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Two way coupling of PDEs

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Hi all, Essentially I have 2 coefficient form PDEs. The first equation solves for dependent variable u, while the second equation solves for dependent variable q. For the sake of simplicity, the equations can be assume as such: du/dt = q+1; gradient q = u. I will be solving this with a time-depdent study. How would one couple these two equations?

Big thanks


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