## Solution to a PDE

Hi guys,

Suppose we have the following PDE for $\displaystyle \psi (x,y,z)$:

$\displaystyle u(z) \frac{\partial \psi}{\partial x} = \frac{\partial}{\partial y}(K_y \frac{\partial \psi}{\partial y}) + \frac{\partial}{\partial z}(K_z \frac{\partial \psi}{\partial z}) + S$

Where:
$\displaystyle S = Q \delta (x) \delta (y - y_s) \delta (z - z_s)$
$\displaystyle u(z) = a z^p$
$\displaystyle K_z (z) = b z^n$
$\displaystyle K_y = 0.5 u(z) \frac {d (\sigma _y ^2 )}{dx}$

I have found in the literature a solution for this PDE:

$\displaystyle \psi (x,y,z) = \frac {Q}{\sqrt{2 \pi} \sigma _y} \cdot exp[- \frac {(y - y_s)^2}{2 \sigma _y ^2}] \cdot \frac{(z z_s)^{(1-n)/2}}{b \alpha x} \cdot exp[- \frac {a(z^2 + z_s ^2)}{b \alpha ^2 x}] \cdot I_{- \nu} [\frac {2 a (z z_s)^{\alpha / 2}}{b \alpha ^2 x}]$

Where:
$\displaystyle \alpha = 2 + p - n$
$\displaystyle \nu = (1-n)/ \alpha$

and $\displaystyle I_{- \nu}$ is the modified Bessel function of order $\displaystyle - \nu$.

I have a reason to doubt this soultion, which comes from physical limitations. It seems to me that the arguments of the last exponent and the Bessel function has different units, providing that $\displaystyle \alpha \not= 1$. can anybody verify if this solution works for $\displaystyle n = 0$, or if it works at all?