Boundary Value 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 $p, q, σ$ be functions of $x$ on open interval $(a, b)$ . Let $X$ be the dependent variable and $λ$ constant. The regular S-L problem is the S-L ODE¹ $\begin{array}{r} \frac{d}{d x} (p X^{'}) + q X + λ σ X = 0 \end{array}$ with boundary conditions $\begin{aligned} β_{1} X (a) + β_{2} X^{'} (a) & = 0 \\ β_{3} X (b) + β_{4} X^{'} (b) & = 0 \end{aligned}$ with coefficients $β_{i} \in R$ . This is a type of boundary value problem.

This problem has nontrivial solutions, called eigenfunctions $X_{n} (x)$ with $n \in Z_{+}$ , corresponding to specific values of $λ = λ_{n}$ 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 $λ_{n}$ for each eigenfunction $X_{n}$
The eigenvalues can be ordered as $λ_{1} < λ_{2} < \dots$
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 $σ$ such that any piecewise continuous function $f : [a, b] \to R$ 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 $β_{2}, β_{4} = 0$ ,
Neumann: for just $β_{1}, β_{3} = 0$ ,
Robin: for all $β_{i} \neq 0$ , and
Mixed: if $β_{1} = 0$ , $β_{3} \neq 0$ ; if $β_{2} = 0$ , $β_{4} \neq 0$ .

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: $\begin{aligned} X (a) & = X (b) \\ X^{'} (a) & = X^{'} (b) . \end{aligned}$

Example 7.1

Consider the differential equation $\begin{array}{r} X^{″} + λ X = 0 \end{array}$ with dirichlet boundary conditions on the boundary of the interval [0,L] $\begin{array}{r} X (0) = 0 and X (L) = 0. \end{array}$ Solve for the eigenvalues and eigenfunctions.

This is a sturm-liouville problem, so we know the eigenvalues are real. The well-known general solution to the ODE is $X (x) = {\begin{cases} k_{1} + k_{2} x & λ = 0 \\ k_{1} e^{j \sqrt{λ} x} + k_{2} e^{- j \sqrt{λ} x} & otherwise \end{cases}$ 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, that for nontrivial solutions, λ ≠ 0. Now let’s check λ < 0. The solution becomes $\begin{aligned} X (x) & = k_{1} e^{- \sqrt{| λ |} x} + k_{2} e^{\sqrt{| λ |} x} \\ = k_{3} \cosh (\sqrt{| λ |} x) + k_{4} \sinh (\sqrt{| λ |} x) \end{aligned}$ where k₃ and k₄ are real constants. Again applying the boundary conditions: $\begin{aligned} X (0) & = 0 \Rightarrow k_{3} \cosh (0) + k_{4} \sinh (0) = 0 \Rightarrow k_{3} + 0 = 0 \Rightarrow k_{3} = 0 \\ X (L) & = 0 \Rightarrow 0 \cosh (\sqrt{| λ |} L) + k_{4} \sinh (\sqrt{| λ |} L) = 0 \Rightarrow k_{4} \sinh (\sqrt{| λ |} L) = 0. \end{aligned}$ However, $\sinh (\sqrt{| λ |} L) \neq 0$ for L > 0, so k₄ = k₃ = 0—again, the trivial solution. Now let’s try λ > 0. The solution can be written $\begin{array}{r} X (x) = k_{5} \cos (\sqrt{λ} x) + k_{6} \sin (\sqrt{λ} x) . \end{array}$ Applying the boundary conditions for this case: $\begin{aligned} X (0) & = 0 \Rightarrow k_{5} \cos (0) + k_{6} \sin (0) = 0 \Rightarrow k_{5} + 0 = 0 \Rightarrow k_{5} = 0 \\ X (L) & = 0 \Rightarrow 0 \cos (\sqrt{λ} L) + k_{6} \sin (\sqrt{λ} L) = 0 \Rightarrow k_{6} \sin (\sqrt{λ} L) = 0. \end{aligned}$ Now, $\sin (\sqrt{λ} L) = 0$ for $\begin{aligned} \sqrt{λ} L & = n π \Rightarrow \\ (n \in Z_{+}) & λ & = {(\frac{n π}{L})}^{2} . \end{aligned}$ Therefore, the only nontrivial solutions that satisfy both the ODE and the boundary conditions are the eigenfunctions $\begin{aligned} X_{n} (x) & = \sin (\sqrt{λ_{n}} x) \\ = \sin (\frac{n π}{L} x) \end{aligned}$ with corresponding eigenvalues $\begin{aligned} λ_{n} & = {(\frac{n π}{L})}^{2} . \end{aligned}$ 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).

Irregular Sturm-Liouville problems

Sturm-Liouville problems that do not satisfy the regularity conditions are called irregular. The nice theorems for regular S-L problems may not hold for irregular S-L problems. However, we can often still solve irregular S-L problems using the same methods as for regular problems.

