Separation of Variables

Separation of Variables

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Wave Equation on an Interval

$$ \begin{cases} \partial_t^2 u - c^2 \partial_x^2 u , x \in [0,l] \\ u(0, t) = 0 = u_t (l,t) \\ u(x,0) = f(x), u_t (0,x) = g(x). \end{cases} $$

How to solve it?

  • Find many solutions of wave equation on $[0,l]$.
  • Build linear combinations.
  • Select combo to satsify initial condition.

Seek Separated Solutions

Look for solutions in the form $$ u(t,x) = X(x) T(t). $$

$\rightarrow$ wave equation yields

$$ X(x) T^{''}(t) = c^2 X^{''}(x) T(t). $$

Seek Separated Solutions

Look for solutions in the form $$ u(t,x) = X(x) T(t). $$

$\rightarrow$ heat equation yields

$$ X(x) T^{'}(t) = k X^{''}(x) T(t). $$

Seek Separated Solutions

Divide by $X(x) T(t)$ to find

$$ \frac{T^{'}(t)}{T(t)} = k \frac{X^{''}(x)}{X(x)}. $$

Since the left side depends only upon $t$ and the right side depends only upon $x$, both sides must be constant. We choose to write the constant as $-\lambda$.

Collapse to ODE

The equation splits into to ODE:

$$ - X^{''} = \lambda X, ~ x \in [0,l], ~ X(0) = X(l) = 0,$$

$$ T^{'} = - \lambda k T. $$

$\implies$

$$ u(t,x) = \sum_{n} A_n e^{-(n \pi /l)^2 t } \sin \frac{n \pi x}{l}. $$

We need to choose the coefficients $A_n$ to satisfy the initial condition.