## Separation of Variables ##
## Separation of Variables ## ![utlogo](http://www.math.toronto.edu/colliand/images/utlogo.jpeg)
## 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 Divide by $-c^2 XT$ to see $$- \frac{T^{''}(t)}{c^2 T (t)} = - \frac{X^{''}(x)}{X(x)}.$$ The left side depends *only* on $t$. The right side depends *only* on $x$. Differentiating with $t$ reveals left side is constant. Differentiating with $x$ reveals right side is constant. Name the constant $\lambda$.
## Collapse to 2 ODE Assume (*why*?) that $\lambda >0$. Write $\lambda = \beta^2.$ $$X^{''} + \beta^2 X =0, ~ T^{''} + \beta^2 c^2 T = 0.$$ For *separated form* solutions, PDE $\rightarrow$ 2 ODE! $$X(x) = C \cos \beta x + D \sin \beta x$$ $$T(t) = A \cos \beta c t + B \sin \beta c t$$
## Boundary Conditions We demand that $X(0) = 0$ so $C = 0.$ We demand that $X(l) = D \sin \beta l = 0.$ We don't want $D=0$! We want $\beta l$ to be a zero of the sine function. This *quantizes* $beta$: $$\beta l = n \pi \implies \beta_n = \frac{n \pi}{l}.$$ We find infintely many solutions in separated form: $$u\_n (t,x) = [A\_n \cos \frac{n \pi ct }{l} + B\_n \sin \frac{n \pi ct}{l} ] \sin \frac{n \pi x}{l}.$$
## Superposition We can form linear combinations. $$u(t,x) = \sum\_{n} [A\_n \cos \frac{n \pi ct }{l} + B\_n \sin \frac{n \pi ct}{l} ] \sin \frac{n \pi x}{l}$$ This expression solves the wave equation and satisfies the Dirichlet boundary conditions. The expression is flexible; we can choose the coefficients $A\_n, B\_n$. Can we choose the coefficients to match the initial conditions?
## Initial Conditions We want $u(0,x) = f(x)$ and $u\_t (0,x) = g(x)$. Therefore we want to choose the coefficients so that $$f(x) = \sum\_{n} [A\_n ] \sin \frac{n \pi x}{l},$$ $$g(x) = \sum\_{n} [ \frac{n \pi c}{l} B\_n ] \sin \frac{n \pi x}{l}.$$ How can we choose the coefficients? Let's postpone this question for a while....
## Heat Equation on an Interval $$\begin{cases} \partial\_t u = k \partial_x^2 u ~, x \in [0,l] \\\\ u(t, 0) = 0 = u(t, l) \\\\ u(x,0) = f(x). \end{cases}$$ ### 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 $$u(t,x) = X(x) T(t).$$ $\rightarrow$ 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.