Re: Tricky differential equation
- To: mathgroup at smc.vnet.net
- Subject: [mg41481] Re: Tricky differential equation
- From: Alois Steindl <Alois.Steindl at jet2web.cc>
- Date: Thu, 22 May 2003 06:53:12 -0400 (EDT)
- Organization: Inst. f. Mechanics II, TU Vienna
- References: <bafqej$6rq$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Luiz Melo <luiz.melo at polymtl.ca> writes: > 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", > Hello, what's about the constant solutions x(r) = integer multiple of pi/2 ? There might of course also be other solutions. > but I'm having at least two problems: > > 1) I don't know how to submit the BC "finite" to Mathematica; You are considering a singular differential equation, which could have bounded and unbounded solutions. If you linearize your equations around the even multiples of pi/2, you have to replace the nonlinear expression sin(x)*cos(x) by x and the resulting linear system has one family of bounded solutions. (You should now really have a closer look at Bessel functions) All these bounded solutions satisfy x(0)=0; that gives you the corresponding boundary condition. (This statement applies only to the linear system; for your equations you should obtain x(0) = k*pi/2.) If you linearize your system at the odd multiples of pi/2, you get oscillating solutions. > 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? > I would think that p isn't the problem, but the singularity at r=0. For such problems I would use collocation methods (after some analytical preparation). You could also play around by starting at r=1 with some value of x(1) and shoot backwards to some small value of r. But I would really suggest to search for articles about singular differential equations (I know that Eva Weinmueller did some investigations in that area.) Alois -- Alois Steindl, Tel.: +43 (1) 58801 / 32558 Inst. for Mechanics II, Fax.: +43 (1) 58801 / 32598 Vienna University of Technology, A-1040 Wiedner Hauptstr. 8-10