Problem set 2, PDE

Problem 1

If, for $latex k=1, 2, \ldots, N$, $latex y_k(x_1, \ldots, x_{n-1},t) = u(x_1 + \ldots + x_{n-1},t)$, then $latex Y = (y_k)$ solves the system

$latex \displaystyle \partial_t y_k = \frac{Mr}{r - (x_1 + \ldots + x_{n-1}) - (y_1 + \ldots + y_N)} \Big( \sum_{i=1}^{n-1}\sum_{j=1}^N \partial_{x_i}y_j + 1 \Big)$

with $latex y_k(x,0)=0$ if and only if $latex u(s,t)$ solves the equation

$latex \displaystyle \partial_t u = \frac{Mr}{r - s - Nu} \big( N(n-1)\partial_su + 1\big)$

with $latex u(x,0)=0$.

Problem 2

The function

$latex \displaystyle u(s,t) = \frac{r - s - \sqrt{(r-s)^2 - 2MNnrt}}{Nn}$

solves the equation of the previous problem near $latex (0,0)$.

Problem 3

Find a solution as a power series expansion of the initial-value problem

$latex \displaystyle \begin{cases} u_{tt} - u_{xx} - u=0,\quad (x,t)\in\R\times(0,\infty)\\ u(x,0) = x,\quad \partial_t u(x,0) = -x. \end{cases}$

Identify this solution.