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I have the system of equations \begin{align} &A \frac{\partial u_1}{\partial t} = 1 - u_1 B \frac{\partial u_2}{\partial y}\\ &\frac{\partial u_2}{\partial t} = \frac{\partial}{\partial y}\left[ e^{u_1} \frac{\partial u_2}{\partial y}\right] \enspace . \end{align}

The initial condition is $u_1(y, 0) = 0$, and $u_2(y<1, 0) = 0$, $u_2(1, 0) = 1$. And the boundary conditions are $u_1(1, t)=1$, $u_2(0, t)=0$, and $u_1(1, t)=0$.

Here, $A$ and $B$ are constants. The value of $A$ is 0.04 and the value of $B$ is 0.9. How can I solve these two PDEs simultaneously in COMSOL?

If it is not possible in COMSOL please suggest me another software.

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Actually that is possible to solve that set of equations in COMSOL.

You can simply select a 1D domain problem, add physics and in particular use the general ODE/DAE interface.

You can add multiple ODE/DAE equations and, for each of them, define all the conditions stated in your picture.

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