I'm trying to solve the heat equation with convection:
$u_t - ku_{xx} + V u_{x} = 0$,
where $V,k$ are constants.
I know in easy way to solve this over the real number line is to transform to the heat equation, and the hint I was given was to use the transformation: $y = x - Vt$, but I'm not able to see how this works, no matter what I try.
Define $\tilde{u}$ by $u(t,x)=\tilde{u}(t,y(t,x))$ where $y(t,x)=x-Vt$. By chain rule, $u_{t}=\tilde{u}_{t}-V\tilde{u}_{y}$, $u_{x}=\tilde{u}_{y}$, and $u_{xx}=\tilde{u}_{yy}$. Substituting these quantities into the PDE yields $$ 0=u_{t}-ku_{xx}+Vu_{x}=\tilde{u}_{t}-k\tilde{u}_{yy}. $$ In other words, $\tilde{u}$ solves the heat equation without advection. Since you already know how to handle this case, you can solve for $\tilde{u}$.