# Splitting PDE into system of PDEs

Hello:

I am wondering if there is a general way of splitting the following PDE into two separate equations. I would like to re-write the second-order spatial derivatives on the LHS as first-order derivatives.

$$$\frac{{\partial p^2 }}{{\partial x^2 }} + \frac{{\partial p^2 }}{{\partial y^2 }} = A\frac{{\partial ^2 p}}{{\partial t^2 }} + B\frac{{\partial p}}{{\partial t}}$$$

So once the above equation is split, there should be two equations involving only first-order spatial derivatives ($$\partial p/\partial x$$ and $$\partial p/\partial y$$).

I think that this equation can be split using the following rules:

(1) The LHS of the wave equation is split by introducing another variable and removing a del operator.

$$$\frac{{\partial \psi }}{{\partial t}} = \nabla p$$$

(2) The RHS of the wave equation is split by removing a derivative:

$$$\nabla \psi = A\frac{{\partial p}}{{\partial t}} + Bp$$$