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Re: NDSolve

• To: mathgroup at smc.vnet.net
• Subject: [mg21557] Re: NDSolve
• From: Alois Steindl <asteindl at mch2pc28.tuwien.ac.at>
• Date: Sat, 15 Jan 2000 02:04:01 -0500 (EST)
• Organization: Inst. f. Mechanics II, TU Vienna
• References: <85esu5$pca@smc.vnet.net> <85mn1s$27t@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

Paul Abbott <paul at physics.uwa.edu.au> writes:

> Klaus Wallmann wrote:
> 
> > Using MATHEMATICA Version 4.0 on a Pentium III PC with Windows NT Version
> > 4, I tried to solve the following ordinary homogeneous non-linear second
> > order differential equation with variable coefficients:
> >
> > y''[t] + p[t] y'[t] + q[t] y[t]/(y[t]+K)  = 0
> >
> > with boundary conditions
> >
> > y[10] = 29
> >
> > y'[6000] = 0
> >
> > and
> >
> > K = 1
> >
> > p[t] = ((-1.15301 (-3 + ln[(0.769 + 0.102 exp(-0.036 t))^(2)]) + 0.2307
> > exp(+0.036 t) (-1 + ln[(0.769 + 0.102 exp(-0.036 t))^(2)])^(2))/(314 (0.871
> > + 0.769 (-1 + exp(+0.036 t) (-1 + ln[(0.769 + 0.102 exp(-0.036 t))^(2)]))
> >
> > q[t] = -[0.00203318 (0.231 - 0.102 exp(-0.036 t)) (0.102 (0.697676 -
> > exp(-0.036 t))+ 0.008316 (93.2 + t))^(-1.142) (1 - ln[(0.769 + 0.102
> > exp(-0.036 t))^(2)])]/(0.769 + 0.102 exp (-0.036 t))
> >
> > I tried NDSolve to solve this equation but I did not succeed.
> >
> > Is it possible to solve this equation or similar differential equations
> > using the numerical procedures implemented in MATHEMATICA?
> 
> The following Notebook outlines one possible approach.
> 
> Cheers,
>     Paul
> 
> .............
> In other words, for the boundary values you have supplied, this \
> problem is not well conditioned. \
> \>", "Text"]
> }
> ]

Hello,
I just have to be a little bit nit-picking: I would guess that the
boundary value problem itself is well conditioned, 
(If you change the BCs to 
y(10) = 29+eps1
y'(6000) = eps2, then the solution should change only by the order of
eps. )
But the solution method by shooting is very ill-conditioned. 
This behaviour is to be expected close to saddle points.  

I think finite difference codes or collocation methods like Colnew
shouldn't encounter severe problems.  

A second thought about this problem: I would guess (Mr. Wallmann
didn't yet react to my query) that the point x=6000 originally means
x=\infty. Maybe the real task is to find the stable solution to the
problem with the initial condition y(10)=29. In this case it seems one
could use a much smaller domain of integration and replace the far
boundary condition by some asymptotic boundary condition, which
requires the endpoint to lie on the stable manifold. 
In any case I would use Colnew to solve the problem.

Best regards
Alois