Re: Re: Tricky differential equation
- To: mathgroup at smc.vnet.net
- Subject: [mg41512] Re: [mg41495] Re: Tricky differential equation
- From: Bobby Treat <drmajorbob+MathGroup3528 at mailblocks.com>
- Date: Fri, 23 May 2003 03:29:55 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
I got your equation (*) this way, but I'm wondering if there's an
easier way:
eq0 = x''[r] + (1/r)*x'[r] + (p - 1/r^2)*Sin[x[r]]*Cos[x[r]] == 0
eq1 = TrigReduce /@ eq0
eq2 = FullSimplify[eq1 //. {x[r] :> y[t]/2,
Derivative[n_][x][r] :> (1/2)*p^(n/2)*
Derivative[n][y][t], r -> t/Sqrt[p]}]
eq3 = (#1/p & ) /@ eq2
There must be!
Bobby
-----Original Message-----
From: Dr. Wolfgang Hintze <weh at snafu.de>
To: mathgroup at smc.vnet.net
Subject: [mg41512] [mg41495] Re: Tricky differential equation
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
>
>
>
>