
Java How to Program
210 solutions
Let \(p\) denote the population density, and let it depend on time \(t\) and age \(r\). The number of people of ages between \(r\) and \(r+d r\) at a certain time \(t\) is given by \(p(t, r) d r\). Define the mortality rate \(\mu(t, r)\) in the following way: \(\mu(t, r) d r d t\) is the fraction of people in the age class \([r, r+d r]\) who die in the time interval \([t, t+d t]\). Based on the infinitesimal equality $$ p(t+d t, r+d t) d r-p(t, r) d r=-\mu p d r d t . $$ Show that \(p\) satisfies the following partial differential equation $$ \frac{\partial p}{\partial r}+\frac{\partial p}{\partial t}=-\mu p $$ Let the initial age distribution be given as $$ p(0, r)=p_{0}(r), \quad 0 \leq r \leq 1 $$ and the birth rate function as the boundary condition $$ p(t, 0)=u(t), \quad t \geq 0 . $$ Here it assumed that the age \(r\) is scaled in such a way that nobody reaches an age \(r>1\). One can consider \(u(t)\) as the input to the system and as output \(y(t)\) for instance the number of people in the working age, say between the ages \(a\) and \(b\), \(0