# How could I compare a SYSTEM of linear second-order partial differential equations with two different functions within them to the heat equation? Please also provide a reference that I can cite in my paper.

## In particular, for a paper, I am looking to compare this equation $i {e}^{4 \omega t} \frac{\partial \Phi}{\partial t} + \frac{{\partial}^{2} \Phi}{\partial {y}^{2}} = 0$ to the forward heat equation in one dimension, $\frac{\partial u}{\partial t} - \frac{{\partial}^{2} u}{\partial {x}^{2}} = 0$, and the backward heat equation in one dimension, $\frac{\partial u}{\partial t} + \frac{{\partial}^{2} u}{\partial {x}^{2}} = 0$, where $\omega$ is a constant and $i$ is the familiar imaginary unit. My problem is, anytime I multiply by $i$, it can look like either the forward or backward heat equation, and I can't just have it look like either one arbitrarily... I tried rewriting $\Phi$ in terms of parts with real and imaginary coefficients: $\setminus \Phi \left(y , t\right) = N \setminus \textrm{\exp} \left(\setminus \frac{i \setminus \epsilon}{4 \setminus \omega} {e}^{- 4 \setminus \omega t}\right) \sin \left(\setminus \sqrt{\setminus \epsilon} y\right)$ $= N \left[\cos \left(\setminus \frac{\setminus \epsilon}{4 \setminus \omega} {e}^{- 4 \setminus \omega t}\right) + i \sin \left(\setminus \frac{\setminus \epsilon}{4 \setminus \omega} {e}^{- 4 \setminus \omega t}\right)\right] \sin \left(\setminus \sqrt{\setminus \epsilon} y\right)$ $= \stackrel{{\Phi}_{r e}}{\overbrace{N \cos \left(\setminus \frac{\setminus \epsilon}{4 \setminus \omega} {e}^{- 4 \setminus \omega t}\right) \sin \left(\setminus \sqrt{\setminus \epsilon} y\right)}} + i \stackrel{{\Phi}_{i m}}{\overbrace{N \sin \left(\setminus \frac{\setminus \epsilon}{4 \setminus \omega} {e}^{- 4 \setminus \omega t}\right) \sin \left(\setminus \sqrt{\setminus \epsilon} y\right)}}$ where $\epsilon$ and $N$ are constants too. I could then write this as: $= \setminus {\Phi}_{r e} + i \setminus {\Phi}_{i m}$ However, when I plug it back into the PDE, I get a system of PDEs with mixed functions... ${e}^{4 \setminus \omega t} \setminus \frac{\setminus \partial \setminus {\Phi}_{i m}}{\setminus \partial t} - \setminus \frac{\setminus {\partial}^{2} \setminus {\Phi}_{r e}}{\setminus \partial {y}^{2}} = 0$ ${e}^{4 \setminus \omega t} \setminus \frac{\setminus \partial \setminus {\Phi}_{r e}}{\setminus \partial t} + \setminus \frac{\setminus {\partial}^{2} \setminus {\Phi}_{i m}}{\setminus \partial {y}^{2}} = 0$ How can I still compare to the forward and/or backward heat equation? Please help soon, this is due by Friday April 28 for a 15-page paper. I am almost done, except for this. Classifying these wasn't a problem (they are both parabolic). It's the comparison to the heat equation that's giving me trouble.

$\text{See explanation}$
$\text{Maybe my answer is not completely to the point, but i know}$
$\text{about the "color(red)("Hopf-Cole transformation").}$
$\text{The Hopf-Cole transformation is a transformation, which maps}$ $\text{the solution of the "color(red)("Burgers equation")" to the "color(blue)("heat equation").}$
$\text{Maybe you can find inspiration there.}$