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MathGroup Archive 2000

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Re: Question: numerical solution of nonlinear differential equation

  • To: mathgroup at smc.vnet.net
  • Subject: [mg26397] Re: Question: numerical solution of nonlinear differential equation
  • From: Alexandra Milik <amilik1 at compuserve.com>
  • Date: Wed, 20 Dec 2000 00:21:25 -0500 (EST)
  • References: <91f7n8$556@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Hi Ronald,

your boundary value problem has no solution, to see this let

z = Exp[-Dx]
y' = w

which yields the linear system of first order differential equations

w' = B/A w -C/A z
z' = -D z
y' = w

with boundary values w(0) = w(E) =0.
The third equation decouples so we are left with a planar linear system
of the form u'=Ju with u=[w z] and matrix
     [ B/A   -C/A ]
J=   |            |
     [   0     -D ]

of which the eigenvalus are B/A and -D, this implies that the trivial
solution w=z=0 is a saddle for positive B/A and D and a node
otherwise.
In case of the node all solutions are lines through the origin of the
(wz)-plane,
in case of the saddle there are two lines through the origin and
hyperbolas in between. None of this solutions crosses the z-achses
twice. Thus there exists no solution to your BVP.

Hope this helps
Alex

Ronald Sastrawan schrieb:

> Hello !
>
> I encountered a problem, trying to numerically solve a differential
> equation.
> My equation looks like:
>
> A y''[x] - B y[x]' + C Exp[-Dx] == 0
> with boundary conditions: y'[0]==0 , y'[E]==0
>
> All constants A to E are known.
>
> Mathematica complains, that the equation is not linear. But in the
> online documentation I saw many examples of nonlinear differential
> equations, which all work fine. What is the difference between the
> examples and my equation ? And is there a possibility to NDSolve my
> equation ?
>
> Any hint on this would be of great help to me.
>
> Thanks a lot,
>
> Ronald
>
> --
> Ronald Sastrawan
>
> Freiburg Materials Research Center
> Stefan-Meier-Str. 21
> D-79104 Freiburg
> Germany
> Tel: ++49/761/203-4802
> FAX: ++49/761/203-4801
> EMAIL: sastra at fmf.uni-freiburg.de
> http://www.fmf.uni-freiburg.de/~biomed/FSZ/forschung-FSZ.html



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