Example 7.2

Consider the ODE called Bessel’s equation [@kreyszig2011, § 5.4], for real independent variable s and dependent variable y(s), s²y″ + sy′ + (s²−ν²)y = 0. This ODE arises in polar coordinate PDE models of circular membranes, where y is the membrane displacement and s = kr, and where r is the radius of the membrane and k is a constant sometimes called the wavenumber.

Consider the following boundary conditions:

At radius r = R, the membrane is fixed, so y = 0.
At radius r = 0, the membrane is free to move, but the displacement is finite, so |y| < ∞.

Prove that the Bessel equation is an irregular Sturm-Liouville problem and solve for the eigenvalues and eigenfunctions for the case ν = 0.

We proceed to show that the Bessel equation with the given boundary conditions is an irregular Sturm-Liouville problem by first showing that the Bessel equation can be written in the form of a Sturm-Liouville problem ODE, then showing that the boundary conditions are not regular. Dividing the Bessel equation by s gives $\begin{aligned} (z \ne 0) & s y^{″} + y^{'} + \frac{s^{2} - ν^{2}}{s} y & = 0 ⟹ \\ \frac{d}{d s} (s y^{'}) + (s - ν^{2} / s) y & = 0. \end{aligned}$ So we see that this is equivalent to the Sturm-Liouville problem’s ODE, $\begin{array}{r} \frac{d}{d x} (p X^{'}) + (λ σ + q) X = 0, \end{array}$ with x = s, X = y, p(s) = s, q(s) = − ν²/s, σ(s) = s, and λ = 1.

Now let us consider the boundary conditions. The second boundary condition cannot be written as a linear combination of y and y′ at s = 0, therefore this is an irregular Sturm-Liouville problem.

Solutions for Bessel’s equation are Bessel functions of the first kind, J_ν(s) (section 6.4), and Bessel functions of the second kind, Y_ν(s) [@kreyszig2011,§ 5.5]. For ν = 0, the Bessel equation simplifies to s²y″ + sy′ + s²y = 0 and the solutions are zeroth-order Bessel functions of the first kind, J₀(s), and of the second kind, Y₀(s). That is, the general solution is y(s) = aJ₀(s) + bY₀(s). However, Y₀(s) is singular at s = 0, so the boundary condition |y(0)| < ∞ requires b = 0. Therefore, the eigenfunctions are y_n(s) = J₀(s) = J₀(k_nr) for eigenvalues λ_n = k_n² and n ∈ ℤ₊. On the boundary, y_n(kR) = 0 implies that k_nR are the zeros of J₀(s), what we called α_0, n in example. Therefore, the eigenvalues are $λ_{n} = {(\frac{α_{0, n}}{R})}^{2} .$

Plotting the eigenfunctions

We proceed in Python. First, load packages.

import numpy as np
import sympy as sp
import scipy
from scipy.special import jn_zeros
import matplotlib.pyplot as plt

The eigenvalues can be computed from the zeros of the Bessel function zeros α_0, n, where n = 1, 2, 3, … as follows:

n = sp.symbols('n', integer=True, positive=True)
k, R = sp.symbols('k, R', positive=True, real=True)
r = sp.symbols('r', nonnegative=True)
J_0 = sp.besselj(0, k * r)
N_zeros = 5
alpha_0_n = jn_zeros(0, N_zeros)
params = {R: 1}  # Set R = 1
lambda_n_ = (alpha_0_n / params[R])**2
print(f"The first {N_zeros} eigenvalues are:")
print(lambda_n_)

The first 5 eigenvalues are:
[  5.78318596  30.47126234  74.88700679 139.04028443 222.93230362]

The eigenfunctions are given by the Bessel functions of the first kind J₀(k_nr).

def k_n(lambda_n): return np.sqrt(lambda_n)
k_n_ = k_n(lambda_n_)

Plot the eigenfunctions.

r_plt = np.linspace(0, 1, 101)
print(J_0)
y_n_fun = sp.lambdify((r, k), J_0, modules=['numpy', 'scipy'])
fig, ax = plt.subplots()
for i in range(5):
    k_i = k_n_[i]
    ax.plot(r_plt, y_n_fun(r_plt, k_i), label=f'$y_{i+1}(k_{i+1} r)$')
ax.spines['bottom'].set_position('zero')
ax.spines['left'].set_position('zero')
ax.set_xlabel('$r$')
ax.set_ylabel('Eigenfunctions $y_n(k_n r)$')
ax.legend()
plt.show()

Figure 7.2: Eigenfunctions for the first few eigenvalues.

The plot shows the eigenfunctions y_n(k_nr) for the first few eigenvalues. Note that the boundary conditions are satisfied and that the eigenfunctions appear to be radial modes of vibration for a circular membrane.

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 $p, q, σ$ are continuous and $p, σ > 0$ 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.