Sturm-liouville problems

Before we introduce an important solution method for PDEs in section 7.3, we consider an ordinary differential equation that will arise in that method when dealing with a single spatial dimension $x$: the sturm-liouville (S-L) differential equation. Let $p, q, \sigma$ be functions of $x$ on open interval $(a,b)$. Let $X$ be the dependent variable and $\lambda$ constant. The regular S-L problem is the S-L ODE¹ \[\begin{align} \frac{\diff} {\diff x} \left( p X' \right) + q X + \lambda \sigma X = 0 \end{align}\] with boundary conditions \[\begin{align}\label{eq:s_l_bcs} \beta_1 X(a) + \beta_2 X'(a) &= 0 \\ \beta_3 X(b) + \beta_4 X'(b) &= 0 \end{align}\] with coefficients $\beta_i \in \mathbb{R}$. This is a type of boundary value problem.

This problem has nontrivial solutions, called eigenfunctions $X_n(x)$ with $n \in \mathbb{Z}_+$, corresponding to specific values of $\lambda = \lambda_n$ called eigenvalues.² There are several important theorems proven about this (see Haberman (2018, sec. 5.3)). Of greatest interest to us are that

there exist an infinite number of eigenfunctions $X_n$ (unique within a multiplicative constant),
there exists a unique corresponding real eigenvalue $\lambda_n$ for each eigenfunction $X_n$,
the eigenvalues can be ordered as $\lambda_1 < \lambda_2 < \cdots$,
eigenfunction $X_n$ has $n-1$ zeros on open interval $(a,b)$,
the eigenfunctions $X_n$ form an orthogonal basis with respect to weighting function $\sigma$ such that any piecewise continuous function $f:[a,b]\rightarrow\mathbb{R}$ can be represented by a generalized fourier series on $[a,b]$.

This last theorem will be of particular interest in section 7.3.

Types of boundary conditions

Boundary conditions of the sturm-liouville kind eq. ¿eq:s_l_bcs? have four sub-types:

dirichlet: for just $\beta_2, \beta_4 = 0$,
neumann: for just $\beta_1, \beta_3 = 0$,
robin: for all $\beta_i \ne 0$, and
mixed: if $\beta_1 = 0$, $\beta_3 \ne 0$; if $\beta_2 = 0$, $\beta_4 \ne 0$.

There are many problems that are not regular sturm-liouville problems. For instance, the right-hand sides of eq. ¿eq:s_l_bcs? are zero, making them homogeneous boundary conditions; however, these can also be nonzero. Another case is periodic boundary conditions: \[\begin{align} X(a) &= X(b) \\ X'(a) &= X'(b). \end{align}\]

Example 7.1

Consider the differential equation $$\begin{aligned} X'' + \lambda X = 0 \end{aligned}$$ with dirichlet boundary conditions on the boundary of the interval [0,L] $$\begin{aligned} X(0) = 0 \quad\text{and}\quad X(L) = 0. \end{aligned}$$ Solve for the eigenvalues and eigenfunctions.

