Given functions $a(t,x,y) > 0, d(t,x) >0$ and constants $b,c > 0$, I would like to find the three functions $u(t,x),v(t,x) , w(t,x,y)$ that solve the system:
\begin{align} u_t - u_x &= -\left(\int_0^x a(t,x,y)\mathrm{d}y+b\right) u(t,x) + c v(t,x) + w_y(t,x,0) + d(t,x)\\ v_t - v_x &= b u(t,x) - c v(t,x) + b \int_0^x w(t,x,y) \mathrm{d} y\\ w_t - w_x - w_y &= a(t,x,y) u(t,x) - b w(t,x,y), \end{align}
and satisfy $u(t,0)=v(t,0)=w(t,x,0) = 0$.
Interpretation : the state of a job allocating platform
- $u,w,v$ - is the number of workers $u$noccupied, $w$orking or on a sick lea$v$e.
- $x$ - is the remaining duration of the working contract, by the end of which a worker leaves the system
- $y$ - is the remaining duration of a job assignment
- $a(t,x,y)$ - is the probability of being assigned a job of duration $y$ at date $t$
- $d(t,x)$ - is the number of new workers joining the platform at date $t$ for the duration $x$
- $b,c$ - is the chance of falling ill and that of recovering.
Once a job assignment is over ($y=0$), the busy worker joins the ranks of the unoccupied (the $w_y(t,x,0)$ term in the $u$-equation). Once the contract is over ($x=0$) the worker leaves the platform.
I kind of found the solution to a linear 1st order PDE system in the Andrei D. Polyanin's bestiary, but am absolutely at loss about what to do with the $w_y(t,x,0)$ and $\int w \mathrm{d}y $ terms.
Although not exactly a model of a physics phenomenon, this system looks like something that the physicists are most likely to be familiar with. I would be immensely grateful for the solution and/or all the pointers to the sources treating this kind of problems (lectures, keywords, etc.).
Update
Noticing that for every worker the date of retirement, $T_x$, and for every job the date of expiry, $T_y$, are constants, I can use the substitution $x = T_x - t$ and $y = T_y - t$ to transform equations 2 and 3 into ODEs, solve them and reverse the substitution to obtain $w,v$ in terms of $u$:
\begin{align*} w(t,x,y) & = \int_0^t \int_{s}^{y+s} \mathrm{e}^{-b s} a_y(t-s,x+s,\xi) u(t-s,x+s) \mathrm{d}\xi \mathrm{d} s \\ v(t,x) & = b \int_0^t \int_{s}^{x + s} \mathrm{e}^{-cs} u_x(t-s,\xi) + W_x(t-s,\xi) \mathrm{d} \xi \mathrm{d} s. \end{align*}
where
\begin{equation*} W(t,x) := \int_0^x w(t,x,y) \mathrm{d} y, \end{equation*}
is the total size of the busy group across all job durations.
Plugging these expressions into the first equation one gets the following PDE in $u$:
\begin{align} u_t =& u_x - \left(\int_0^x a\mathrm{d}y +b \right) u \\ & + cb \int_0^t \int_{s}^{x + s} \mathrm{e}^{-cs} u_x(t-s,\xi)\mathrm{d} \xi \mathrm{d} s \\ & + cb \int_0^t \int_{s}^{x + s} \mathrm{e}^{-cs} W_x(t-s,\xi) \mathrm{d} \xi \mathrm{d} s \\ & + \int_0^t \mathrm{e}^{-bs} a_y(t-s,x+s,0) u(t-s, x+s) \mathrm{d}\xi \\ & + d(t,x). \end{align}
I think this is a nice example of a system of equations solvable by the Laplace transform (I suppose that we are interested only in $x,y>0$ here): $$ \bar{u}(p,t)=\int_0^{+\infty}dx u(x,t)e^{-px}, etc. $$ This should reduce it to a system of linear ordinary differential equations in respect to time, which can be solved. Then one uses the Laplace transform table to try to obtain the solution in $(x,t)$ variables.
Update. Another rather general approach that works particularly well for the first order PDEs is the method if characteristics (see the section "Two-dimensional theory" here).