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How to make fitting code using NDSolve?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg127761] How to make fitting code using NDSolve?
  • From: JYeom <cygnis at hanmail.net>
  • Date: Mon, 20 Aug 2012 04:14:27 -0400 (EDT)
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Hello,
I am trying to fit some basic data to a partial differential equation- here are the data points:

data = {
  {2.113, 2.915*^-7}, {2.256, 2.758*^-7}, {2.303, 2.567*^-7}, {2.493,
   2.343*^-7}, {2.635, 2.085*^-7}, {2.825, 1.838*^-7}, {3.015,
   1.614*^-7}, {3.205, 1.412*^-7}, {3.538, 1.199*^-7}, {3.775,
   1.03*^-7}, {4.108, 8.173*^-8}, {4.487, 6.266*^-8}, {4.915,
   4.695*^-8}, {5.295, 3.349*^-8}, {5.912, 2.34*^-8}, {6.672,
   1.554*^-8}, {7.527, 7.692*^-9}, {8.524, 5.448*^-9}, {9.426,
   4.326*^-9}, {10.23, 3.205*^-9}, {10.99,
   2.083*^-9}, {11.99, -1.608*^-10}
  }

And there are four parameters to fit in our fitting equation:

model[ka_?NumberQ, kr_?NumberQ, kd_?NumberQ,
  kb_?NumberQ] := (model[ka, kr, kd, kb] =
   First[Subscript[Dye, 0] /. NDSolve[{
       Derivative[1][Subscript[Dye, 0]][
         t] ==\[VeryThinSpace]-ka Subscript[Dye, 0][t] +
         kr  Subscript[Dyes, 0][t], Subscript[Dye, 0][0] == 1.22 10^-6,
       Derivative[1][Subscript[Dye, 1]][
         t] == -ka Subscript[Dye, 1][t] + kr  Subscript[Dyes, 1][t],
       Subscript[Dye, 1][0] == 0,
      
       Derivative[1][Subscript[Dyes, 0]][t] ==
        ka  Subscript[Dye, 0][t] - kr  Subscript[Dyes, 0][t] -
         kd  Subscript[CI, 0][t] Subscript[Dyes, 0][t] -
         kd /2 * Subscript[CI, 1][t] Subscript[Dyes, 1][t],
       Subscript[Dyes, 0][0] == 0,
       Derivative[1][Subscript[Dyes, 1]][t] ==
        ka Subscript[Dye, 1][t] -
         kd Subscript[CI, 1][t] Subscript[Dyes, 0][t] -
         kr Subscript[Dyes, 1][t] -
         kd  Subscript[CI, 0][t] Subscript[Dyes, 1][t],
       Subscript[Dyes, 1][0] == 0,
      
       Derivative[1][Subscript[CI, 0]][
         t] == -kd Subscript[CI, 0][t] Subscript[Dyes, 0][t] -
         kd/2*  Subscript[CI, 1][t] Subscript[Dyes, 1][t] -
         kb Subscript[CI, 0][t] Subscript[HDye, 0][t] -
         kb/2* Subscript[CI, 1][t] Subscript[HDye, 1][t],
       Subscript[CI, 0][0] == 3.18 10^-3,
       Derivative[1][Subscript[CI, 1]][
         t] == -kd Subscript[CI, 1][t] Subscript[Dyes, 0][t] -
         kd/2*  Subscript[CI, 0][t] Subscript[Dyes, 1][t] -
         kb Subscript[CI, 1][t] Subscript[HDye, 0][t] -
         kb/2*Subscript[CI, 0][t] Subscript[HDye, 1][t],
       Subscript[CI, 1][0] == 0,
      
       Derivative[1][Subscript[HDye, 0]][t] ==
        kd  Subscript[CI, 0][t] Subscript[Dyes, 0][t] +
         kd/2* Subscript[CI, 1][t] Subscript[Dyes, 1][t] -
         kb  Subscript[CI, 0][t] Subscript[HDye, 0][t] -
         kb /2* Subscript[CI, 1][t] Subscript[HDye, 1][t],
       Subscript[HDye, 0][0] == 0,
       Derivative[1][Subscript[HDye, 1]][t] ==
        kd  Subscript[CI, 1][t] Subscript[Dyes, 0][t] +
         kd  Subscript[CI, 0][t] Subscript[Dyes, 1][t] -
         kb Subscript[CI, 1][t] Subscript[HDye, 0][t] -
         kb  Subscript[CI, 0][t] Subscript[HDye, 1][t],
       Subscript[HDye, 1][0] == 0
       }, {Subscript[Dye, 0][t], Subscript[Dyes, 0][t],
       Subscript[CI, 0][t], Subscript[HDye, 0][t],
       Subscript[Dye, 1][t], Subscript[Dyes, 1][t],
       Subscript[CI, 1][t], Subscript[HDye, 1][t]}, {t, 0, 12}]])

In the NDSolve, what we want to fit is Dye0[t]. So we wrote code for fitting using FindFit:

fit = FindFit[data,
  model[ ka, kr, kd, kb][Subscript[Dye,
   0]], {{ka, 10^(-4)}, {kr, 10^(-4)}, {kd, 500}, {kb, 500}},
  Subscript[Dye, 0]]

However, we encountered the error message like this:

FindFit::nrlnum: "The function value {-2.915*10^-7+Subscript[Dye, 0][2.113],-2.758*10^-7+Subscript[Dye, 0][2.256],-2.567*10^-7+Subscript[Dye, 0][2.303],<<16>>,-3.205*10^-9+Subscript[Dye, 0][10.23],-2.083*10^-9+Subscript[Dye,0][10.99],1.608*10^-10+Subscript[Dye, 0][11.99]} is not a list of real numbers with dimensions {22} at {ka,kr,kd,kb} = {0.0001,0.0001,500.,500.}."

The desirable solution for parameter is around ka= 0.67, kr = 10^(-3), kd=1600, kb=1600.

Does anyone know what I am doing wrong?



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