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MathGroup Archive 1996

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Re: Problem with discontinuity in NDSolve[]

  • To: mathgroup at smc.vnet.net
  • Subject: [mg4404] Re: Problem with discontinuity in NDSolve[]
  • From: rubin at msu.edu (Paul A. Rubin)
  • Date: Fri, 19 Jul 1996 03:46:47 -0400
  • Organization: Michigan State University
  • Sender: owner-wri-mathgroup at wolfram.com

In article <4qqnms$49m at dragonfly.wolfram.com>,
   a_kowald at chemie.fu-berlin.de (Axel Kowald) wrote:
->Hi everybody,
->
->I have a problem with NDSolve if my equation contains a discontinuity.
->If I try
->
->NDSolve[{y'[t] == 10 - y[t] * If[y[t]-15 <= 0, 0, 1], 
->         y[0] == 0}, y, {t,0,20}]
->
->I always get the error:  Maximum number of steps reached.
->The solution should be a linear increase until y = 15 followed by an 
->exponential decay until a steady state is reached by y = 10
->
->
->Funny enough if I try 
->
->NDSolve[{y'[t] == 10 - y[t] * If[t-15 <= 0, 0, 1], 
->         y[0] == 0}, y, {t,0,20}]
->
->
->causing the discontinuity to be at t = 15 instead of y = 15 NDSolve has 
no
->problems at all. (But of course this is not what I want.)
->
->
->ANY SOLUTIONS ???     ANY IDEAS  ???
->
->
->
->Many thanks
->
->
->	Axel Kowald
->
->
->P.S. I'm using Mma 2.2 on a Mac with 7.5.3 and 24 MB RAM

I tried this in Mma 2.2.2/Windows, and got the same thing.  I'm not sure 
what Mma is using to solve the ODE, but I suspected at first that it was 
just jamming at the point where the derivative of y[t] is undefined 
(t=1.5).  The difference between your first and second versions is that in 
the second version, since it knows t exactly, the error in estimating y'[t] 
is proportional to the error in estimating y[t], whereas in the first 
method small errors in y[t] produce relatively enormous errors in y'[t] 
(which essentially flip-flops between 10 and approximately -5).

I still think that's part of the problem, but there may be something about 
how Mma does its function evaluations at work.  I tried replacing your If[] 
factor with Max[ 0, Min[ 1, (y[t] - 15)/eps ] ], where eps was given a 
small positive value (.0001) beforehand.  This was an attempt to smooth the 
transition from 0 to 1 in the If[].  NDSolve produced no diagnostics, but 
the solution it gave rose linearly from t=0, y=0 to t=1.5, y=15, and then 
leveled out as a constant function (y=15 for t>1.5), which is utterly 
wrong!

Paul

**************************************************************************
* Paul A. Rubin                                  Phone: (517) 432-3509   *
* Department of Management                       Fax:   (517) 432-1111   *
* Eli Broad Graduate School of Management        Net:   RUBIN at MSU.EDU    *
* Michigan State University                                              *
* East Lansing, MI  48824-1122  (USA)                                    *
**************************************************************************
Mathematicians are like Frenchmen:  whenever you say something to them,
they translate it into their own language, and at once it is something
entirely different.                                    J. W. v. GOETHE

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