MATH 461: Fourier Series and Boundary Value Problems Chapter V: Sturm–Liouville Eigenvalue Problems Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2015 [email protected] MATH 461 – Chapter 5 1 Outline 1 Introduction 2 Examples 3 Sturm–Liouville Eigenvalue Problems 4 Heat Flow in a Nonuniform Rod without Sources 5 Self-Adjoint Operators and Sturm–Liouville Eigenvalue Problems 6 The Rayleigh Quotient 7 Vibrations of a Nonuniform String 8 Boundary Conditions of the Third Kind 9 Approximation Properties [email protected] MATH 461 – Chapter 5 2 Introduction So far all of our problems have involved the two-point BVP '00(x) + λϕ(x) = 0 which – depending on the boundary conditions – leads to a certain set of eigenvalues and eigenfunctions: e.g., 1 '(0) = '(L) = 0: nπ 2 nπx λ = ;' (x) = sin ; n = 1; 2; 3;::: n L n L 2 '0(0) = '0(L) = 0: nπ 2 nπx λ = ;' (x) = cos ; n = 0; 1; 2;::: n L n L 3 '(−L) = '(L) and '0(−L) = '0(L): nπ 2 nπx nπx λ = ;' (x) = c cos + c sin ; n = 0; 1; 2;::: n L n 1 L 2 L [email protected] MATH 461 – Chapter 5 4 Introduction Remark The eigenfunctions in the examples on the previous slide were subsequently used to generate 1 Fourier sine series, 2 Fourier cosine series, or 3 Fourier series. In this chapter we will study problems which involve more general BVPs and then lead to generalized Fourier series. [email protected] MATH 461 – Chapter 5 5 Examples Heat Flow in a Nonuniform Rod Recall the general form of the 1D heat equation: @u @ @u c(x)ρ(x) (x; t) = K (x) (x; t) + Q(x; t): @t @x 0 @x The separation of variables technique is likely to be applicable if the PDE is linear and homogeneous. Therefore, we assume Q(x; t) = α(x)u(x; t) with x-dependent proportionality factor α. The resulting PDE @u @ @u c(x)ρ(x) (x; t) = K (x) (x; t) + α(x)u(x; t) (1) @t @x 0 @x is linear and homogeneous and we will derive the corresponding BVP resulting from separation of variables below. [email protected] MATH 461 – Chapter 5 7 Examples Remark Note that @ @u @u @2u K (x) (x; t) = K 0 (x) (x; t) + K (x) (x; t): @x 0 @x 0 @x 0 @x2 Therefore, a PDE such as @u @ @u c(x)ρ(x) (x; t) = K (x) (x; t) + α(x)u(x; t) @t @x 0 @x arises, e.g., as convection-diffusion-reaction equation in the modeling of chemical reactions (such as air pollution models) with 0 @u convection term: K0(x) @x (x; t) 2 ( ) @ u ( ; ) diffusion term: K0 x @x2 x t reaction term: α(x)u(x; t) [email protected] MATH 461 – Chapter 5 8 Examples We now assume u(x; t) = '(x)T (t) and apply separation of variables to @u @ @u c(x)ρ(x) (x; t) = K (x) (x; t) + α(x)u(x; t): @t @x 0 @x This results in d c(x)ρ(x)'(x)T 0(t) = K (x)'0(x)T (t) + α(x)'(x)T (t): dx 0 Division by c(x)ρ(x)'(x)T (t) gives T 0(t) 1 d α(x) = K (x)'0(x) + = −λ. T (t) c(x)ρ(x)'(x) dx 0 c(x)ρ(x) Remark As always, we choose the minus sign with λ so that the resulting ODE T 0(t) = −λT (t) has a decaying solution for positive λ. [email protected] MATH 461 – Chapter 5 9 Examples From T 0(t) 1 d α(x) = K (x)'0(x) + = −λ. T (t) c(x)ρ(x)'(x) dx 0 c(x)ρ(x) we see that the resulting ODE for the spatial BVP is d K (x)'0(x) + α(x)'(x) + λc(x)ρ(x)'(x) = 0 dx 0 and it is in general not known how to solve this ODE eigenvalue problem analytically. [email protected] MATH 461 – Chapter 5 10 Examples Circularly Symmetric Heat Flow in 2D The standard 2D-heat equation in polar coordinates is given by @u (r; θ; t) = kr2u(r; θ; t); @t where 1 @ @u 1 @2u r2u = r + : r @r @r r 2 @θ2 If we assume circular symmetry, i.e., no dependence on θ, then @2u @θ2 = 0 and we have (see also HW 1.5.5) @u k @ @u (r; t) = r (r; t) : @t r @r @r [email protected] MATH 461 – Chapter 5 11 Examples We assume u(r; t) = '(r)T (t) and apply separation of variables to @u k @ @u (r; t) = r (r; t) @t r @r @r to get k d '(r)T 0(t) = r'0(r)T (t) r