Method of characteristic to solve Sharpe-Lokta model

Question

The conservation law for the population is,

$$ \underbrace{\frac{\partial}{\partial t} x(t,a) + \frac{\partial}{\partial a} x(t,a)}_{\text{directional derivative}} = -\mu(a) x(t,a) dt\tag1 $$ where $x(t,a)$ is the density of individuals of age $a$ and time $t$, and $\mu$ is the death function. Now with initial condition, $$x(0,a)=\phi(a)\tag2$$the population at time $t=0$ has a given age distribution $\phi(a)$. The boundary condition come from the birth rate function $\gamma(a)$, $$x(t,0)=\int_0^\infty \gamma(a)x(t,a)da\tag3$$ Now, they directly said the solution as,

$$x(t,a)=\begin{cases} \phi(a-t)\exp\left(-\int_{a-t}^a \mu(s)ds\right), & a>t \\ x(t-a,0)\exp\left(-\int_{0}^a \mu(s)ds\right), & a<t \end{cases} $$

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I didn't understand how they solve the PDE? I try to use characteristic method, $$\frac{dt}{ds}=1,\quad\frac{da}{ds}=1,\quad\frac{dx}{ds}=-\mu(a)x$$ And got, $$dt=da\implies t-a=C_1,\quad x=C_2\exp\left( -\int \mu ds \right)$$ And using the initial condition I could reach, $$x=\phi(a-t)\exp\left(- \int \mu ds \right)$$ I don't understand how they get the solution and the piecewise thing. It will be a great help if anyone explain it. Thanks in advance.

Why $x(t,0)=\int_0^\infty \gamma(a)x(t,a)da$ is called boundary condition? Is it mean newborn at time $t$?

Answer 1

The reason behind the solution's piecewise nature is that for $a > t$, information is being carried from the initial condition (at $t=0$), while for $a < t$, information is being carried from the boundary condition (at $a=0$). You might observe this by sketching the characteristic curves ($a = t + c.$) on the positive quadrant of the $(t,a)$ plane. Intuitively, the number of 10 year olds depends on the the birth rate 10 years ago rather than the number of -10 year olds 20 years ago.

Here is a derivation. Let $x(a,t) = X(a,s)$ where $s = a-t$. Here $s$ is constant on the characteristics. Then $$ \frac{\partial X}{\partial a} = -\mu(a) X \qquad \Rightarrow \qquad X = C(s)\exp\left(\int_{c_1(s)}^a \mu(\alpha)\,\mathrm d\alpha\right) $$ Here I have expressed the arbitrary constant both in a prefactor and a lower limit in the definite integral (note that this corresponds to a single arbitrary constant, however). We need to use either the initial or boundary condition to work out the arbitrary constant, as appropriate. For $a > t$, we need $$ X(a, s=a) = \phi(a) $$ so applying this condition: $$ X = \phi(s)\exp\left(\int_s^a \mu(\alpha)\,\mathrm d\alpha\right). $$ When $a < t$, the boundary condition we require is $$ X(a=0, s=-t) = x(0,t) = \psi(t) $$ (here I've given this birth function a name). Applying this boundary condition results in $$ X = \psi(-s)\exp\left(\int_0^a \mu(\alpha)\,\mathrm d\alpha\right). $$ Substitute in $s=a-t$ for your solution.