Consider the following type of functional differential equations: $$\begin{align} \frac{du}{dt} &= F(x,t,u(x,t),u_{x,t}), & (x,t) &\in [a,b] \times [0,T] \end{align}$$ where $u(x,t)$ is the unknown function, which satisfies the initial-boundary value condition: $$\begin{align} u(x,t) &= \psi(x,t), &(x,t) &\in \Omega = \Omega_1 \cup \Omega_0 \cup \Omega_2 \end{align}$$ $$\begin{align} \Omega_1 &= [a - \eta, a] \times [-\tau,T], &\Omega_0 &= [a, b] \times [-\tau,0], &\Omega_2 &= [b, b + \eta] \times [-\tau,T] \end{align}$$ $x$ is a spatial vector, $a, b, \eta$ and $T, \tau$ are constants, $u_{x,t}$ is the prehistory of $u(x,t)$: $$u_{x,t} = \{u(x+\xi,t+s) | -\eta \le \xi \le \eta, -\tau \le s \le 0 \}$$
It is also assumed that the functional $F(\cdot)$ and the initial-boudary function $\psi(x,t)$ are set appropriately, so that a solution of the equation exists and is unique.
Help me to collect an exemplary problems for the specified type of FDE.
For instance, it is known that the following PDE with delay
$$\frac{\partial u}{\partial t}(x,t)+a\frac{\partial u}{\partial x}(x,t)=f(x,t,u(x,t),u_t(x,\cdot))$$
$$u_t(x,\cdot)=\{u(x,t+s) | -\tau \le s \le 0 \}$$
can be transfromed (through the replacement of variables) to the FDE
$$\frac{du}{dt} = F(x,t,u(x,t),u_{x,t})$$
where $t \ge 0$, $x \in [-t,t]$, $u_{x,t}=\{u(x-as,t+as) | -\tau \le s \le 0 \}$
and the corresponding numerical method will be simpler and more effective.
Another interesting example equation of the specified type is $$ \frac{du}{dt} = \alpha(t) u(x,t) + \beta(t) u(x, t-\tau) + \gamma \int_{-\eta}^{\eta} u(x+\xi,t)d\xi + R(x,t) $$ where $\alpha(t), \beta(t), \gamma, R(x,t)$ are problem specific.