Re: Re: NDSolve help needed for nonlinear eq.

• To: mathgroup at smc.vnet.net
• Subject: [mg26045] Re: [mg26033] Re: NDSolve help needed for nonlinear eq.
• From: Reza Malek-Madani <research at usna.edu>
• Date: Sat, 18 Nov 2000 23:08:00 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```Dear Bill:  I don't know how to solve nonlinear boundary value problems in
general but I have had some success with BVPs that appear in boundary
layer theory in fluid dynamics. The following program finds the solution
to
y1'=y2, y2'=y3, y3'=-1/2 y1 y3; y1(0) = y2(0) = 0, y2(10) = 1;

My program uses NDSolve with FindRoot to implement a simple shooting
technique. (I have a few other examples of the shooting method and the
spectral method on my homepage at http://web.usna.navy.mil/~rmm).

****************************
\$TextStyle = {FontFamily -> "Times", FontSize -> 14};
b1 = 1;
b2 = 0.4; b3 = 2.4; (* Starting guesses for FindRoot *)
f[y1_, y2_, y3_] = y2; g[y1_, y2_, y3_] = y3;
h[y1_, y2_, y3_] = -1/2 y1 y3;
a = 10;
sol[b_] := NDSolve[{y1'[x] == f[y1[x], y2[x], y3[x]],
y2'[x] == g[y1[x], y2[x], y3[x]],
y3'[x] == h[y1[x], y2[x], y3[x]],
y1[0] == 0, y2[0] == 0, y3[0] == b},
{y1, y2, y3}, {x, 0, a}];
F[b_] := Module[{shoot}, shoot = sol[b];
out1 = First[Evaluate[y2[x] /. shoot /. x -> a]]];
shoot1 = FindRoot[F[b] - b1, {b, b2, b3}];
solution = sol[b /. shoot1]
graph = Plot[y1'[x] /. solution, {x, 0, a},
PlotLabel -> llabel["Graph of y_2"], PlotRange -> All];
******************************

Reza.

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On Fri, 17 Nov 2000, Bill Bertram wrote:

>
> Jens-Peer Kuska <kuska at informatik.uni-leipzig.de> wrote in message
> news:qDKQ5.1279\$b16.93558 at ralph.vnet.net...
> > Hi,
> >
> > did you have a differential equation too ?
> > And not only the boundary conditions. Otherwise
> > A=0, B=const is your solution. You realy need Mathematica
> > for that ?
>
>
>
> The original poster stated quite clearly that he was trying to solve two
> non-linear ordinary DEs. It doesn't matter what these DEs are, Mathematica
> V4 doesn't solve non-linear boudary value problems.
>
> Cheers,
>
>       Bill
>
>
>
>

```

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