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MathGroup Archive 2008

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integration - Interpolating Function

  • To: mathgroup at smc.vnet.net
  • Subject: [mg92015] integration - Interpolating Function
  • From: marion70 at gmx.de
  • Date: Wed, 17 Sep 2008 04:28:38 -0400 (EDT)

Hi,

I have two different problems concerning Mathematica:
First: to calculate a characteristic function  \[CurlyPhi][u_] for
each u I have to evaluate a solution of a system of ordinary
differential equations depending on u:


solution[u_] := NDSolve[
  {z1'[t] == -\[Kappa] \[Theta]y y1[
       t] - (ly \[Mu] (y1[
           t] - \[Mu]  y1[t]^2 - \[Mu] y2[t]^2))/((1 - \[Mu] y1[
             t])^2 + \[Mu]^2 y2[t]^2),
   z2'[t] == -\[Kappa] \[Theta]y y2[
       t] - (ly \[Mu]  y2[
         t])/((1 - \[Mu] y1[t])^2 + \[Mu]^2 y2[t]^2),
   y1'[t] == \[Kappa] y1[t] - (\[Sigma]^2)/2 (y1[t]^2 - y2[t]^2),
   y2'[t] == \[Kappa] y2[t] - \[Sigma]^2 y1[t] y2[t] - u,
   z1[T] == 0,
   z2[T] == 0,
   y1[T] == 0,
   y2[T] == 0}, {z1, z2, y1, y2}, {t, 0, T}]
a1[u_] := z1[0] /. solution[u][[1, 1]]
a2[u_] := z2[0] /. solution[u][[1, 2]]
b1[u_] := y1[0] /. solution[u][[1, 3]]
b2[u_] := y2[0] /. solution[u][[1, 4]]
\[CurlyPhi][u_] := E^(a1[u] + I a2[u] + b1[u] + I b2[u] x0)

as a result I get a set of Interpolating Functions, so
I can evaluate the characteristic functions in specific points u. Next
I have to integrate over this characteristic function:

Needs["FourierSeries`"];
f[z_]:=NInverseFourierTransform[\[CurlyPhi][u_]  ,
u,z,FourierParameters=84_{1,1}]

but Mathematica cannot solve this problem.

Does anybody know an answer or a way to fix this problem?

Second: Can anybody give me advice, how to implement efficiently a
recursion, that depends on two variables? (Does Mathematica have
another function besides RSolve which solves this kind of problems?)

I would be really grateful if someone could help me.

Best regards,
Marion


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