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How to relate two functions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg127468] How to relate two functions
  • From: William Duhe <williamduhe at hotmail.com>
  • Date: Fri, 27 Jul 2012 04:57:35 -0400 (EDT)
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  • Delivered-to: mathgroup-newout@smc.vnet.net
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So here I have several functions plotted, a[t], B[t],Rm[t]. What I need is syntax that will allow me to see the value of a[t] when T[t] is 1. This should be easy to see that at first when T[t] is 1 a[t] is also one, but as the function turns around T[t] becomes 1 again, at about t = 575, and a[t] has risen by a certain amount. I need to quantify this amount.I have plotted a parametric plot of a[t] vs T[t] and what I need to do is find the Y-INTERCEPTS of that plot in order to get the data I need, but other more sophisticated methods would also be welcome. Thanks ahead of time for any help!

The syntax is provided bellow.


M = 100;
g = 1;
\[Alpha] = 1;
ti = 0;(*initial time*)
tf = 700; (*final plot time*)
m = 1;
RB[t] = (m*T[t]^3*g)/(2*\[Pi])^(3/2)*(m/T[t])^(3/2)*E^(-m/T[t]);
R = 0.0005;
V = 5*10^(-12)*M^4;
i = Sqrt[(g*m^4 - V)/(3*M^2)]*m;
rR = g*T[t]^4;(*Energy Density-Radiation*)
pR = g*T[t]^4;(*Pressure-Radiation*)
pM = 0;(*Pressure-Phi Particle*)
r = rR + Rm[t] - V;(*Overall Energy Density*)
p = pR + V;(*Overall Pressure*)
s1 = NDSolve[{a''[t] == -a[t]*1/(6*M^2) (2*g*T[t]^4 + 2 V + Rm[t]), 
   T'[t] == (R*Rm[t])/
     T[t]^3 - (Sqrt[3/2]*\[Alpha]^2)/(
      Sqrt[T[t]]*m^(
       11/2))*(((g*T[t]^3*m)/(2*\[Pi])^(3/2)*(m/T[t])^(3/2)*
          E^(-m/T[t]))^2 - Rm[t]^2) - (a'[t]*T[t])/a[t], 
   Rm'[t] == -R*Rm[t] + (Sqrt[3/2]*\[Alpha]^2*T[t]^(5/2))/m^(
      11/2)*((g/(2*\[Pi])^(3/2)*(m/T[t])^(3/2)*E^(-m/T[t]))^2 - 
        Rm[t]^2) - 3 a'[t]/a[t]*Rm[t], T[0] == m, 
   Rm[0] == g/(2*\[Pi])^(3/2)*E^-1, a'[0] == i, a[0] == 1}, {a[t], 
   T[t], Rm[t]}, {t, ti, tf}, Method -> "BDF"]

Plot[Evaluate[Rm[t]/RB[t] /. s1], {t, ti, tf}, PlotRange -> Automatic,
  AxesLabel -> {Time, Rm[t]}]
Plot[Evaluate[T[t] /. s1], {t, ti, tf}, PlotRange -> Automatic, 
 AxesLabel -> {Time, T[t]}]
Plot[Evaluate[a[t] /. s1], {t, ti, tf}, PlotRange -> Automatic, 
 AxesLabel -> {Time, a[t]}]
Plot[{Evaluate[a[t] /. s1], Evaluate[T[t] /. s1]}, {t, ti, tf}, 
 PlotRange -> Automatic, AxesLabel -> {Time, a[t]}]
ParametricPlot[Evaluate[{T[t], a[t]} /. s1], {t, ti, tf}, 
 PlotRange -> Automatic]



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