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I am trying to solve the following system of coupled PDEs but I am getting an error.

eqns = {D[Z[t, x], t] == W[t, x],
        4*D[U[t, x], {x, 2}] + 4*D[Z[t, x], {x, 2}]*D[W[t, x], x] + 
          4*D[Z[t, x], x]*D[W[t, x], {x, 2}] == 0, 
        -D[W[t, x], {x, 4}]/3 + 4*D[W[t, x], x]*D[Z[t, x], x]*D[Z[t, x], {x, 2}] + 
          4*D[Z[t, x], {x, 2}]*D[U[t, x], x] - 1 == 0};

bc = {U[t, -1] == 0, U[t, 1] == 0, W[t, -1] == 0, W[t, 1] == 0, 
       Derivative[0, 1][W][t, -1] == 0, Derivative[0, 1][W][t, 1] == 0, 
       Z[t, -1] == 0, 
       Z[t, 1] == 0, U[0, x] == 0, Z[0, x] == 0, W[0, x] == -(1 - x^2)^2/8}

NDSolve[{eqns, bc}, {U, W, Z}, {x, -1, 1}, {t, 0, 1}]

In the paper where these equations are taken from, it is mentioned that, to expedite the computation and to preserve the form of the equations, they added the term Re*dW/dt to the left side of third equation (where Re is the Reynold's number) and the term e^2*Re*dU/dt (where e<<1) to the left side of the second equation.

I tried to run the code above but I got the error

"Repeated convergence test failure at t == 0.000025`; unable to continue".

Could anybody help me please? Thank you very much.

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  • $\begingroup$ A numerical solution is more than enough. I tried indeed with NDSolve but nothing... $\endgroup$ – acalore88 Jun 18 '18 at 13:27
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    $\begingroup$ Your boundary conditions should also be expressed as equations using ==, rather than assignments using =. Make sure to restart your kernel to clean out any spurious assigned values before you evaluate the corrected code. If I do that with your code, I do get a result. $\endgroup$ – MarcoB Jun 18 '18 at 13:45
  • $\begingroup$ thank you very much. my apologizes for the noob mistake. $\endgroup$ – acalore88 Jun 19 '18 at 7:29
  • $\begingroup$ I updated my question as the situation has changed. $\endgroup$ – acalore88 Jun 19 '18 at 8:59
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    $\begingroup$ Add Method -> {"IndexReduction" -> Automatic} to NDSolve should help.And Then see solution: Plot3D[#[t, x] /.% , {x, -1, 1}, {t, 0, 1}] & /@ {U, W, Z} $\endgroup$ – Mariusz Iwaniuk Jun 19 '18 at 9:30

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