| 48 | | [[latex(d\tau_\nu = \chi_\nu(s) ds )]] which gives [[latex(\tau_\nu(s) = \int\limits_0^s \chi_\nu(s') ds')]] |
| 49 | | |
| 50 | | [[latex(s(\tau_\nu) = \int\limits_0^\tau_\nu \frac{1}{\chi_\nu} d\tau'_\nu )]] |
| 51 | | |
| 52 | | There are a few important dimensionless numbers to consider: |
| | 48 | [[latex(d\tau_\nu = \chi_\nu(s) ds )]] |
| | 49 | which gives |
| | 50 | |
| | 51 | || [[latex(\tau_\nu(s) = \int\limits_0^s \chi_\nu(s') ds')]] || |
| | 52 | || [[latex(s(\tau_\nu) = \int\limits_0^\tau_\nu \frac{1}{\chi_\nu} d\tau'_\nu )]] || |
| | 53 | |
| | 54 | we can write the transport equation in the simplest form |
| | 55 | |
| | 56 | [[latex(\frac{dI_\nu}{d\tau_\nu} = S_\nu(\tau_\nu) - I_\nu(\tau_\nu))]] |
| | 57 | |
| | 58 | although the RHS is now more difficult to evaluate as |
| | 59 | |
| | 60 | [[latex( f \left ( \tau_\nu \right ) = f \left ( s \left ( \tau_nu \right ) \right ) = f \left ( \mathbf{x} \left (s \left ( \tau_nu \right ) \right ), t \left ( s \left ( \tau_nu \right ) \right ) \right ) )]] |
| | 61 | |
| | 62 | There are also a few important dimensionless numbers to consider: |
