The derivation goes as follows; assume that we have a renewal process with age distribution u and hazard rate f. After a short time dt you will have that units with age s+dt at time t+dt are exactly the units that at time t had age s and that did not produce an event. Locally on time, the probability of not emitting an event is
P(\mbox{no event in (t,t+dt}) \approx e^{-\phi(s(t)) dt}
so, plugging this into the age distribution we obtain
u(t+dt,s+dt) \approx e^{-\phi(s) dt} u(t,s)
Fine. We now subtract u(t,s+dt) from both sides of the equation and expand the exponential function up to first order. This leads to
u(t+dt,s+dt)-u(t,s+dt) \approx (1-\phi(s) dt) u(t,s) - u(t,s+dt)
Rearrangin and dividing by dt leads to
\frac{u(t+dt,s+dt)-u(t,s+dt)}{dt} \approx \frac{u(t,s)- u(t,s+dt)}{dt}- \phi(s) u(t,s)
Of course this means
\partial_t u(t,s) \approx -\partial_s u(t,s) - \phi(s) u(t,s)
complemented with some initial and boundary condition.
