Convert second order PDE $u_{tt} = u_{xx} + u$ to a system of first order PDE's

Question

I am attempting to convert $$ u_{tt} = u_{xx} + u$$ to a system of first order PDE's. I believe that the system will require 3 equations, one for each of $u, u_t,$ and $u_x.$ Here is my attempt: \begin{equation} \frac{\partial}{\partial t} \begin{pmatrix} u \\ u_t \\ u_x \end{pmatrix} - \frac{\partial}{\partial x} \begin{pmatrix} u \\ u_x \\ u_t \end{pmatrix} = \begin{pmatrix} u_t - u_x \\ u \\ 0 \end{pmatrix} \end{equation}

This doesn't feel correct. If anyone is familiar with a standard way to do this, any help would be appreciated.

Answer 1

This is a wave equation and you need to change variables as follows: p = x+t, q = x-t. You will obtain: $$ -4\frac{d^2u}{dpdq}=u(p,q)$$ And then set $$ v= \frac{du}{dq} $$ and another equation $$\frac{dv}{dp} = -u(p, q)/4 $$

Answer 2

You should introduce two more functions $v$ and $w$, so that your system of three equations does indeed contain three unknown functions. Then you specify that $v$ and $w$ are, respectively, the first time-derivative and the first space-derivative of $u$. So I would rather write something like this $$ \begin{cases} v=\partial_{t}u\\ w=\partial_{x}u\\ u=\partial_{t}v-\partial_{x}w. \end{cases} $$