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PAGE 1

Heat eq

\[ u_t = k u_{xx} \]
\( 0 < x < L \)     \( t > 0 \)

\[ u(0, t) = u(L, t) = 0 \]

or

\[ u_x(0, t) = u_x(L, t) = 0 \]

\[ u(x, 0) = f(x) \]

Separation of variables:

\[ u(x, t) = X(x) T(t) \]

\[ u_t = X T' \quad u_{xx} = X'' T \]

\[ u_t = k u_{xx} \rightarrow X T' = k X'' T \]
\[ \frac{X''}{X} = \frac{T'}{kT} = -\lambda \] (\(\lambda\) is ok, too)

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

\[ X(x) = A \cos(\sqrt{\lambda} x) + B \sin(\sqrt{\lambda} x) \]

PAGE 2

if \( u(0, t) = u(L, t) = 0 \rightarrow X(0) = X(L) = 0 \)

\[ 0 = A \]

\[ 0 = B \sin(\sqrt{\lambda} L) \] \( B \neq 0, \lambda > 0 \)

\[ \sin(\sqrt{\lambda} L) = 0 \]

\[ \sqrt{\lambda} L = n\pi \] \( n = 1, 2, 3, \dots \)
\[ \lambda_n = \frac{n^2 \pi^2}{L^2} \]
\[ X_n = \sin\left(\frac{n\pi}{L}x\right) \]

if \( u_x(0, t) = u_x(L, t) = 0 \rightarrow X'(0) = X'(L) = 0 \)

\[ X'(x) = -\sqrt{\lambda} A \sin(\sqrt{\lambda} x) + \sqrt{\lambda} B \cos(\sqrt{\lambda} x) \]

\[ 0 = \sqrt{\lambda} B \rightarrow B = 0 \]

\[ 0 = -\sqrt{\lambda} A \sin(\sqrt{\lambda} L) \rightarrow \sin(\sqrt{\lambda} L) = 0 \rightarrow \]
\[ \lambda_n = \frac{n^2 \pi^2}{L^2} \]
\( n = 1, 2, 3, \dots \)
\[ X_n = \cos\left(\frac{n\pi}{L}x\right) \]
PAGE 3

Solving Differential Equations with Boundary Conditions

if \(\lambda = 0\),

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

\[ X = Ax + B \]

\[ X'(0) = X'(L) = 0 \rightarrow A = 0 \rightarrow \]

\[ X_0 = 1 \]
\[ \lambda_0 = 0 \]

if

  • \( u(0, t) = 0 \)
  • \( u_x(L, t) = 0 \)

\( \downarrow \)

  • \( X(0) = 0 \)
  • \( X'(L) = 0 \)

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

if \( \lambda > 0 \), \( X = A \cos(\sqrt{\lambda}x) + B \sin(\sqrt{\lambda}x) \)

\( X(0) = 0 \rightarrow A = 0 \)

\( X = B \sin(\sqrt{\lambda}x) \)

\( X' = \sqrt{\lambda} B \cos(\sqrt{\lambda}x) \)

\( X'(L) = 0 \rightarrow 0 = \sqrt{\lambda} B \cos(\sqrt{\lambda}L) \)

\( B \neq 0 \)

\[ \cos(\sqrt{\lambda}L) = 0 \]

\[ \sqrt{\lambda}L = \frac{(2n+1)}{2}\pi \]

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

\[ = \frac{(2n-1)}{2}\pi \]

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

\[ \lambda_n = \frac{(2n-1)^2 \pi^2}{4L^2} \]

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

\[ X_n = \sin(\sqrt{\lambda_n}x) \]
PAGE 4

Laplace's eq

\[ u_{xx} + u_{yy} = 0 \quad 0 < x < a \quad 0 < y < b \]

  • \( u(x, 0) = 0 \)
  • \( u(x, b) = 0 \)
  • \( u_x(0, y) = 0 \)
  • \( u(a, y) = f(y) \)
A coordinate graph showing a rectangular domain in the first quadrant with vertices at (0,0), (a,0), (a,b), and (0,b). The right boundary at x=a is labeled with f(y).