This is a sturm-liouville problem, so we know the eigenvalues are real. The well-known general solutions to the ODE is $$X(x) = \begin{cases} k_1 + k_2 x & \lambda = 0 \\ k_1 e^{j\sqrt{\lambda} x} + k_2 e^{-j\sqrt{\lambda} x} & \text{otherwise} \end{cases}$$ with real constants k₁, k₂. The solution must also satisfy the boundary conditions. Let’s apply them to the case of λ = 0 first: $$\begin{aligned} X(0) &= 0 \Rightarrow k_1 + k_2 (0) = 0 \Rightarrow k_1 = 0 \\ X(L) &= 0 \Rightarrow k_1 + k_2 (L) = 0 \Rightarrow k_2 = -k_1/L. \end{aligned}$$ Together, these imply k₁ = k₂ = 0, which gives the trivial solution X(x) = 0, in which we aren’t interested. We say, then, for nontrivial solutions λ ≠ 0. Now let’s check λ < 0. The solution becomes $$\begin{aligned} X(x) &= k_1 e^{-\sqrt{\abs{\lambda}} x} + k_2 e^{\sqrt{\abs{\lambda}} x} \\ &= k_3 \cosh(\sqrt{\abs{\lambda}}x) + k_4 \sinh(\sqrt{\abs{\lambda}}x) \end{aligned}$$ where k₃ and k₄ are real constants. Again applying the boundary conditions: $$\begin{aligned} X(0) &= 0 \Rightarrow k_3 \cosh(0) + k_4 \sinh(0) = 0 \Rightarrow k_3 + 0 = 0 \Rightarrow k_3 = 0 \\ X(L) &= 0 \Rightarrow 0 \cosh(\sqrt{\abs{\lambda}}L) + k_4 \sinh(\sqrt{\abs{\lambda}}L) = 0 \Rightarrow k_4 \sinh(\sqrt{\abs{\lambda}}L) = 0. \end{aligned}$$ However, $\sinh(\sqrt{\abs{\lambda}}L) \ne 0$ for L > 0, so k₄ = k₃ = 0—again, the trivial solution. Now let’s try λ > 0. The solution can be written $$\begin{aligned} X(x) = k_5 \cos(\sqrt{\lambda}x) + k_6 \sin(\sqrt{\lambda}x). \end{aligned}$$ Applying the boundary conditions for this case: $$\begin{aligned} X(0) &= 0 \Rightarrow k_5 \cos(0) + k_6 \sin(0) = 0 \Rightarrow k_5 + 0 = 0 \Rightarrow k_5 = 0 \\ X(L) &= 0 \Rightarrow 0 \cos(\sqrt{\lambda}L) + k_6 \sin(\sqrt{\lambda}L) = 0 \Rightarrow k_6 \sin(\sqrt{\lambda}L) = 0. \end{aligned}$$ Now, $\sin(\sqrt{\lambda}L) = 0$ for Therefore, the only nontrivial solutions that satisfy both the ODE and the boundary conditions are the eigenfunctions with corresponding eigenvalues $$\begin{aligned} \lambda_n &= \left(\frac{n \pi} {L}\right)^2. \end{aligned}$$ Note that because λ > 0, λ₁ is the lowest eigenvalue.

Plotting the eigenfunctions

import numpy as np
import matplotlib.pyplot as plt

Set L = 1 and compute values for the first four eigenvalues lambda_n and eigenfunctions X_n.

L = 1
x = np.linspace(0, L, 100)
n = np.linspace(1, 4, 4, dtype=int)
lambda_n = (n*np.pi/L)**2
X_n = np.zeros([len(n), len(x)])
for i,n_i in enumerate(n):
  X_n[i, :] = np.sin(np.sqrt(lambda_n[i])*x)

Plot the eigenfunctions.

fig, ax = plt.subplots()
for i, n_i in enumerate(n):
  ax.plot(x, X_n[i,:], linewidth=2,label='$n = '+str(n_i)+'$')
plt.legend()
plt.show()

We see that the fourth of the S-L theorems appears true: n − 1 zeros of X_n exist on the open interval (0,1).

Haberman, R. 2018. Applied Partial Differential Equations with Fourier Series and Boundary Value Problems (Classic Version). Pearson Modern Classics for Advanced Mathematics. Pearson Education Canada.

For the S-L problem to be regular, it has the additional constraints that $p, q, \sigma$ are continuous and $p, \sigma > 0$ on $[a,b]$. This is also sometimes called the sturm-liouville eigenvalue problem. See Haberman (2018, sec. 5.3) for the more general (non-regular) S-L problem and Haberman (2018, sec. 7.4) for the multi-dimensional analog.↩︎
These eigenvalues are closely related to, but distinct from, the “eigenvalues” that arise in systems of linear ODEs.↩︎

Online Resources for Section 7.2

No online resources.