Ecuaciones en derivadas parciales con los no-locales

Question

No estoy seguro si he usado la terminología correcta aquí ("no-local"). Creo que la falta de conocimiento de la terminología correcta es la razón por la que no he sido capaz de encontrar ninguna información sobre mi consulta.

Estoy interesado, en particular, los sistemas de semi-lineal de primer orden funcional (?) Ecuaciones en derivadas parciales. Un ejemplo en donde puedo buscar soluciones definidas en $\mathbb{R}_+ \times \mathbb{R}_+$ es:

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

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

al $z > 0$; y:

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

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

al $z = 0$. Las condiciones de contorno son $F_1(z,0) = \sigma(z)$ $F_2(z,0) = 0$ para todos los $z$.

La cosa que aparece para hacer estos especiales es la presencia de la "no-local" términos de $F_1(0,t)$. Supongo que podría intentar resolver el sistema usando los métodos de las diferencias finitas, pero me preguntaba, ¿hay algo mejor que se puede hacer aquí, por ejemplo, puede que el método de las características todavía se puede utilizar? Yo no soy especialmente familiarizado con ecuaciones en derivadas parciales por lo que cualquier ayuda sería recibido con gratitud --- do sistemas como este incluso tiene un nombre, hay referencias que debo ver?

Gracias!

Answer 1

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

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

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

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

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

Para $(2)$, esta es una ODA.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\therefore\begin{cases}F_1(z,t)=\lambda\sigma(z)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}\int_0^te^{\lambda u+\beta\int_0^uf(\tau)~d\tau}~du+\sigma(z)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}\\F_2(z,t)=\sigma(z)-\lambda\sigma(z)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}\int_0^te^{\lambda u+\beta\int_0^uf(\tau)~d\tau}~du-\sigma(z)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}\end{cases}~\text{when}~z=0$ donde $f(t)$ es la solución de $\lambda\sigma(0)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}\int_0^te^{\lambda u+\beta\int_0^uf(\tau)~d\tau}~du+\sigma(0)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}=f(t)$

$\begin{cases}F_1(0,t)=\lambda\sigma(0)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}\int_0^te^{\lambda u+\beta\int_0^uf(\tau)~d\tau}~du+\sigma(0)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}\\F_2(0,t)=\sigma(0)-\lambda\sigma(0)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}\int_0^te^{\lambda u+\beta\int_0^uf(\tau)~d\tau}~du-\sigma(0)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}\end{cases}$ donde $f(t)$ es la solución de $\lambda\sigma(0)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}\int_0^te^{\lambda u+\beta\int_0^uf(\tau)~d\tau}~du+\sigma(0)e^{-\lambda t-\beta\int_0^tf(\tau)~d\tau}=f(t)$