Date: Jan 8, 2013 11:39 PM
Author: Oliver Ruebenkoenig
Subject: Re: system of differential equations mathematica help

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