       Re: Coupled second-order nonlinear ODEs

• To: mathgroup at smc.vnet.net
• Subject: [mg90504] Re: Coupled second-order nonlinear ODEs
• From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
• Date: Fri, 11 Jul 2008 02:03:13 -0400 (EDT)
• Organization: The Open University, Milton Keynes, UK
• References: <g54olq\$f15\$1@smc.vnet.net>

```dkjk at bigpond.net.au wrote:

> I'm trying to solve the following system
>
> \[Lambda]1 = 1;
> \[Lambda]2 = 1;
> M = 1;
> v = 1;
> h = 1;
> m = 20;
> s = NDSolve[{-1/2 (-M^2 + h \[Eta][y]^2) \[CurlyPhi][y] -
>      2 \[Lambda]1 \[CurlyPhi][y]^3 + \[CurlyPhi]''[y] ==
>     0, -4 \[Eta][y] (-v^2 + \[Eta][y]^2) \[Lambda]2 -
>      h \[Eta][y] \[CurlyPhi][y]^2 + \[Eta]''[y] == 0, \[Eta] ==
>     0, \[Eta]' == 1, \[CurlyPhi] == 1, \[CurlyPhi]' ==
>     0}, {\[Eta], \[CurlyPhi]}, {y, -m, m}, WorkingPrecision -> 20]
> Plot[\[CurlyPhi][y] /. s, {y, -m, m}]
> Plot[\[Eta][y] /. s, {y, -m, m}]
>
> but keep running into the error ``step size is effectively zero;
> singularity or stiff system suspected''.  I tried increasing
> is able to solve this?

Your system is really stiff, I mean *really* stiff, on the given range
for y. Do you need m as large as twenty? Everything is fine up to m
about 1.4. Setting m = 7/5, we do not need to add any options to NDSolve
and it does not return any warning message.

\[Lambda]1 = 1; \[Lambda]2 = 1; M = 1; v = 1; h = 1; m = 7/5;

s = NDSolve[{-1/2 (-M^2 + h \[Eta][y]^2) \[CurlyPhi][y] -
2 \[Lambda]1 \[CurlyPhi][y]^3 + \[CurlyPhi]''[y] ==
0, -4 \[Eta][y] (-v^2 + \[Eta][y]^2) \[Lambda]2 -
h \[Eta][y] \[CurlyPhi][y]^2 + \[Eta]''[y] == 0, \[Eta] ==
0, \[Eta]' == 1, \[CurlyPhi] == 1, \[CurlyPhi]' ==
0}, {\[Eta], \[CurlyPhi]}, {y, -m, m}]

Plot[\[CurlyPhi][y] /. s, {y, -m, m}]

Plot[\[Eta][y] /. s, {y, -m, m}]

Also, you may want to explore some of the numerous solvers available for
NDSolve. See,

http://reference.wolfram.com/mathematica/tutorial/NDSolveStiffnessSwitching.html

http://reference.wolfram.com/mathematica/tutorial/NDSolveOverview.html

HTH,
-- Jean-Marc

```

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