Hi guys,

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

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


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

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

and I_{- \nu} is the modified Bessel function of order - \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 \alpha \not= 1. can anybody verify if this solution works for n = 0, or if it works at all?