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equation solving - Numerically Integrating to find a Maximum using NDSolve



I am trying to numerically find an equilibrium (maximum) of a function using its differential. The following is a simplified version.


myEquilibrium[p_]:=Last[{myPreviousStep=1;NDSolve[{s'[t]==p-s[t],s[0]==myPreviousStep,WhenEvent[s[t]-myPreviousStep<10^-4||s[t]<10^-4,"StopIntegration"]},{s},{t,0,Infinity},StepMonitor:>(myPreviousStep=s[t])];myPreviousStep}]

myEquilibrium[.5] // 0.5 which is correct


The function should be constrained to positive s, which is why I included the s[t]<10^-4 requirement. However this does not seem to work.


myEquilibrium[-.5] // -0.5 but should be 10^-4


The NDSolve should also not go to 0 exactly, as the real, non-simplified version contains a 1/s[t] in the differential. That's another reason I want the procedure to stop at s[t]=10^-4, before s[t]=0.


myEquilibrium[0] // 0. but should be 10^-4


Finally, I often get NDSolve::mxst: Maximum number of 10000 steps reached at the point t == 6.57563031118913721074693815431*^4952. >> and NDSolve::ndsz: At t == 1.79769313486*^308, step size is effectively zero; singularity or stiff system suspected. >>. What would typically solve this issue?


Thanks.




Answer



Because s[t] is decreasing whenever p < 1, s[t] - myPreviousStep < 10^-4 will always be True. WhenEvent[cond, action] evaluates action when the condition changes from False to True; however, the condition is always True when p < 1. You need something like Abs[s[t] - myPreviousStep] < 10^-4, instead. Note that if p is closer to 1 than 10^-4, then the same thing happens. Also note this condition controls the accuracy in an indirect way, dependent on the convergence rate of NDSolve on your particular ODE.)


myEquilibrium[p_] := 
 Last[{myPreviousStep = 1; 
   NDSolve[{s'[t] == p - s[t], s[0] == myPreviousStep,
      WhenEvent[s[t] < 10^-4 || Abs[s[t] - myPreviousStep] < 10^-4, 
       "StopIntegration"]}, {s}, {t, 0, Infinity}, 
     StepMonitor :> (myPreviousStep = s[t])]; myPreviousStep}]

If you want to handle initial values p that happen to land close to an equilibrium position, then the following forces the condition to be False on the second step. The function step redefines itself after the first step. The first definition forces the condition Abs[s[t] - myPreviousStep] < 10^-4 to be False.



SetAttributes[step, HoldFirst];
myEquilibrium[p_] := Last[{
   step[var_, s_] := (var = s + 2. 10^-4; 
     step[var2_, s2_] := var2 = s2;); myPreviousStep = 1; 
   NDSolve[{s'[t] == p - s[t], s[0] == myPreviousStep,
     WhenEvent[s[t] < 10^-4 || Abs[s[t] - myPreviousStep] < 10^-4, 
      "StopIntegration"]}, {s}, {t, 0, Infinity}, 
    StepMonitor :> step[myPreviousStep, s[t]]]; myPreviousStep}]

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