Solving PDE by changing coordinate system

Question

So we have a system of equations

$\partial_{t}u(x,t)=-\sigma\partial_{x}u(x,t)-\frac{D}{H_{u}}(u(x,t)-v(x,t))$,

$\partial_{t}v(x,t)=-\sigma\partial_{x}v(x,t)-\frac{D}{H_{v}}(u(x,t)-v(x,t))$.

Using the Method of Characteristics, I derived a new coordinate system $(r,s)$ where

$s:=\frac{x}{\sigma}$,

$r:=t-s=t-\frac{x}{\sigma}$.

How can I rewrite my system of equations so that they are under the new coordinate system?

Answer 1

You use the chain rule. With these variables you have

$$\frac{\partial f}{\partial x} = \frac{\partial f}{\partial s} \frac{\partial s}{\partial x} + \frac{\partial f}{\partial r} \frac{\partial r}{\partial x} \\ \frac{\partial f}{\partial t} = \frac{\partial f}{\partial s} \frac{\partial s}{\partial t} + \frac{\partial f}{\partial r} \frac{\partial r}{\partial t}$$

In your case you have the simple form:

$$\frac{\partial f}{\partial x} = \frac{\partial f}{\partial s} \frac{1}{\sigma} - \frac{\partial f}{\partial r} \frac{1}{\sigma} \\ \frac{\partial f}{\partial t} = \frac{\partial f}{\partial r}.$$

Now plug these into your equations and simplify.