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Not sure if I've used the correct terminology here (`non-local'). I think the lack of knowing the correct terminology is why I haven't been able to find any information about my query thus far.

I'm interested in particular systems of semi-linear first-order (functional?) PDEs. One example where I seek solutions defined on $\mathbb{R}_+ \times \mathbb{R}_+$ is:

$\frac{\partial F_1(z,t)}{\partial t} + \gamma \frac{\partial F_1(z,t)}{\partial z} = \lambda F_2(z,t) - \beta F_1(z,t) F_1(0,t)$

$\frac{\partial F_2(z,t)}{\partial t} = \beta F_1(z,t) F_1(0,t) - \lambda F_2(z,t)$

when $z > 0$; and:

$\frac{\partial F_1(z,t)}{\partial t} = \lambda F_2(z,t) - \beta F_1(z,t) F_1(0,t)$

$\frac{\partial F_2(z,t)}{\partial t} = \beta F_1(z,t) F_1(0,t) - \lambda F_2(z,t)$

when $z = 0$. Boundary conditions are $F_1(z,0) = \sigma(z)$ and $F_2(z,0) = 0$ for all $z$.

The thing which appears to make these special is the presence of the `non-local' terms $F_1(0,t)$. I guess I could try and solve the system using finite difference methods, but I was wondering, is there anything better that can be done here, e.g. can the method of characteristics still be used? I'm not particularly familiar with PDEs so any help would be gratefully received --- do systems like this even have a name, are there any references I should see?

Thanks!

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1 Answer

Suppose $\gamma,\lambda,\beta\neq0$ :

$\begin{cases}\begin{cases}\dfrac{\partial F_1(z,t)}{\partial t}+\gamma\dfrac{\partial F_1(z,t)}{\partial z}=\lambda F_2(z,t)-\beta F_1(z,t)F_1(0,t)\\\dfrac{\partial F_2(z,t)}{\partial t}=\beta F_1(z,t)F_1(0,t)-\lambda F_2(z,t)\end{cases}&\text{when}~z>0\\\begin{cases}\dfrac{\partial F_1(z,t)}{\partial t}=\lambda F_2(z,t)-\beta F_1(z,t)F_1(0,t)\\\dfrac{\partial F_2(z,t)}{\partial t}=\beta F_1(z,t)F_1(0,t)-\lambda F_2(z,t)\end{cases}&\text{when}~z=0\end{cases}$

$\therefore\begin{cases}\dfrac{\partial F_2(z,t)}{\partial t}=-\dfrac{\partial F_1(z,t)}{\partial t}-\gamma\dfrac{\partial F_1(z,t)}{\partial z}&\text{when}~z>0\\\dfrac{\partial F_2(z,t)}{\partial t}=-\dfrac{\partial F_1(z,t)}{\partial t}&\text{when}~z=0\end{cases}$

But $\begin{cases}\dfrac{\partial^2F_1(z,t)}{\partial t^2}+\gamma\dfrac{\partial^2F_1(z,t)}{\partial t\partial z}=\lambda\dfrac{\partial F_2(z,t)}{\partial t}-\beta F_1(0,t)\dfrac{\partial F_1(z,t)}{\partial t}-\beta\dfrac{\partial F_1(0,t)}{\partial t}F_1(z,t)&\text{when}~z>0\\\dfrac{\partial^2F_1(z,t)}{\partial t^2}=\lambda\dfrac{\partial F_2(z,t)}{\partial t}-\beta F_1(0,t)\dfrac{\partial F_1(z,t)}{\partial t}-\beta\dfrac{\partial F_1(0,t)}{\partial t}F_1(z,t)&\text{when}~z=0\end{cases}$

$\therefore\begin{cases}\dfrac{\partial^2F_1(z,t)}{\partial t^2}+\gamma\dfrac{\partial^2F_1(z,t)}{\partial t\partial z}=-\lambda\dfrac{\partial F_1(z,t)}{\partial t}-\gamma\lambda\dfrac{\partial F_1(z,t)}{\partial z}-\beta F_1(0,t)\dfrac{\partial F_1(z,t)}{\partial t}-\beta\dfrac{\partial F_1(0,t)}{\partial t}F_1(z,t)&\text{when}~z>0~......(1)\\\dfrac{\partial^2F_1(z,t)}{\partial t^2}=-\lambda\dfrac{\partial F_1(z,t)}{\partial t}-\beta F_1(0,t)\dfrac{\partial F_1(z,t)}{\partial t}-\beta\dfrac{\partial F_1(0,t)}{\partial t}F_1(z,t)&\text{when}~z=0~......(2)\end{cases}$

For $(2)$, this is an ODE.

