I'd like to know if there is some litterature about the cylindrical Mason-Weaver equation.

The basic Mason-Weaver equation is treated in [Wikipedia](https://en.wikipedia.org/wiki/Mason%E2%80%93Weaver_equation#Solution_of_the_Mason%E2%80%93Weaver_equation) :

$$\partial_t f(t,x)=\partial_x f(t,x)+\partial_{xx}^2 f(t,x)=\nabla\mathbf{v}  $$
 
and the boundary conditions at $x=x_b$ : $$f(t,x)+\partial_{x} f(t,x)=0  $$

It is treated with the separation of variables : $f(x,t)=F(t)G(x)$

But I'm wondering how to proceed with the cylindrical symmetry case :

$$\partial_t f(t,z,r)=\partial_z f(t,z,r)+\partial_{zz}^2 f(t,z,r) +\frac{1}{r}\partial_r(r\partial_r f(t,z,r)) =\nabla\mathbf{v}$$

and with boundary conditions and the boundary $s(x,y)=0$ :

$$\cos(\theta)(f(t,z,r)+\partial_{z} f(t,z,r) )-\sin(\theta)\partial_r f(t,z,r)=0$$

where the vector $\mathbf{n}=(\sin(\theta),-\cos(\theta))$ is perpendicular to the boundary so that $\mathbf{v}.\mathbf{n}=0$

One can also attack the problem with the separation of variables : $f(t,z,r)=F(t)G(r)H(z)$ and solve each equation. 

We then obtain : 

$$f(t,z,r)=\sum C_{\beta,k,\lambda} e^{-\beta t} e^{-\frac{1}{2}z}(a\cos(\omega_k z)+b\sin(\omega_k z))\operatorname{J}_0(r/\lambda)$$

where $\omega_k^2=1+4(\kappa-\beta)$ and $\lambda=1/\sqrt{\kappa}$ and $\operatorname{J}_0$ the cylindrical Bessel function (Bessel Function of the First Kind) - whose derivative at $r=0$ is zero.

But I'm then stuck how to find the good constants in order to satisfy the boundary conditions.

Would you have an advice to tackle this problem or recommend articles if this case has already been treated in the litterature please ?