PDE support developer notes

Depth distributions

The inputs to the system have to be described by a differential across depth. The following derivation provides an exponential function $f(z;b) = a e^{bz}$, such that $\int_{z_m}^0 f(z) dz = 1$. Such a function can be used to distribute total inputs across depth.

Derivation: for positive $x = -z$:

\[\begin{aligned} f(x) &= a e^{-bx} \\ F(x) &= \frac{a}{-b} e^{-bx} \\ \left[ F(x) \right]_0^{x_m} &= \frac{a}{-b} \left( e^{-b x_m} -1 \right) = 1 \\ a &= \frac{b}{1 - e^{-bx_m}} \\ f(x) &= \frac{b}{1 - e^{-bx_m}} e^{-bx} \\ F(x) &= \frac{1}{e^{-bx_m} - 1} \left( e^{-bx} -1 \right) \end{aligned}\]