Download original notes
PAGE 1

Sturm-Liouville Theory

1-D Heat eq:

\( u_t = k u_{xx} \)

\( 0 < x < L \)

we've seen that if \( u(0, t) = u(L, t) = 0 \) (ends frozen)

then \( \lambda_n = \frac{n^2 \pi^2}{L^2} \)

with \( X_n = \sin(\sqrt{\lambda} x) = \sin\left(\frac{n\pi}{L}x\right) \)

\( n = 1, 2, 3, \dots \)

if \( u_x(0, t) = u_x(L, t) = 0 \) (ends insulated)

then \( \lambda_n = \frac{n^2 \pi^2}{L^2} \)

with \( X_n = \cos(\sqrt{\lambda} x) = \cos\left(\frac{n\pi}{L}x\right) \)

\( n = 0, 1, 2, 3, \dots \)

in both cases, the eigenvalues \( \lambda_n \) give us the frequencies of the modes of solutions \( \rightarrow \) all integer multiples of \( \frac{\pi}{L} \)

and the eigenfunctions are all mutually orthogonal

\[ \int_0^L X_n X_m \, dx = 0 \quad \text{if } m \neq n \]

is this always the case w/ other kinds of boundary conditions?

PAGE 2

let's look at this one:

\[ \begin{aligned} u_t &= k u_{xx} \quad 0 < x < L \\ u(0,t) &= 0 \quad \text{left end frozen} \\ u_x(L,t) &= -h u(L,t) \quad h > 0 \end{aligned} \]

\(\underbrace{\hspace{5cm}}_{\text{models a heat exchange at the right end}}\)
(from Newton's Law of Cooling)

after separating the variables, we get to

\[ \begin{aligned} \bar{X}'' + \lambda \bar{X} &= 0 \\ \bar{X}(0) &= 0 \\ \bar{X}'(L) + h \bar{X}(L) &= 0 \end{aligned} \]

after using \(\bar{X}(0) = 0\), we get

\[ \begin{aligned} \bar{X} &= B \sin(\sqrt{\lambda} x) \longrightarrow \bar{X}_n = \sin(\sqrt{\lambda_n} x) \\ \bar{X}' &= \sqrt{\lambda} B \cos(\sqrt{\lambda} x) \end{aligned} \]
\[ \bar{X}'(L) + h \bar{X}(L) = 0 \longrightarrow \sqrt{\lambda} B \cos(\sqrt{\lambda} L) + h B \sin(\sqrt{\lambda} L) = 0 \]
PAGE 3

need: \( \lambda \neq 0, B \neq 0 \)

\[ \sqrt{\lambda} \cos(\sqrt{\lambda} L) = -h \sin(\sqrt{\lambda} L) \]

\[ \tan(\sqrt{\lambda} L) = -\frac{\sqrt{\lambda}}{h} \]

Solve for \( \lambda \)

this is a transcendental equation (variable we are interested in is on both sides and cannot be isolated)

the graphical interpretation:

let \( z = \sqrt{\lambda} L \)

that equation becomes \( \tan(z) = -\frac{z}{hL} \) \( (h > 0, L > 0) \)

→ \( z \) is the intersection of \( \tan(z) \) and the line

\[ -\frac{z}{hL} \]
PAGE 4
A coordinate graph showing the intersection of the tangent function tan(z) with a line of negative slope passing through the origin. The tangent curves have vertical asymptotes at odd multiples of pi over 2. Intersection points are labeled as z1 and z2 on the positive z-axis.

we want to find all positive intersections \( z = z_1, z_2, \dots \)

whatever \( z_n \) is, it is clearly NOT integer multiples of \( \frac{\pi}{L} \)

\( X_n = \sin(\sqrt{\lambda_n} x) \) frequencies are not \( \frac{n\pi}{L} \) anymore

as \( n \to \infty, z_n \to \) the left asymptote of each tangent cycle

→ eventually resembling one of the two basic cases

PAGE 5

Sturm-Liouville Theory

What if diffusivity is not constant? \( u_t = k(x) u_{xx} \)?

Sturm-Liouville theory gives us a big picture understanding of the solutions to \( X'' + \lambda X = 0 \).

Sturm-Liouville Problem

\[ \frac{d}{dx} \left[ p(x) \frac{dy}{dx} \right] + q(x)y + \lambda w(x)y = 0 \]
\( a < x < b \)

Subject to BCs:

\( \alpha_1 y(a) + \alpha_2 y'(a) = 0 \) \( \alpha_1, \alpha_2 \) not both zero
\( \beta_1 y(b) + \beta_2 y'(b) = 0 \) \( \beta_1, \beta_2 \) not both zero

Notice if:

\( p=1, q=0, w=1 \)
we get \( y'' + \lambda y = 0 \) (Fourier)
\( p=x, q=-\frac{n^2}{x}, w=x \)
we get \( xy'' + y' + (\lambda x - \frac{n^2}{x})y = 0 \)

Solutions are Bessel functions (waves of a circular drum)

\( p=1-x^2, q=0, w=1 \)
we get \( y''(1-x^2) - 2xy' + \lambda y = 0 \)
PAGE 6

Solutions are Legendre polynomials

(steady state solution of a heated sphere)

Big picture: different \( p, q, w \) \( \rightarrow \) different heat/wave eq. situations