# Marshak Wave.

1. Apr 17, 2010

### MathematicalPhysicist

I have this PD equation:
$$T_t=\nabla ^2 T^4$$
where T=T(r,t) the above laplacian is spherical symmetrical (i.e only the spherical radial coordinate of the operator should be taken into account).
and $$Q_0=\int_{0}^{\infty}T(r,t=0)dr$$.

So I tried solving it by seperation of variables but I get a tough ODE of the radial part.

Here's what I got
$$T(r,t)=F(r)G(t) \newline \frac{\frac{dG}{dt}}{G^4}=\frac{\nabla ^2 F^4}{F}=\lambda$$
Now after some manipulations I get for the radial equation the next equation:
$$F''(r)+3(F'(r))^2/F(r)+2F'(r)/r-\lambda/(2F(r))^2=0$$

And this is where I am stuck, any suggestion as to how to untie this equation, is even possible?