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Numerically solving a system of PDEs where one function is composed with the other
I'm trying to solve the following system using NDSolve:
$$
\begin{align}
u_t(x,t) &= u_{xx}(x,t) - v(x,t) \\
v_t(x,t) &= u(v(x,t),t)
\end{align}
$$
with
$$
u_x(-1,t) = 0, \quad u_x(1,t) = 0, \quad …