Chain Rule for Partial Derivatives in convection-diffusion

Question

Suppose I have the following transformation: $$P(x,t)=G(x-vt,t)$$

How can I apply this to: \begin{equation} \frac{\partial P(x,t)}{\partial t}=-v\frac{\partial P(x,t)}{\partial x}+D \frac{\partial^2 P(x,t)}{\partial x^2} \end{equation}

Answer 1

This is an advection-diffusion equation. Let us write the change of variable as follows: $$ P(x,t) =G(\xi,t),\qquad \xi = x-vt. $$ Then, use the chain rule \begin{aligned} P_t &= G_\xi\xi_t + G_t = -v G_\xi + G_t \\ P_x &= G_\xi\xi_x = G_\xi \\ P_{xx} &= G_{\xi\xi}\xi_x = G_{\xi\xi} . \end{aligned} Finally, the heat equation $G_t = DG_{\xi\xi}$ is recovered.