Differential Equations Brannan Copyright © 2010 by John Wiley & Sons, Inc. All rights reserved. Chapter 10: Boundary Value Problems and Sturm– Liouville Theory

Chapter 10 Chapter 10 Boundary Value Problems and Sturm–Liouville Theory As a result of separating variables in a partial differential equation in Chapter 9, we repeatedly encountered the differential equation X'' + λX = 0, 0 < x < L, with the boundary conditions X(0) = 0, X(L) = 0. This boundary value problem is the prototype of a large class of problems that are important in applied mathematics. These problems are known as Sturm– Liouville boundary value problems. In this chapter we discuss the major properties of Sturm–Liouville problems and their solutions; in the process we are able to generalize somewhat the method of separation of variables for partial differential equations.

Chapter 10 - Chapter 10 - Boundary Value Problems and Sturm–Liouville Theory 10.1 The Occurrence of Two-Point Boundary Value Problems 10.2 Sturm–Liouville Boundary Value Problems 10.3 Nonhomogeneous Boundary Value Problems 10.4 Singular Sturm–Liouville Problems 10.5 Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion 10.6 Series of Orthogonal Functions: Mean Convergence

10.1 The Occurrence of Two- Point Boundary Value Problems \we extend and generalize the results of Chapter 9. Our main goal is to show how the method of separation of variables can be used to solve problems somewhat more general. We are interested in three types of generalizations. 1. Consider more general partial differential equations—for example, r (x)u t = [p(x)u x ] x − q(x)u + F(x, t). 2. Allow more general boundary conditions. In particular, we wish to consider boundary conditions of the form u x (0,t)−h 1 u(0, t) = 0, u x (L, t) + h 2 u(L, t) = Change the geometry of the region in which the problem is posed.

Method of separation of variables Consider r (x)u t = [p(x)u x ] x − q(x)u Set u(x, t) = X(x)T(t), Use separation constant by −λ & simplify [p(x)X']'−q(x)X + λr(x)X = 0, T' + λT = 0. The boundary conditions for constant h 1, and h 2 are X'(0) − h 1 X(0) = 0, X'(L) + h 2 X(L) = 0. For certain values of λ, called eigenvalues, there are nontrivial solutions, X(x) called eigenfunctions. These eigenfunctions form the basis for series solutions.

Example Find the eigenvalues and the corresponding eigenfunctions of the boundary value problem y'' + λy = 0, y(0) = 0, y'(1) + y(1) = 0. Answer If λ > 0, then the eigenvalues are λ n =(2n − 1) 2 π 2 /4 for n = 4, 5,... and the eigenfunction corresponding to the eigenvalue λn is φ n (x, λ n ) = k n sin √λ n x; n = 1, 2,..., where the constant kn remains arbitrary. There are no zero, negative, or complex eigenvalues.

10.2 Sturm–Liouville Boundary Value Problems Sturm–Liouville problem (1), (2) Consider the class consists of a differential equation of the form [p(x)y']' − q(x)y + λr (x)y = 0 (1) on the interval 0 < x < 1, together with the separated (each with one boundary point) boundary conditions α 1 y(0) + α 2 y'(0) = 0, β 1 y(1) + β 2 y'(1) = 0 (2) at the endpoints. Define the linear homogeneous differential operator L by L[ y] = − [p(x)y']' + q(x)y and obtain L[ y] = λr(x)y.

Lagrange’s Identity Let u and v be functions having continuous second derivatives on the interval 0 ≤ x ≤ 1. Then If u and v satisfy the boundary conditions (2), then Lagrange’s identity reduces to Using the inner product (u, v), We get

Theorems THEOREM All the eigenvalues of the Sturm–Liouville problem (1), (2) are real. THEOREM If φ 1 and φ 2 are two eigenfunctions of the Sturm–Liouville problem (1), (2) corresponding to eigenvalues λ 1 and λ 2, respectively, and if λ 1 ≠ λ 2, then

Orthogonality of eigenfunctions Theorem expresses the property of orthogonality of the eigenfunctions with respect to the weight function r. normalization condition, Orthonormal set satisfy the orthogonality relation and normalization condition. Using Kronecker delta, δ mn ={0, if m ≠ n, {1, if m = n.

