Exact solution to advection equation in spherical coordinates

[lostidentity]

I've been trying to find the exact solution to the advection equation in spherical coordinates given below

$$\frac{\partial{\phi}}{\partial{t}} + \frac{u}{r^2}\frac{\partial{}}{\partial{r}}r^2\phi = 0$$

Where the velocity u is a constant. First I tried to expand the second term using product rule and then apply the separation of variables, which gives me the following.

Separation of variables

$$\phi = R(r)T(t)$$

$$\frac{1}{T}\frac{dT}{dt} = -\frac{u}{R}\frac{dR}{dr} - \frac{2u}{r} = -\lambda^2$$

And the final answer is

$$\phi = Ae^{\frac{\lambda}{u}(r-ut)-2}$$

Wonder if this is correct? For Cartsesian coordinates I know the solution is very simple just $$\phi=\phi_0(x-ut)$$ where $$\phi_0$$ is the initial profile for phi.

 

[jackmell]

Wonder if this is correct?

Hi. You need to remember always if needed and no complications, you can back-substitute your presumptive solution into the DE. If it satisfies the DE, then it is a correct solution, barring any initial, boundary or other requirements. Do you mean the general solution? If so than I think it would be some arbitrary function of a function such as C(g(x,y)) as normally encountered in first order PDEs.