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 > > > >