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Plot a solution of DSolveValue

Posted 1 year ago
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Hello, please I calculated the solution of a particular heat equation with control, now I have the solution but I can't plot it if someone can help me please:

(* this first part is just  for building the control, you can ignore it*)

T = 2 \[Pi];
l = 7 \[Pi]/4;
a = 1/100;
w1 = Sin;
(*Définit la fonction indicatrice*)
indicator[x_] := Piecewise[{{1, 0 < x < \[Pi]/2}}, 0];
(*Semi groupe S[f,t][x]*)
S[t_, f_, x_] := 
Sum[ Exp[- t  n^2] Integrate[f[s] Sin[s n], {s, 0, \[Pi]}]  Sin[x n], {n, 5}]
v1[x_] := w1[x];
L[t_, x_] := S[T - t, v1, x];
Q[x_] := indicator[x] Integrate[L[s, x]^2, {s, T - l, T}];
g1[x_] := a v1[x] + Q[x];
g2[t_, x_] := indicator[x] v1[x] L[t, x];
u[t_, x_] := g2[t, x]/g1[x];

Now that we have our control u, we pass to the calculation of solution of our heat equation

homogen =  D[f[x, t], t] - D[f[x, t], {x, 2}] -  indicator[x] u[x, t] == 0;
 (*données initiales propres *)
ic = {f[x, T - l] == 0};
 (* Condition aux bord de Dirichlet*)
bc = {f[0, t] == 0, f[\[Pi], t] == 0};
(*résolution analytique de l'équation *)
sol1 = DSolveValue[{homogen, ic, bc}, f[x, t], {x, 0, \[Pi]}, {t, T - l, T}]

1) now we have the solution I want to plot the solution when t = T (fix) and x varies between 0 and $\pi$?

I would do it this way:

sol1 = NDSolveValue[{homogen, ic, bc}, f, {x, 0, \[Pi]}, {t, T - l, T}]
Plot[sol1[x, T], {x, 0, Pi}]
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