THEOREM The eigenvalues of the Sturm–Liouville problem (1), (2) are all simple; that is, to each eigenvalue there corresponds only one linearly independent eigenfunction. Further, the eigenvalues form an infinite sequence and can be ordered according to increasing magnitude so that λ1 < λ2 < λ3 < …< λn < …. Moreover λ n →∞as n→∞.

Example Determine the normalized eigenfunctions of the problem: y'' + λy = 0, y(0) = 0, y(1) = 0. Answer: The normalized eigenfunctions of the given boundary value problem are φ n (x) =√2 sin nπx, n = 1, 2, 3,....

Generalized Fourier Series THEOREM Let φ 1, φ 2,..., φ n,... be the normalized eigenfunctions of the Sturm–Liouville problem (1), (2): [p(x)y']'−q(x)y + λr(x)y = 0, α 1 y(0) + α 2 y'(0) = 0, β 1 y(1) + β 2 y'(1) = 0. Let f and f' be piecewise continuous on 0≤x≤1. Then the series whose coefficients c m aregiven by Eq. converges to [ f (x+) + f (x−)]/2 at each point in the open interval 0 < x < 1.

Example Expand the function f (x) = x, 0 ≤ x ≤ 1 in terms of the normalized eigenfunctions φ n (x) =√2 sin nπx in previous example. Answer:

Self-Adjoint Problems Consider the boundary value problem consisting of the differential equation L[y]=λr(x)y, 0 < x < 1, where and n linear homogeneous boundary conditions at the endpoints. If Lagrange Identity is valid for every pair of sufficiently differentiable functions that satisfy the boundary conditions, then the given problem is said to be self-adjoint.

10.3 Nonhomogeneous Boundary Value Problems The boundary value problem consisting of the nonhomogeneous differential equation L[ y] = −[p(x)y']' + q(x)y = μr (x)y + f (x), (1) where μ is a given constant and f is a given function on 0 ≤ x ≤ 1, and the boundary conditions are α 1 y(0) + α 2 y'(0) = 0, β 1 y(1) + β 2 y'(1) = 0. (2) Formal Solution (3) where

THEOREM The nonhomogeneous boundary value problem (1), (2) has a unique solution for each continuous f whenever μ is different from all the eigenvalues of the corresponding homogeneous problem; the solution is given by Eq. (3), and the series converges for each x in 0 ≤ x ≤ 1. If μ is equal to an eigenvalue λ m of the corresponding homogeneous problem, then the nonhomogeneous boundary value problem has no solution unless f is orthogonal to φ m, that is, unless the condition holds. In that case, the solution is not unique and contains an arbitrary multiple ofφ m (x).

THEOREM Fredholm alternative Theorem. For a given value of μ, either the nonhomogeneous problem (1), (2) has a unique solution for each continuous f (if μ is not equal to any eigenvalue λ m of the corresponding homogeneous problem), or else the homogeneous problem L[ y] = λr(x)y and (2) has a nontrivial solution (the eigenfunction corresponding to λ m ).

Example Solve the boundary value problem y'' + 2y = −x, y(0) = 0, y(1) + y'(1) = 0. Answer where

Nonhomogeneous Heat Conduction Problems Eigenfunction expansions can be used to solve nonhomogeneous problems for partial differential equations, let us consider the generalized heat conduction equation r (x)u t = [p(x)u x ] x − q(x)u + F(x,t) (25) with the boundary conditions u x (0,t)−h 1 u(0,t)= 0, u x (1,t)+h 2 u(1, t)=0 (26) and the initial condition u(x,0) = f (x). (27)

Method 1. Find the eigenvalues λ n and the normalized eigenfunctions φ n of the homogeneous problem. 2. Calculate the coefficients B n and γ n (t) via 3. Evaluate the integral and determine b n (t). 4. Sum the infinite series

Example Find the solution of the heat conduction problem u t = u xx + xe −t, u(0,t) = 0, u x (1,t) +u(1,t) = 0, u(x, 0) = 0. Answer

10.4 Singular Sturm–Liouville Problems We use the term singular Sturm–Liouville problem to refer to a certain class of boundary value problems for the differential equation (1) L[ y] = −[p(x)y']' + q(x)y = λr(x)y, 0<x<1, in which the functions p, q, and r satisfy the regular conditions on the open interval 0 < x < 1, but at least one of these functions fails to satisfy regular conditions at one or both of the boundary points. Regular means that, p is differentiable, q and r are continuous, and p(x) > 0 and r (x) > 0 at all points in the closed interval. Singular problems also occur if the interval is unbounded, for example, 0 ≤ x < ∞.