dr or 1 T 0(t) 1 d = r'0(r) = −λ. k T (t) r'(r) dr [email protected] MATH 461 – Chapter 5 12 Examples From 1 T 0(t) 1 d = r'0(r) = −λ k T (t) r'(r) dr we see that the ODE for the spatial (radial) BVP problem is d r'0(r) + λr'(r) = 0: dr Again, we don’t yet know how to solve this ODE. Contrary to the previous problem, this equation can be solved using Bessel functions (more later). In earlier work (see Chapter 2.5) we encountered the steady-state solution of this equation, i.e., Laplace’s equation. Potential BCs therefore are: On an annulus, with BCs u(a; t) = u(b; t) = 0 or '(a) = '(b) = 0. On a circular disk, with BCs u(b; t) = 0 and ju(0; t)j < 1, i.e., '(b) = 0 and j'(0)j < 1. [email protected] MATH 461 – Chapter 5 13 Sturm–Liouville Eigenvalue Problems A general form of an ODE that captures all of the examples discussed so far is the Sturm–Liouville differential equation d p(x)'0(x) + q(x)'(x) + λσ(x)'(x) = 0 dx with given coefficient functions p, q and σ, and parameter λ. We now show how this equation covers all of our examples. [email protected] MATH 461 – Chapter 5 15 Sturm–Liouville Eigenvalue Problems Example If we let p(x) = 1, q(x) = 0 and σ(x) = 1 in d p(x)'0(x) + q(x)'(x) + λσ(x)'(x) = 0 dx we get '00(x) + λϕ(x) = 0 which led to the standard Fourier series earlier. [email protected] MATH 461 – Chapter 5 16 Sturm–Liouville Eigenvalue Problems Example If we let p(x) = K0(x), q(x) = α(x) and σ(x) = c(x)ρ(x) in d p(x)'0(x) + q(x)'(x) + λσ(x)'(x) = 0 dx we get d K (x)'0(x) + α(x)'(x) + λc(x)ρ(x)'(x) = 0 dx 0 which is the ODE for the heat equation in a nonuniform rod. [email protected] MATH 461 – Chapter 5 17 Sturm–Liouville Eigenvalue Problems Example If we let p(x) = x, q(x) = 0 and σ(x) = x in d p(x)'0(x) + q(x)'(x) + λσ(x)'(x) = 0 dx and then replace x by r we get d r'0(r) + λr'(r) = 0 dr which is the ODE for the circularly symmetric heat equation. [email protected] MATH 461 – Chapter 5 18 Sturm–Liouville Eigenvalue Problems Example If we let p(x) = T0, q(x) = α(x) and σ(x) = ρ0(x) in d p(x)'0(x) + q(x)'(x) + λσ(x)'(x) = 0 dx we get 00 T0' (x) + α(x)'(x) + λρ0(x)'(x) = 0 which is the ODE for vibrations of a nonuniform string (see HW 5.3.1). [email protected] MATH 461 – Chapter 5 19 Sturm–Liouville Eigenvalue Problems Boundary Conditions A nice summary is provided by the table on p.156 of [Haberman]: [email protected] MATH 461 – Chapter 5 20 Sturm–Liouville Eigenvalue Problems Regular Sturm–Liouville Eigenvalue Problems We will now consider the ODE d p(x)'0(x) + q(x)'(x) + λσ(x)'(x) = 0; x 2 (a; b) (2) dx with boundary conditions 0 β1'(a) + β2' (a) = 0 0 (3) β3'(b) + β4' (b) = 0 where the βi are real numbers. Definition If p, q, σ and p0 in (2) are real-valued and continuous on [a; b] and if p(x) and σ(x) are positive for all x in [a; b], then (2) with (3) is called a regular Sturm–Liouville problem. Remark Note that the BCs don’t capture those of the periodic or singular type. [email protected] MATH 461 – Chapter 5 21 Sturm–Liouville Eigenvalue Problems Facts for Regular Sturm–Liouville Problems We pick the well-known example '00(x) + λϕ(x) = 0 '(0) = '(L) = 0 nπ 2 nπx with eigenvalues λn = L and eigenfunctions 'n(x) = sin L , n = 1; 2; 3;::: to illustrate the following facts which hold for all regular Sturm–Liouville problems. Later we will study the properties and prove that they hold in more generality. [email protected] MATH 461 – Chapter 5 22 Sturm–Liouville Eigenvalue Problems 1 All eigenvalues of a regular SL problem are real. nπ 2 For our example, obviously λn = L is real for any value of the integer n. Remark This property ensures that when we search for eigenvalues of a regular SL problem it suffices to consider the three cases λ > 0; λ = 0 and λ < 0: Complex values of λ are not possible.