$\dfrac{\partial^2F_1(z,t)}{\partial t^2}=-\lambda\dfrac{\partial F_1(z,t)}{\partial t}-\beta F_1(0,t)\dfrac{\partial F_1(z,t)}{\partial t}-\beta\dfrac{\partial F_1(0,t)}{\partial t}F_1(z,t)$

$\dfrac{\partial^2F_1(z,t)}{\partial t^2}+\lambda\dfrac{\partial F_1(z,t)}{\partial t}=-\beta\dfrac{\partial(F_1(0,t)F_1(z,t))}{\partial t}$

$\dfrac{\partial F_1(z,t)}{\partial t}+\lambda F_1(z,t)=-\beta F_1(0,t)F_1(z,t)+C_1(z)$

$\dfrac{\partial F_1(z,t)}{\partial t}+(\lambda+\beta F_1(0,t))F_1(z,t)=C_1(z)$

Let $F_1(0,t)=f(t)$ ,

Then $\dfrac{\partial F_1(z,t)}{\partial t}+(\lambda+\beta f(t))F_1(z,t)=C_1(z)$

I.F. $=e^{\lambda t+\beta\int_0^tf(\tau)~d\tau}$

$\therefore\dfrac{\partial}{\partial t}(e^{\lambda t+\beta\int_0^tf(\tau)~d\tau}F_1(z,t))=C_1(z)e^{\lambda t+\beta\int_0^tf(\tau)~d\tau}$

$e^{\lambda t+\beta\int_0^tf(\tau)~d\tau}F_1(z,t)=C_1(z)\int_0^te^{\lambda u+\beta\int_0^uf(\tau)~d\tau}~du+C_2(z)$

$F_1(z,t)=C_1(z)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}\int_0^te^{\lambda u+\beta\int_0^uf(\tau)~d\tau}~du+C_2(z)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}$

$\therefore C_1(z)-C_1(z)(\lambda+\beta f(t))e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}\int_0^te^{\lambda u+\beta\int_0^uf(\tau)~d\tau}~du-C_2(z)(\lambda+\beta f(t))e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}=\lambda F_2(z,t)-\beta C_1(z)f(t)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}\int_0^te^{\lambda u+\beta\int_0^uf(\tau)~d\tau}~du-\beta C_2(z)f(t)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}$

$F_2(z,t)=\dfrac{C_1(z)}{\lambda}-C_1(z)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}\int_0^te^{\lambda u+\beta\int_0^uf(\tau)~d\tau}~du-C_2(z)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}$

Hence $\begin{cases}F_1(z,t)=C_1(z)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}\int_0^te^{\lambda u+\beta\int_0^uf(\tau)~d\tau}~du+C_2(z)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}\\F_2(z,t)=\dfrac{C_1(z)}{\lambda}-C_1(z)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}\int_0^te^{\lambda u+\beta\int_0^uf(\tau)~d\tau}~du-C_2(z)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}\end{cases}~\text{when}~z=0$

$F_1(z,0)=\sigma(z)$ and $F_2(z,0)=0$ :

$\begin{cases}C_2(z)=\sigma(z)\\\dfrac{C_1(z)}{\lambda}-C_2(z)=0\end{cases}~\text{when}~z=0$

$\begin{cases}C_1(z)=\lambda\sigma(z)\\C_2(z)=\sigma(z)\end{cases}~\text{when}~z=0$

$\therefore\begin{cases}F_1(z,t)=\lambda\sigma(z)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}\int_0^te^{\lambda u+\beta\int_0^uf(\tau)~d\tau}~du+\sigma(z)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}\\F_2(z,t)=\sigma(z)-\lambda\sigma(z)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}\int_0^te^{\lambda u+\beta\int_0^uf(\tau)~d\tau}~du-\sigma(z)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}\end{cases}~\text{when}~z=0$ , where $f(t)$ is the solution of $\lambda\sigma(0)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}\int_0^te^{\lambda u+\beta\int_0^uf(\tau)~d\tau}~du+\sigma(0)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}=f(t)$

$\begin{cases}F_1(0,t)=\lambda\sigma(0)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}\int_0^te^{\lambda u+\beta\int_0^uf(\tau)~d\tau}~du+\sigma(0)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}\\F_2(0,t)=\sigma(0)-\lambda\sigma(0)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}\int_0^te^{\lambda u+\beta\int_0^uf(\tau)~d\tau}~du-\sigma(0)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}\end{cases}$ , where $f(t)$ is the solution of $\lambda\sigma(0)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}\int_0^te^{\lambda u+\beta\int_0^uf(\tau)~d\tau}~du+\sigma(0)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}=f(t)$

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