I am solving this PDE numerically (dimensions: spatial variable x and time t). I know how to solve my problem numerically (for a given range of time). The solution reaches its equilibrium after a certain time depending on other parameters of the problem.
ClearAll[uind, x, t];
Du = 1;
alpha = 4;
T = 100;
pde = D[uind[t, x], t] == Du*D[uind[t, x], x, x]-D[uind[t,x],x]-alpha;
bc = {uind[t, 0]== 3, (D[uind[t, x], x] /. x -> 1) == 0};
ic = uind[0, x] == 3;
usol = NDSolve[{pde, ic, bc}, uind, {t, 0, T}, {x, 0, 1}]
Plot3D[{Evaluate[uind[t, x]] /. usol}, {t, 0, T}, {x, 0, 1}, PlotRange -> All, AxesLabel -> {"t", "x", "Sol"}]
ClearAll[uind, x, t];
Du = 1;
alpha = 4;
T = 100;
pde = D[uind[t, x], t] == Du*D[uind[t, x], x, x]-D[uind[t,x],x]-alpha;
bc = {uind[t, 0]== 3, (D[uind[t, x], x] /. x -> 1) == 0};
ic = uind[0, x] == 3;
usol = NDSolve[{pde, ic, bc}, uind, {t, 0, T}, {x, 0, 1}]
Plot3D[{Evaluate[uind[t, x]] /. usol}, {t, 0, T}, {x, 0, 1},
PlotRange -> All, AxesLabel -> {"t", "x", "Sol"}]
I need to calculate the time it takes to reach the equilibrium. Are there any standard MathematicaMathematica techniques for that? Any examples in MathematicaMathematica are appreciated.
I found this solution, but it does not help me because it is an ODE.
