Change of variables in a differential equation with partial derivatives

Question

I have a doubt about the change of variables in differential equation.

Suppose to have:

$\frac{\partial f}{\partial t} + a \frac{\partial f}{\partial x} = b \frac{\partial^2 f}{\partial x^2}$

Now i want to use $y= x-at$ $P(x,t) \rightarrow \psi(y,t)$

Do I sobstitute the partial derivatives with the chain rule?

$\frac{\partial}{\partial x} = \frac{\partial}{\partial y} ( \frac{\partial y}{\partial x} + \frac{\partial y}{\partial t} ) $

and for

$ \frac{\partial^2}{\partial x^2} $ ?

THanks

Answer 1

As $x = y + at$, we have that $\frac{\partial f}{\partial x} = \frac{\partial f}{\partial y} \cdot \frac{\partial y}{\partial x} + \frac{\partial f}{\partial t} \cdot \frac{\partial t}{\partial x} = \frac{\partial f}{\partial y} + \frac{1}{a} \cdot \frac{\partial f}{\partial t} $ by the chain rule.

I don't understand where your equality $\frac{\partial}{\partial x} = ...$ comes from. The left hand side is an operation, the right hand side a function.

The case $\frac{\partial^2 }{\partial x^2}$ is just applying the chain rule twice.