In "Basha, H. A., & El‐Habel, F. S. (1993). Analytical solution of the one‐dimensional time‐dependent transport equation. Water Resources Research, 29(9), 3209-3214."
My question might be naive, but I do not know how to solve it. I have the following PDE:
$$\frac{\partial C_1}{\partial T} = D \frac{\partial ^2 C_1}{\partial X^2} - U \frac{\partial C_1}{\partial X} + Q\; exp[\mu T]$$
A change from a fixed coordinate into a moving coordinate system is done:
$$\xi = X - UT$$
Now the equation becomes:
$$\frac{\partial C_1}{\partial T} = D \frac{\partial ^2 C_1}{\partial \xi^2} + Q\; exp[\mu T]$$
I do not know how to apply this change. Can someone explain this in more detail?
Let $C_1(T,X)=F(T,X-UT)$. Plugging this in and using the chain rule gives a PDE for $F(T,\xi)$, which they relabeled as $C_1(T,\xi)$.