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