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Help!! I can't find the error in this program? (InverseFunction)

Lina Lili, faculty of science
Posted 5 years ago

Hello, please I 'm really stuck in this program, I can't find the error, I really need to solve this problem If anyone can help me, please

T = 2 \[Pi];
l = \[Pi];
a = 1/2;
w1 = 2 \[Pi];
(*Définit la fonction indicatrice*)
indicator[x_] := Piecewise[{{1, 0 < x < \[Pi]/2}}, 0];
S[x_, t_] := Sum[ Exp[- t  n^2] Integrate[x Sin[s n^2], {s, 0, \[Pi]}]  Sin[x n^2], {n, 5}]
(*controllability map*)
G[x_, t_] := indicator[x] S[x, T - t]
(*Grammian operator Q*)
Q[x_, t_] := (indicator[x])^2  Integrate[(S[x, T - s])^2, {s, T - l, T}]
(*inverse function*)
v[x_, t_] := InverseFunction[a + Q][x, t];
(*Test*)
v[w1 - S[1, T],T]
POSTED BY: Lina Lili
8 Replies
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Hmmmm, what do you understand by

InverseFunction[a + Q]

????

I assume that this is nonsense.

Look at 

ContourPlot[a + Q[x, y], {x, .`5, 1}, {y, 4, 5}]

and you see that there are really many combinations of x and y giving the same value of a + Q [x, y], so the inverse function is not really existing.

For example

a + Q[.5397, 4.651]
a + Q[.87921, 4.243]

So what do expect as result of InverseFunction[ a + Q ] ?
POSTED BY: Hans Dolhaine

InverseFunction[ a + Q ] for me is the inverse of 1/2+Q.

OK, but this does (in general) not exist. So - what do you expect of a non-existing function in your equations?

S looks indeed somewhat peculiar, but I can't see why there should be a problem.

S[x_, t_] :=  Sum[-Exp[-t n^2] (\[Pi] Cos[\[Pi] n^2])/(n^2) Sin[x n^2], {n, 5}]
Plot3D[S[x, t], {x, 0, \[Pi]}, {t, 0, T}, PlotRange -> All,  PlotPoints -> 50]

Why don't you tell us about the physical problem you want to describe. I got the feeling that your equations are somewhat strange.
POSTED BY: Hans Dolhaine
Rohit Namjoshi
Rohit Namjoshi
Posted 5 years ago

Duplicate of this question.
POSTED BY: Rohit Namjoshi

OK. But as this inverse does not exist there seems to be something severely wrong with your model.
POSTED BY: Hans Dolhaine

Hi Mr. Hans, I just edit my question, I've just noticed that there are some mistakes in my original question, I've changed it completely. but I still have in the inverse !!
POSTED BY: Lina Lili

Hi Hans, As you said InverseFunction[ a + Q ] for me is the inverse of 1/2+Q. But it doesn't have any sense I think the problem is in S[x,t], I just test It's not working !! can you see it, please!

Best regards,
POSTED BY: Lina Lili

Sorry, I forget to add indicator[x]. I edit the question:

(*Définit la fonction indicatrice*)
indicator[x_] := Piecewise[{{1, 0 < x < \[Pi]/2}}, 0];

do you have any idea why it doesn't work ? this is error i get :

NDSolve::underdet: There are more dependent variables, {f[x,t],InverseFunction[1/2+Q][x,t]}, than equations, so the system is underdetermined.
POSTED BY: Lina Lili
Rohit Namjoshi
Rohit Namjoshi
Posted 5 years ago

What is indicator[x]?
POSTED BY: Rohit Namjoshi
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