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Using NDSolve in an Iteration

Posted 9 years ago
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Hey Folks,

I have a question regarding the usage of NDSolve/ParametricNDSolve in an iteration. Probably I just struggle with the right formating of the code, but here is a simple idea of what I want to do:

In the first step, I want to solve a differential equation N'[t] with NDSolve which looks like this: (1)

d/dt N[t] = a* I[t]^6

(this is no problem at all and easily feasible) but then I want to use the result in the next step to modify I[t] and use it to solve the differential equation again: (2)

d/dt N[t] = a* (I[t] - b*N[t] (previous result) )^6.

and so on...

So my question is how to use it in an as simple as possible iteration without recursion to take place.

Maybe you guys could give me a simple example of how to properly use NDSolve in an iteration.

Thanks in advance!

Thomas

POSTED BY: Thomas Winkler

Hello, I know you asked for no recursion. However, this problem lends itself perfectly for a recursive step, that I have put it in anyway....

ellFunc[t_] := t Exp[-t] (*example for your l[t]*)

Define the first RHS of your ODE:

rhs[0][a_, b_, ell_] = a ell[t]^6

Compute the the base numerical solution to your ODE (I've arbitrarily integrated for 0 < t < 10 and set l[0]=0) sol[i] is the solution at the ith iteration; a and b are from your definition, ell is the choice for l[t]

sol[0][a_?NumericQ, b_?NumericQ, ell_] :=  
 sol[0][a, b, ell] = 
  enn /. NDSolve[{D[enn[t], t] == rhs[0][a, b, ell], enn[0] == 1}, 
     enn, {t, 0, 10}][[1]]

example:

sol[0][1, 2, ellFunc]
Plot[sol[0][1, 2, ellFunc][t], {t, 0, 10}]

The RHS for the ith iteration, need the solution from the previous iteration.

rhs[i_][a_?NumericQ, b_?NumericQ, ell_] := 
 rhs[i][a, b, ell] = a (ell[t] - b enn[t]) sol[i - 1][a, b, ell][t] 

The solution at the ith iteration uses the RHS for that iteration:

sol[i_][a_?NumericQ, b_?NumericQ, ell_] := 
 sol[i][a, b, ell] = 
  enn /. NDSolve[{D[enn[t], t] == rhs[i - 1][a, b, ell], enn[0] == 1},
      enn, {t, 0, 10}][[1]]

Example:

Plot[Evaluate[sol[1][1, 2, ellFunc][t]], {t, 0, 10}]

A list of solutions:

solutions = Table[sol[i][1, 2, ellFunc][t], {i, 0, 10}]

and plotting them:

Plot[solutions, {t, 0, 10}]
POSTED BY: W. Craig Carter
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