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Re: NDSolve -- n-body indexing ([[]]) problem

  • To: mathgroup at smc.vnet.net
  • Subject: [mg112237] Re: NDSolve -- n-body indexing ([[]]) problem
  • From: Greylander <greylander at gmail.com>
  • Date: Mon, 6 Sep 2010 04:14:37 -0400 (EDT)
  • References: <i5vnri$2e1$1@smc.vnet.net>

Hello Themis,

Thanks for the reply.  I think you mis-interpreted my problem.  The
differential equation(s) are not stochastic.  The RandomReal[]
function in the example is only used for the initial conditions.  I
should have chosen an example that was easier to read.  See my new
thread which states the problem more clearly:  ""NDSolve -- indexing
of dependent variable that is arbitrary list".

I have found a workaround, which I will post on that thread.

Scott


On Sep 5, 5:28 am, Themis Matsoukas <tmatsou... at me.com> wrote:
> The problem is that your differential equations are stochastic. I asked wolfram about it (see email below) using the following as a minimal example (which doesn't work):
>
> Remove@g1
> g1[t_?NumberQ] := (RandomReal[] - 0.5)*t;
> sln = NDSolve[{
>    y'[t] == g1[t],
>    y[0] == 0
>    }, {y}, {t, 0, 1}]
> Plot[Evaluate[y /. sln[[1]]][t], {t, 0, 1}, PlotRange -> All,
>  AxesOrigin -> {0, 0}]
>
> Wolfram's answer is basically that you cannot do this NDSolve. Tech support suggested to use RecurrenceTable:
>
> Remove@g1 ; Clear [a, pp];
> g1[t_?NumberQ] := (RandomReal[] - 0.5)*t; dt = 0.0001;
>
> pp = RecurrenceTable[{a[n + 1] - a[n] == dt*g1[n], a[0] == 0},
>    a[n], {n, 1, Round[1/dt]}] ;
> ListPlot[pp, PlotRange -> All]
>
> This, however,  forces you to apply a simple Euler integration and that's unfortunate because you cannot take advantage of advanced integration algorithms (stiff systems, adjustable time steps, etc) unless you code them yourself. I hope that Mathematica will come up with a module similar to NDSolve that is truly numerical, i.e. it would work with tabulated data, not just functions that can be expressed in analytic form.
>
> Themis
>
> ===========================
> From: Themis Matsoukas
> Date: Wed, 16 Jun 2010 11:42:03 -0500
> Subject: Premier Service Help Form
> To: supp... at wolfram.com
>
> Suggestion or Bug:
> I want to solve a differential equation of the form,
>
> {y'[t]==G[t], y[0]==y0}
>
> where G[t] is a stochastic function. In the problem I have in mind, the value of G[t] is calculated by running a Monte Carlo simulation, so its actual value at each point is subject to some noise. However, I found out that NDSolve does not accommodate ODEs whose right-hand side contains stochastic terms. In a minimal example (which I am attaching), the definition of G is
>
> G[t_] := (RandomReal[] - 0.5) t;
>
> When G is used inside NDSolve, it seems that the random number is calculated just once and not each time an evaluation of G[t] is called for. That is, NDSolve treats G as if
>
> G[t_] := A t
>
> where A is a constant  that is randomly picked at the beginning of the calculation. Is there a way to force NDSolve to use the actual values that would returned by the delayed assignment in the definition of G?
>
> Thanks
>
> Themis Matsoukas
> ==========================
>
> Hello Mr. Matsoukas,
>
> Thank you for the email. Please refer to the attached notebook. Essentially
> NDSolve is not well suited to solve Stochastic type differential equations.
> The use of RecurrenceTable is a more appropriate option.
>
> Sincerely,
>
> Paritosh Mokhasi, Ph.D.
> Technical Support
> Wolfram Research, Inc.http://support.wolfram.com



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