An age structured population with distribution $u(a,t)$ over age $a$ has a death rate increasing linearly with time and constant birth rate $b$, $u(\alpha,0) = u_{0}(\alpha)$.

Model is $u_{\alpha} + u_{t} = -\mu t u, t>0$ $u(a,0) = u_{0}(a), a \geq 0$ $u(0,t) = F(t) = b\int_{0}^{\infty} u(a,t) da$(which is the non-local boundary condition), $\mu,b$ are constants.

We need to find the solution $u(\alpha,t)$ for $\alpha > t > 0$ and for $0<a<t$.

Seeing the non-local boundary condition I thought of $F(t) = b\int_{0}^{t} u(a,t)da +b\int_{t}^{\infty} u(a,t) da$ (in case it may be useful)

I am clueless about how to obtain $u(a,t)$.

Thinking of method of characteristics, we have $\frac{d\alpha}{1} = \frac{dt}{1} = \frac{du}{-\mu t}$ and taking last equality we have $u = -\mu t^2 +c$


Since the equation is linear, let's define $$u(\alpha,t)=f(\alpha)g(t)$$then by substituting in the equation we obtain$$f'(\alpha)g(t)+f(\alpha)g'(t)=-\mu t f(\alpha)g(t)$$after dividing by $u(\alpha ,t)$ and solving the outcome ODEs we finally conclude that:$$u(\alpha,t)=Ae^{k(\alpha-t)}e^{-{1\over 2}\mu t^2}$$where $k$ and $A$ are complex constants. Since the equation is linear, then any linear combination of the above answers is also itself an answer, therefore$$u(\alpha,t)=\int_{0}^{\infty}A(\omega)e^{k(\omega)\cdot (\alpha-t)}e^{-{1\over 2}\mu t^2}d\omega$$Now if we decide to have a real answer with the assumption that $u(a,0)$ has a Fourier Transform, then only the values of $k(\omega)$ that are pure imaginary are needed. In that case, $$\Re\{A(-\omega)\}=\Re\{A(\omega)\}\\\Im\{A(-\omega)\}=-\Im\{A(\omega)\}$$Also without loss of generality, we assume $k(\omega)=i\omega$. These constraints assure that our final answer has the following real-valued form:$$u(\alpha,t)=e^{-{1\over 2}\mu t^2}\int_0^\infty C(\omega)\cos \omega (\alpha-t)+D(\omega)\sin \omega (\alpha-t)d\omega$$now if we expand $u(a,0)$ evenly around zero we obtain$$D(\omega)=0\\C(\omega)={2\over \pi}\int_0^\infty u(a,0)\cos \omega ada$$and the final answer becomes:$$u(\alpha,t)=e^{-{1\over 2}\mu t^2}\int_0^\infty C(\omega)\cos \omega (\alpha-t)d\omega$$


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