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MathGroup Archive 2000

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Re: Question: nonlinear differential equation with boundary conditions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg26491] Re: [mg26429] Question: nonlinear differential equation with boundary conditions
  • From: Reza Malek-Madani <research at usna.edu>
  • Date: Thu, 28 Dec 2000 02:52:29 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

Dear Ronald:  I am not aware that Mathematica has an internal function
that can handle nonlinear BVPs--as you know, nonlinear BVPs sometimes have
no solutions and sometimes multiple solutions, which makes it difficult to
write a simple program to take care of all cases. But when there is a
unique (and stable) solution, a simple shooting method does the job. Here
is how I would approach your BVP (I am not sure if the values I chose are
typical for your case):

e=1.2;a=b=c=d=1;

F[z_] := Module[{sol} , 
          sol = NDSolve[{a y''[x] - b y[x]^2 + c Exp[-d x]==0,
	              y[0] == z, y'[0]==0}, y, {x, 0, e}];
          output = First[Evaluate[y'[x] /. sol/. x -> e]]]
		
shoot=FindRoot[F[z], {z, 1, 0.9}];

sol = NDSolve[{a y''[x] - b y[x]^2 + c Exp[-d x]==0,
	 y[0] == z/. shoot, y'[0]==0}, y, {x, 0, e}];

Plot[Evaluate[y[x]/. sol], {x,0,e}]

Regards,

Reza.




-------------------------------------------------------------------------
Reza Malek-Madani               Director of Research
Research Office, MS 10m         Phone: 410-293-2504 (FAX -2507)
589 McNair Road                 DSN:      281-2504
U.S. Naval Academy              Nimitz Room 17 in ERC
Annapolis MD 21402-5031         Email: research at usna.edu

--------------------------------------------------------------------------


On Wed, 20 Dec 2000, Ronald Sastrawan wrote:

> Hello!
> Sorry, in my last post there was a mistake in my equation.
> (There should be a y^2 in the equation). The real question was:
> 
> I encountered a problem, trying to numerically solve a differential
> equation.
> My equation looks like:
> 
> A y''[x] - B (y[x])^2 + 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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