How do the initial conditions change when I achieve transformation on the heat equation

Question

Let us consider 1D heat equation: $u_t = u_{xx}, x∈[0,2]$ subjected to zero Neumann Boundary conditions: $u_x (0,t) = u_x (2,t) = 0$. The initial conditions: $u(x,0) = x$.

Now, let us introduce a new variable $\xi$, where $\xi = (1/2) x$ and $\xi∈[0,1]$. Considering the new variable and the chain rule of derivative the previous heat equation becomes: $$u_t = (1/4) u_{\xi\xi}$$ How will the initial conditions become after we achieved the transformation?

Answer 1

Well, if you call $\hat u(\xi,t)$ your new function, $\hat u(\xi,t)=u(x,t)$, you have $\hat u(\xi,0)=u(x,0)=x=2\xi$.