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Re: system of differential equations mathematica help

  • To: mathgroup at smc.vnet.net
  • Subject: [mg129375] Re: system of differential equations mathematica help
  • From: Oliver Ruebenkoenig <ruebenko at wolfram.com>
  • Date: Tue, 8 Jan 2013 23:39:45 -0500 (EST)
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On Mon, 7 Jan 2013, 01af wrote:

> Before anyone starts to type: I have read the top post on a similar matter, but the reply makes no sense to me so if anyone can give my issue a read through I would be grateful. Thanks.
>
> ok I am modelling airflow in the upper airway for application i obstructive sleep apnoea, but I have hit a brick wall with mathematica. I have a system of 3 differential equations with boundary conditions, and I need to solve to find 3 unknown functions numerically so that they may be plotted in various graphs.
>
> The equations are as follows:
>
> D[a[x]*u[x], x] == 0,
> u[x] u'[x] == -p'[x],
> p[x] - 1 == 2 (1 - ((a[x])^(-3/2))) - 50 (a''[x]).
>
> with boundary conditions:
>
> u[0] == 0.1, a[0] == 1, a[10] == 1, p[10] == 1.
>
> so initially I tried to use NDSolve like so..
>
> Code:
>
> NDSolve[{D[a[x]*u[x], x] == 0, u[x] u'[x] == -p'[x],
>  p[x] - 1 == 2 (1 - ((a[x])^(-3/2))) - 50 (a''[x]), u[0] == 0.1,
>  a[0] == 1, a[10] == 1, p[10] == 1}, {a}, {x, 0, 10}]
>
> but mathematica does this:
>
> Code:
>
> Power::infy: "Infinite expression 1/0.^(3/2) encountered. "
> Infinity::indet: Indeterminate expression 0. ComplexInfinity encountered. >>
> Infinity::indet: Indeterminate expression 0. ComplexInfinity encountered. >>
> General::stop: Further output of Infinity::indet will be suppressed during this calculation. >>
> NDSolve::ndnum: Encountered non-numerical value for a derivative at x == 0.`. >>
>
> which is super annoying, any pointers as to where I'm going wrong would be great. I'm not even sure if I should be using NDSolve so let me know what you think.
> thanks in advance
> a.
>
>

Hi 01af,

to be quite honest I did not see what the issue was and had to resort 
asking colleagues myself. What you can do is:

NDSolve[{D[a[x]*u[x], x] == 0, u[x] u'[x] == -p'[x],
   p[x] - 1 == 2 (1 - ((a[x])^(-3/2))) - 50 (a''[x]), u[0] == 0.1,
   a[0] == 1, a[10] == 1, p[10] == 1}, {a}, {x, 0, 10},
  Method -> {"BoundaryValues" -> {"Shooting",
      "StartingInitialConditions" -> {a[0] == 1, a'[0] == 0, p[0] == 0,
         u[0] == .1}}}]


The issue is that the default starting conditions produce singularities - 
so I filed this as a suggestion for future improvement that in such a case 
other starting conditions are attempted.

Hope this helps,

Oliver



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