Problem set 2, PDE

Problem 1

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

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

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

with u(x,0)=0.

Problem 2

The function

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

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

Problem 3

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

\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.

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