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d'Alembert solution

\( a = 4 \), infinite string, \( g(x) = 0 \) (initial velocity)

\( f(x) = e^{-x^2} \), find \( y(1, 2) \)

\[ y(x, t) = \frac{1}{2} \left[ f(x - at) + f(x + at) \right] + \frac{1}{2a} \int_{x - at}^{x + at} g(s) \, ds \]
\[ = \frac{1}{2} \left[ e^{-(x - 4t)^2} + e^{-(x + 4t)^2} \right] \]

Sturm-Liouville (orthogonality)

\( y'' + \lambda y = 0 \)

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

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

\( a < x < b \)

solve for \( y_n \) for different cases of \( \lambda \) (\( \lambda < 0 \), \( \lambda = 0 \), \( \lambda > 0 \))

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the \( y_n \) are mutually orthogonal \( \rightarrow \int_a^b y_n y_m \, dx = 0 \) if \( n \neq m \)

they can be used to expand some function \( f(x) \)

\[ f(x) = \sum_{n=1}^{\infty} c_n y_n \quad \text{(just like w/ Fourier series)} \]

to find \( c_n \), we use orthogonality of \( y_n \)

multiply by \( y_m \)

\[ f(x) y_m = \sum_{n=1}^{\infty} c_n y_n y_m \]
\[ = c_1 y_1 y_m + c_2 y_2 y_m + \dots \]

integrate over \( a < x < b \)

\[ \int_a^b f(x) y_m \, dx = \underbrace{\int_a^b c_1 y_1 y_m \, dx + \int_a^b c_2 y_2 y_m \, dx + \dots}_{\text{all zero except when } n = m} \]
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\[ \int_{a}^{b} f(x) y_n \, dx = \int_{a}^{b} C_n (y_n)^2 \, dx \rightarrow C_n = \frac{\int_{a}^{b} f(x) y_n \, dx}{\int_{a}^{b} (y_n)^2 \, dx} \]

\[ = \frac{2}{L} \int_{0}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) dx \]

or \( \cos\left(\frac{n\pi x}{L}\right) \) if \( y_n = \sin\left(\frac{n\pi x}{L}\right) \) or \( \cos\left(\frac{n\pi x}{L}\right) \)

2-D Heat

\[ u_t = k(u_{xx} + u_{yy}) \]

\[ 0 < x < 1, \quad 0 < y < 1 \]

A square domain in the xy-plane with vertices at (0,0) and (1,1). The boundary conditions are labeled: 100 on the bottom edge, and 0 on the top, left, and right edges.
  • \( u(x, 0) = 100 \)
  • \( u(x, 1) = 0 \)
  • \( u(0, y) = 0 \)
  • \( u(1, y) = 0 \)

\[ u(x, y) = \sum_{n=1}^{\infty} C_n \sin(n\pi x) \sinh(n\pi(1-y)) \]

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Solution has no time \( \rightarrow \) steady state

A square region on a coordinate system with y-axis and x-axis. The top edge is at y=1 and labeled 0. The bottom edge is at y=0 and labeled 100. The left and right edges are at x=0 and x=1 respectively, both labeled 0.

\[ u_{xx} + u_{yy} = 0 \]

\[ X''Y + XY'' = 0 \]

\[ \frac{X''}{X} = -\frac{Y''}{Y} = -\lambda \]

\[ X'' + \lambda X = 0 \]

\[ \lambda_n = n^2\pi^2 \]

\[ X(0) = X(1) = 0 \]

\[ X_n = \sin(n\pi x) \]

\[ Y'' - \lambda Y = 0 \]

\[ Y'' - n^2\pi^2 Y = 0 \]

\( Y(1) = 0 \rightarrow Y(z=0) = 0 \)

\[ Y = A \cosh(n\pi y) + B \sinh(n\pi y) \]

let \( y = 1 - z \), \( z = 1 - y \)

\[ Y(z) = A \cosh(n\pi(1-z)) + B \sinh(n\pi(1-z)) \rightarrow \dots \rightarrow \text{Sinh survives} \]

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Solving for Coefficients in Steady-State Solutions

\[ u(x,y) = \sum C_n \sin(n\pi x) \sinh(n\pi(1-y)) \]

\[ u(x,0) = 100 = \sum_{n=1}^{\infty} C_n \sin(n\pi x) \sinh(n\pi) \]

\[ 100 = \sum_{n=1}^{\infty} [C_n \sinh(n\pi)] \sin(n\pi x) \]
Sine series \( L=1 \)

\[ C_n \sinh(n\pi) = \frac{2}{1} \int_{0}^{1} 100 \sin(n\pi x) dx \]

\[ C_n = \frac{2}{\sinh(n\pi)} \int_{0}^{1} 100 \sin(n\pi x) dx \]

if not steady-state: \[ u(x,y,t) = \sum C_n T_{nm} X_n Y_m \]

exponential vs hyperbolic

\(\downarrow\)

dealing w/ \( \infty \)

\(\hookrightarrow\)

finite domain

\[ \cosh(x) = \frac{1}{2}(e^x + e^{-x}) \]

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Separation of Variables for Laplace's Equation

A square domain in the first quadrant of a coordinate system. The x-axis and y-axis are shown. The square has vertices at (0,0), (1,0), (1,1), and (0,1). Boundary conditions are written along the edges: u=0 on the left edge, u_y=0 on the top and bottom edges, and 100 on the right edge.

\[ \frac{X''}{X} = -\frac{Y''}{Y} = \lambda \]

solve \( Y \) first: want \( Y'' + \lambda Y = 0 \)

\[ Y'' + \lambda Y = 0 \quad Y'(0) = Y'(1) = 0 \]

\[ \lambda_n = \frac{n^2 \pi^2}{1^2} = n^2 \pi^2 \]

\[ Y_n = \cos(n\pi y) \quad n=0, 1, 2, 3, \dots \]

\[ X'' - \lambda X = 0 \]

\[ X'' - n^2 \pi^2 X = 0 \]

\[ X = A \cosh(n\pi x) + B \sinh(n\pi x) \]

\[ X'(0) = 0 \]

\[ X' = n\pi A \sinh(n\pi x) + n\pi B \cosh(n\pi x) \]

\[ 0 = n\pi B \rightarrow B = 0 \]

\[ X_n = \cosh(n\pi x) \]