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I am trying to solve the following coupled system:

$$ \left\{\begin{aligned} U_t + \lambda_1 U_x &= 0 \\ V_t - \lambda_2 V_x &= c(x) U(x,t) \\ U(0,t) + k_{0} V(0,t) &= 0 \\ V(L,t) - k U(L,t) &= 0 \end{aligned}\right. $$

where $\lambda_{1} = 70, \lambda_{2} = 20, c(x) = - 24e^{-12x/35}, k_{0} = 2/7$ and $k=e^{-12/35}$.

I would like to discretize these coupled PDEs and I tried writing out using finite difference methods, but how do I deal with $U(x,t)$ on the right side of the second-line equation of the system?

Matthew Cassell
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Ziwei Xu
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    The PDE in $U$ is decoupled from $V$ so solve that equation first and then solve the problem in $V$ which, as you know the exact form of $U$, is just a first order, inhomogeneous, linear PDE. – Matthew Cassell Jul 20 '23 at 13:15
  • @MatthewCassell Oh, it explains the matter. Thank you very much! But if the right side of the first equation is V(x,t), then how do I solve it? – Ziwei Xu Jul 20 '23 at 13:39
  • You solve it the same way both times, using the method of characteristics. However, when there is a dependence on $V$ on the right hand side of the equation in $U$, you would usually try to diagonalise the system in the same manner as here though I believe that this can be quite difficult as the right hand side of the system will be dependent on $(V, U)$ and not derivatives of. Give it a try, see how it goes. – Matthew Cassell Jul 20 '23 at 23:51

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