How to "embed" Piecewise in NDSolve in Mathematica - wolfram-mathematica

Question

I am using NDSolve to solve a non-linear partial differential
equation.
I'd like one of the variables (Kvar) to be a function
of the time step currently being solved and hence and using
Piecewise.
Mathematica generates an error message saying:
SetDelayed::write: Tag Real in 0.05[t_] is Protected. >>
NDSolve::deqn: Equation or list of equations expected instead of
$Failed in the first argument ....
ReplaceAll::reps: ....
I haven't included the entire error message for ease of reading.
My code is as follows:
Needs["VectorAnalysis`"]
Needs["DifferentialEquations`InterpolatingFunctionAnatomy`"];
Clear[Eq4, EvapThickFilm, h, S, G, E1, K1, D1, VR, M, R]
Eq4[h_, {S_, G_, E1_, K1_, D1_, VR_, M_, R_}] := \!\(
\*SubscriptBox[\(\[PartialD]\), \(t\)]h\) +
Div[-h^3 G Grad[h] +
h^3 S Grad[Laplacian[h]] + (VR E1^2 h^3)/(D1 (h + K1)^3)
Grad[h] + M (h/(1 + h))^2 Grad[h]] + E1/(
h + K1) + (R/6) D[D[(h^2/(1 + h)), x] h^3, x] == 0;
SetCoordinates[Cartesian[x, y, z]];
EvapThickFilm[S_, G_, E1_, K1_, D1_, VR_, M_, R_] :=
Eq4[h[x, y, t], {S, G, E1, K1, D1, VR, M, R}];
TraditionalForm[EvapThickFilm[S, G, E1, K1, D1, VR, M, R]];
And the second cell where I am trying to implement Piecewise in NDSolve:
L = 318; TMax = 7.0;
Off[NDSolve::mxsst];
(*Ktemp = Array[0.001+0.001#^2&,13]*)
hSol = h /. NDSolve[{
(*S,G,E,K,D,VR,M*)
Kvar[t_] := Piecewise[{{0.01, t <= 4}, {0.05, t > 4}}],
EvapThickFilm[1, 3, 0.1, Kvar[t], 0.01, 0.1, 0, 160],
h[0, y, t] == h[L, y, t],
h[x, 0, t] == h[x, L, t],
(*h[x,y,0] == 1.1+Cos[x] Sin[2y] *)
h[x, y, 0] ==
1 + (-0.25 Cos[2 \[Pi] x/L] - 0.25 Sin[2 \[Pi] x/L]) Cos[
2 \[Pi] y/L]
},
h,
{x, 0, L},
{y, 0, L},
{t, 0, TMax}
][[1]]
hGrid = InterpolatingFunctionGrid[hSol];
PS: I am sorry but the first cell block doesn't display so well here. And thanks to not having enough "reputation", I can't post images.
The error message occurs when using the NDSolve cell block.

Answer 1

Define the function Kvar outside of a set of equations in NDSolve, like
Off[NDSolve::mxsst];
(*Ktemp=Array[0.001+0.001#^2&,13]*)
Kvar[t_] := Piecewise[{{0.01, t <= 4}, {0.05, t > 4}}];
hSol = ...
and remove it from the list in NDSolve, so that it starts as NDSolve[{(*S,G,E,K,D,VR,M*)EvapThickFilm[..., and it will work. It gives warnings, but those are related to possible singularities in your equation.
Also, your original error indicates that your Kvar was assigned a value of 0.05. So, add Clear[Kvar] before anything else in the second cell.