RE: fixed point in a function of two variables

• To: mathgroup at smc.vnet.net
• Subject: [mg28438] RE: [mg28336] fixed point in a function of two variables
• From: "Higinio Ramos" <higra at gugu.usal.es>
• Date: Thu, 19 Apr 2001 03:26:26 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```When I receive an e-mail of the MathGroup the first thing that I do is
to reproduce the situation that is indicated. I have to apologize
because after receiving the message of Tomas Garza  I have tried to do
the operations  that I sent and I have had problems with [ Mu][T _ ]
etc.  [ Mu][T _ ] was a function that measure the viscosity of a fluid
(oil) and the two following functions newT[Pwf2 _ ] and fonRep[{T _, mu
_ } ] were to obtain the final temperature of working of an axis that
turn within a cylinder and have a lubrication by oil. fonRep[{T _, mu _
} ] is a function that have a point fix but there are  problems in
calculate it  because after applying the Mathematica function FixedPoint
this enter in a cycle and is impossible to obtain. I show the problem
again changing the name [ Mu][T _ ] by vis[T_]:

r = 0.05; b0 = 0.03 10^(-3); b1 =
0.066 10^(-3); n = 500; T0 = 80; Ta = 20;
U = 2 Pi n r/60;
PPi = Pv - 10^5;
e = (b1 - b0)/2;
R = r + (b0 + b1)/2;
b = Sqrt[R^2 - e^2 Sin[t]^2] + e Cos[t] - r;
sol = Solve[
6 r  mu U NIntegrate[1/b^2, {t, 0, 2Pi}] -
r 12 mu Q NIntegrate[1/b^3, {t, 0, 2Pi}] == 0, Q]; Q = Q
/. sol[[1]]

Out[9]=
0.0000504476

vis[T_] := (10^(-3))*10^(10^(((Log[10, Log[10, 200]] - Log[10, Log[10,
7.55]])(120 -
T))/(120 - 40) + Log[10, Log[10, 7.55]]));

newT[Pwf2_] :=
If[2 r <= 0.1, (-Pwf2 + 20(35 + 30 r)10^(-3) Ta)/(20(35 + 30
r)10^(-3)),
(-Pwf2 + 20(25 + 40 r^2 10^3)10^(-3) Ta)/(20(25 + 40 r^2
10^3)10^(-3))];

fonRep[{T_, mu_}] :=
(\[Tau][t_] := -4 mu U /b + 6 mu Q/b^2;
M = r^2NIntegrate[\[Tau][t], {t, 0, 2Pi}];
Pwf = M U/r; {newT[Pwf], vis[newT[Pwf]]});

And the problem:
FixedPoint[fonRep,{Ta,vis[Ta]}]

and why:
NestList[fonRep, {Ta, vis[Ta]}, 20]

Despite that, I think I have solved the problem: like the viscosity
depends also of the temperature T, I have expressed the function fonRep
only as function of this temperature T.

fonRep[T_] :=   (\[Tau][t_] := -4 vis[T] U /b +
6 vis[T] Q/b^2;
M = r^2NIntegrate[\[Tau][t], {t, 0, 2Pi}]; Pwf = M U/r; newT[Pwf])

I obtain the interpolation function of a  collection of point of this
function  fonRep[T] and making the intersection with the diagonal we see
that there is only one fixed point:

int = Interpolation[Table[{T, fonRep[T]}, {T, Ta, 300, 5}]];
Plot[int[T], {T, Ta, 300}];
finalT = FindRoot[int[T] == T, {T, Ta}]; T /. finalT