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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)

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

eigenfunctions \( X_n = \sin\left(\frac{n\pi}{L}x\right) \) \( n=1,2,3... \)

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

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

\( X_n = \cos\left(\frac{n\pi}{L}x\right) \) \( n=0,1,2,3... \)

in both cases, the eigenvalues are the frequencies of each mode of the solutions \( \rightarrow \) integer multiples of \( \frac{\pi}{L} \)

also, the eigenfunctions are mutually orthogonal: \( \int_0^L X_n X_m dx = 0 \) if \( n \neq m \)

will these still be true if we used more complicated boundary conditions?

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for example,

\[ u_t = k u_{xx} \quad 0 < x < L \]
\[ u(0,t) = 0 \] left end frozen
\[ u_x(L,t) = -h u(L,t) \quad h > 0 \]
\[ \underbrace{\phantom{u_x(L,t) = -h u(L,t) \quad h > 0}}_{\text{this models a heat exchange at } x=L} \]

(from Fourier's law and Newton's law of cooling)

after separation of variables, we get

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

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

\[ \bar{X} = B \sin(\sqrt{\lambda} x) \] \[ \bar{X}' = \sqrt{\lambda} B \cos(\sqrt{\lambda} x) \]
\( \rightarrow \) eigenfunctions are \( \bar{X}_n = \sin(\sqrt{\lambda_n} x) \)
\[ \bar{X}'(L) + h \bar{X}(L) = 0 \rightarrow \sqrt{\lambda} B \cos(\sqrt{\lambda} L) + B h \sin(\sqrt{\lambda} L) = 0 \]
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we require \( \lambda \neq 0 \), \( B \neq 0 \)

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

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

solve for \( \lambda \)

( transcendental eq: variable we want is on both sides )

let's try to interpret the solution graphically

let's define \( z = \sqrt{\lambda} L \) \( \quad (z > 0) \)

then \( \tan(z) = -\frac{z}{hL} \) is describing the intersections

of \( y = \tan(z) \) and \( y = -\frac{1}{hL} z \)

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A coordinate graph showing the intersection of the tangent function y = tan(z) and a line with a negative slope y = -z/(hL). The tangent curves have vertical asymptotes at odd multiples of pi/2. The intersection points are labeled z1, z2, and z3 on the positive z-axis.

there are infinitely-many intersections \( z_1, z_2, z_3, \dots \)

whatever they are, \( z = \sqrt{\lambda} L \) no longer leads to

\[ \lambda_n = \frac{n^2 \pi^2}{L^2} \]

but notice as \( n \to \infty \), \( z_n \) gets closer to multiples of \( \frac{\pi}{2} \)

( left asymptote of each cycle of tangent )

\( \rightarrow \) eventually start to resemble the \( \lambda \) of the two basic cases

what if the diffusivity is not constant?

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\( u_t = k(x) u_{xx} \)

what would \( X_n \) look like? \( \lambda_n \)?

collectively, this is what the Sturm-Liouville theory can answer

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

\( \alpha_1 y(a) + \alpha_2 y'(a) = 0 \)

\( \beta_1 y(b) + \beta_2 y'(b) = 0 \)

\( \alpha_1, \alpha_2 \) not both zero

\( \beta_1, \beta_2 \) not both zero

notice if \( p=1, q=0, w=1 \) we get \( y'' + \lambda y = 0 \) (\( X'' + \lambda X = 0 \))

Solutions are Fourier series

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

(models waves of a circular drum)

Solutions are Bessel functions

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

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(models the steady state solution of a heated sphere)

Solutions are Legendre polynomials

big picture view: choosing \( p, q, w \) can give us a wide variety of heat / wave situations