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Oct 23rd, 2014
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  1. eqns = {D[c[x, t], t] == (2 c[x, t] + 1)*D[v[x, t], x] -
  2. D[c[x, t], x]*v[x, t] - D[v[x, t], x] c[x, t],
  3. 2 D[v[x, t], t] == (c[x, t] + 1)*(8 c[x, t]*D[c[c, t], x] + 4 D[c[x, t], x])*
  4. D[v[x, t], x] + (c[x, t] + 1)*(4 c[x, t]^2 + 4 c[x, t] + 1)*D[v[x, t], x, x] +
  5. 4 (c[x, t] + 1)*D[c[x, t], x]*mu[x, t] + (2 c[x, t] + 1)*(c[x, t] + 1)*
  6. D[mu[x, t], x] + (c[x, t] + 1)*D[v[x, t], x, x] 11/3 - 2 v[x, t]*D[v[x, t], x],
  7. mu[x, t] - log[c[x, t]] - 1 - 4 c[x, t] + 3 D[c[x, t], x, x] == 0};
  8.  
  9. bcs = {v[0, t] == 0, v[1, t] = 0, Derivative[1, 0][c][0, t] == 0, Derivative[1, 0][c][1, t] == 0};
  10. ics = {c[x, 0] == 0.3, v[x, 0] == 0, mu[x, 0] == 0.6960};
  11.  
  12.  
  13. sol = NDSolveValue[{eqns, bcs, ics}, {c, v,mu}, {x, 0, 1}, {t, 0, 3},
  14. Method -> {"IndexReduction" -> Automatic, "EquationSimplification" -> "Residual",
  15. "PDEDiscretization" -> {"MethodOfLines",
  16. "SpatialDiscretization" -> {"TensorProductGrid", "MinPoints" -> 200, "MaxPoints" -> 200}}}]
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