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

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

since \( y \) has 0 BC

we want to solve \( Y'' + \lambda Y = 0 \)

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

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

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

\[ Y_n = \sin\left(\frac{n\pi}{b}y\right) \]

\[ \frac{X''}{X} = \lambda = \frac{X''}{X} = \frac{n^2 \pi^2}{b^2} \]

PAGE 5
\[ X'' - \frac{n^2 \pi^2}{b^2} X = 0 \]
\[ X'(0) = 0 \]
\[ X = A \cosh\left(\frac{n\pi}{b}x\right) + B \sinh\left(\frac{n\pi}{b}x\right) \] \[ X' = A \cdot \frac{n\pi}{b} \sinh\left(\frac{n\pi}{b}x\right) + B \cdot \frac{n\pi}{b} \cosh\left(\frac{n\pi}{b}x\right) \]
\[ 0 = B \]
\[ X_n = \cosh\left(\frac{n\pi}{b}x\right) \]
\[ u(x, y) = \sum_{n=1}^{\infty} C_n \cosh\left(\frac{n\pi}{b}x\right) \sin\left(\frac{n\pi}{b}y\right) \] \[ u(a, y) = f(y) = \sum_{n=1}^{\infty} \left[ C_n \cosh\left(\frac{n\pi a}{b}\right) \right] \sin\left(\frac{n\pi}{b}y\right) \] \[ C_n \cosh\left(\frac{n\pi a}{b}\right) = \frac{2}{b} \int_{0}^{b} f(y) \sin\left(\frac{n\pi y}{b}\right) dy \]
PAGE 6

2D Wave

\[ u_{tt} = a^2(u_{xx} + u_{yy}) \]
  • \( u(x, 0) = 0 \)
  • \( u(x, b) = 0 \)
  • \( u(a, y) = 0 \)
  • \( u(0, y) = 0 \)
  • \( u(x, y, 0) = f(x) \)
  • \( u_t(x, y, 0) = g(x) \)
\( 0 < x < a \) \( 0 < y < b \)
A simple 2D coordinate graph showing a rectangle defined by boundaries on the x and y axes, representing the domain 0 to a and 0 to b.

Separation of Variables

\[ u(x, y, t) = X Y T \] \[ X Y T'' = a^2(X'' Y T + X Y'' T) \] \[ \frac{T''}{T} = a^2\left( \frac{X''}{X} + \frac{Y''}{Y} \right) \] \[ \frac{T''}{a^2 T} = \frac{X''}{X} + \frac{Y''}{Y} \] \[ \frac{X''}{X} = \frac{T''}{a^2 T} - \frac{Y''}{Y} = -\lambda \]
\[ X'' + \lambda X = 0 \] \[ \lambda_n = \frac{n^2 \pi^2}{a^2} \]
\[ X(0) = X(a) = 0 \] \[ X_n = \sin\left(\frac{n\pi}{a}x\right) \]
PAGE 7

Separation of Variables for Wave Equation

\[ \frac{T''}{a^2 T} - \frac{Y''}{Y} = -\lambda \]

\[ Y'' + \mu Y = 0 \]

\[ \mu_m = \frac{m^2 \pi^2}{b^2} \]

\[ \frac{Y''}{Y} = \frac{T''}{a^2 T} + \lambda = -\mu \]

\[ Y(0) = Y(b) = 0 \]

\[ Y_m = \sin\left(\frac{m\pi}{b} y\right) \]

\[ \frac{T''}{a^2 T} = -(\lambda + \mu) \]

\[ T'' + a^2(\lambda + \mu)T = 0 \]

\[ T = A \cos(a\sqrt{\lambda + \mu} \, t) + B \sin(a\sqrt{\lambda + \mu} \, t) \]

\[ T_{nm} \]

General Solution

\[ u(x, y, t) = \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} C_{nm} T_{nm} X_n Y_m \]