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Re: Delay Differential Equations
*To*: mathgroup at smc.vnet.net
*Subject*: [mg105232] Re: [mg105198] Delay Differential Equations
*From*: Daniel Lichtblau <danl at wolfram.com>
*Date*: Wed, 25 Nov 2009 02:31:53 -0500 (EST)
*References*: <200911241048.FAA00661@smc.vnet.net> <4B0C33E2.5070102@wolfram.com> <5bb7300a0911241148o69013c9br1871eb1cb1a73c73@mail.gmail.com>
Yun Zhao wrote:
> Hi Daniel,
>
> Thanks for your reply. But my question is I was trying to solve an
> equation, and the solution just doesnt make much sense, and I looked in
> the help files, but they didn't really help me much. The problem is this:
>
> I have some cells of type A, and at some point in time, cells of type A
> are going to slowly change into cells of type B. So, population size of
> cell type A will decrease over time due to two factors: (1) becoming
> cells of type B and (2) dying over time. So for a differential
> equation, i have two rates, one is the transformation rate (0.0367), one
> is the death rate (0.0123). But I want to delay the transformation by
> 37 hours. So from time = 0 to time = 37, all that is changing in cells
> of type A is that they are dying at the death rate, then after 37 hours,
> they are also disappearing at the transformation rate.
>
> So my differential equation is like this:
>
> *solrandom1=NDSolve[{p'[t]*
>
> *==0.0367*p[t-37]-0.0123*p[t],p[t/;t<0]=30000},p,{t,0,120}]*
>
> What I expect when I plot p(t) over t is a gradual decrease from t=0 to
> t=37. Then more decrease from t=37 to t=120. p(t) should never be less
> than zero. But when I ran that code in Mathematica, and plotted it
> using this command
>
> *Plot[Evaluate[p[t]/.solrandom1],{t,0,120},PlotRange**ï?®Automatic]*
>
> p(t) was actually less than zero. I am not able to figure out why this
> is.
>
> Please tell me what I did wrong. Thanks.
First is that the delay term you have is not negative. So I'd expect an
increase. When I do the following I indeed see a rising population.
solrandom1 =
NDSolve[{p'[t] == 0.0367*p[t - 37] - 0.0123*p[t],
p[t /; t <= 0] == 30000}, p, {t, 0, 120}]
Plot[Evaluate[p[t] /. solrandom1], {t, 0, 120},
PlotRange -> Automatic]
Second is that your claim of no transformations for 37 hours is wrong.
You are modeling that by 0.0367*p[t - 37], and p[t] is 30000 for t<=0.
So this contribution is not going to be zero (whether you use a positive
or negative thereof). You can repair this by using the system below.
solrandom1 =
NDSolve[{p'[t] == -0.0367*p[t - 37]*UnitStep[t - 37] - 0.0123*p[t],
p[t /; t <= 0] == 30000}, p, {t, 0, 120}]
Next is that there is nothing to stop this solution from going negative.
For that you'd need an EventHandler. Could do as below.
solrandom1 =
NDSolve[{p'[t] == -0.0367*p[t - 37]*UnitStep[t - 37] - 0.0123*p[t],
p[t /; t <= 0] == 30000}, p, {t, 0, 120},
Method -> {"EventLocator", "Event" -> p[t]}]
Last is that these methods are things in the documentation I had noted
in my last reply.
Daniel Lichtblau
Wolfram Research
> On Tue, Nov 24, 2009 at 1:28 PM, Daniel Lichtblau <danl at wolfram.com
> <mailto:danl at wolfram.com>> wrote:
>
> Yun Zhao wrote:
>
> Hi,
>
> Does anyone have experience working with delay differential
> equations in
> Mathematica 7? I found some help files on wolfram and
> Mathematica help, but
> the information available there were very limited. If anyone can
> refer me to
> other online or textual sources, I would really appreciate it.
> Thank you
> very much.
>
> Mike
>
>
> Not sure what you saw in the Documentation Center. These are perhaps
> the most useful, worth checking if you have not encountered them
> already.
>
> howto/SolveDelayDifferentialEquations
> tutorial/NDSolveDelayDifferentialEquations
>
> Daniel Lichtblau
> Wolfram Research
>
>
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