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Re: Tricky differential equation

  • To: mathgroup at smc.vnet.net
  • Subject: [mg41495] Re: Tricky differential equation
  • From: "Dr. Wolfgang Hintze" <weh at snafu.de>
  • Date: Thu, 22 May 2003 06:57:59 -0400 (EDT)
  • References: <bafqej$6rq$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Luiz,

not a solution but just a hint:

Make the differential equation dimensionless by setting

r = t/Sqrt[p]
y = 2x

leading to

(*) y'' + y'/t + (1-1/t^2) Sin[y] ==0

Now we can see that

(1) for y->0 we have Sin[y] -> y, the dgl is the Bessel-Dgl with the 
solution J_1[t]

(2) for t>>1 (*) is the pendulum equation

(**) y'' + Sin[y] ==0

with a solution in terms of elliptic integrals.

Hope this helps.

Regards,
Wolfgang

Luiz Melo wrote:

> Hello everyone,
> 
> I'm trying to find the numerical solution of the following
> differential equation (r is the independent variable):
> 
> x''[r] + 1/r x'[r] + (p - 1/r^2)*Sin[x[r]]*Cos[x[r]] == 0 ,
> 
> with boundary conditions: x'[1] == 0 , and x[0] -> "has to be finite",
> 
> but I'm having at least two problems:
> 
> 1) I don't know how to submit the BC "finite" to Mathematica;
> 2) The coefficient p is about 10^4. For this reason, it seems
> that the Runge-Kutta method usually used for numerical
> integration of ordinary differential equations turns out
> to be unsuccessfull in our case. Do we need a special method
> to solve this?
> 
> The solution of this equation gives the internal magnetic structure
> of a cylinder. The function x[r] is the angle between the
> magnetization and the axial direction, and it depends on the radial
> direction, r.
> 
> I would like to plot the Cossine of the result as a function of r
> (which varies from 0 to 1), for several values of p.
> 
> Any help will be very appreciated!
> Thank you
> 
> Luiz Melo
> 
> Ecole Polytechnique de Montreal,
> Montreal, Quebec
> luiz.melo at polymtl.ca
> 
> 
> 
> 



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