Self - Adjoint A singular boundary value problem for Eq. (1) is said to be self-adjoint if Eq. (17) is valid, possibly as an improper integral, for each pair of functions u and v with the following properties: they are twice continuously differentiable on the open interval 0 < x < 1, they satisfy a boundary condition of the form α 1 y(0) + α 2 y(0) = 0 at each regular boundary point, and they satisfy a boundary condition sufficient to ensure Eq. (21) if x = 0 is a singular boundary point, or Eq. (22) if x = 1 is a singular boundary point. If at least one boundary point is singular, then the differential equation (1), together with two boundary conditions of the type just described, is said to form a singular Sturm–Liouville problem.

Continuous spectrum. The most striking difference between regular and singular Sturm–Liouville problems is that in a singular problem the eigenvalues may not be discrete. That is, the problem may have nontrivial solutions for every value of λ, or for every value of λ in some interval. In such a case the problem is said to have a continuous spectrum. A singular problem could have a mixture of discrete eigenvalues and also a continuous spectrum.

10.5 Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion The Vibrations of a Circular Elastic Membrane To study the motion of a circular membrane, write it in polar coordinates If u is independent of θ, we get (3) The boundary condition at r = 1 is (4) and the initial conditions (5-6) are where f (r ) describes the initial configuration of the membrane. For consistency we also require that f (1) = 0. Finally, we state explicitly the requirement that u(r, t) be bounded for 0 ≤ r ≤ 1.

Solution The fundamental solutions of this problem, satisfying the partial differential equation (3), the boundary condition (4), and boundedness condition, are The solution of the partial differential equation (3) satisfying the boundary condition (4) and the initial conditions (5) and (6) is given by where

10.6 Series of Orthogonal Functions: Mean Convergence Method of collocation. Choose n points x 1,..., x n in the interval 0 ≤ x ≤ 1 and require that have the same value as f (x) at each of these points. The coefficients a 1,..., a n are found by solving the set of linear algebraic equations where the set of functions φ 1, φ 2,..., φ n, are continuous on the interval 0 ≤ x ≤ 1 and satisfy the orthonormality condition,

THEOREM The eigenfunctions φ i of the Sturm–Liouville problem are complete with respect to mean convergence for the set of functions that are square integrable on 0 ≤ x ≤ 1. That is, given any square integrable function f, the series (10), whose coefficients are given by Eq. (9), converges to f (x) in the mean square sense.

Example Let f (x) = 1 for 0 < x < 1. Expand f (x) using the eigenfunctions φ n (x)=√2 sin nπx and discuss the pointwise and mean square convergence of the resulting series.

Summary Section 10.1 The Occurrence of Two-Point Boundary Value Problems Section 10.2 Sturm–Liouville Boundary Value Problems Many of the partial differential equations that can be solved by the method of separation of variables require the solution of an eigenvalue problem that lies in the class of regular Sturm–Liouville boundary value problems: [p(x)y']' − q(x)y + λr (x)y = 0, α<x < β, α 1 y(α) + α 2 y'(α) = 0, β 1 y(β) + β 2 y'(β) = 0, where the functions p, p, q, and r are continuous on the interval α ≤ x ≤ β and where p(x) > 0 and r (x) > 0 at all points in α ≤ x ≤ β. Regular Sturm–Liouville problems constitute a special case of a general class known as self-adjoint problems.

Section 10.2 Sturm– Liouville Boundary Value Problems

Section 10.3 Nonhomogeneous Boundary Value Problems

Section 10.4 Singular Sturm– Liouville Problems

Section 10.5 A Bessel Series Expansion

Section 10.6 Series of Orthogonal Functions: Mean Convergence