φnis complete.A regular Stu rm-Liouville eigenvalue problem:φ′′+ λφ = 0, φ(0) = φ(L) = 0.(p = σ = 1, q = 0, [a, b] = [0, L])Eigenvalues: λn= (nπL)2, n = 1, 2, . . .Eigenfunctions: φn(x) = sinnπxL.The zeros of φndivide the interval [0, L] into nequal parts.Property 3a. Suppose x1< x2< . . . < xn−1arezeros of the eigenfunction φnin (a, b). Then φn+1has exactly one zero in each of the followingintervals: (a, x1), (x1, x2), (x2, x3), . . . , (xn−2, xn−1),(xn−1, b).Eigenfunctions φnOrthogonality:ZL0sinnπxLsinmπxLdx = 0, n 6= m.Rayleigh quotie nt: λ =ZL0|φ′(x)|2dxZL0|φ(x)|2dx.Fourier sine series: f ∼P∞n=1cnsinnπxL,where cn=2LZL0f (x) sinnπxLdx.Note thatZL0sinnπxL2dx