Re: NDSolve

*To*: mathgroup at smc.vnet.net*Subject*: [mg21557] Re: NDSolve*From*: Alois Steindl <asteindl at mch2pc28.tuwien.ac.at>*Date*: Sat, 15 Jan 2000 02:04:01 -0500 (EST)*Organization*: Inst. f. Mechanics II, TU Vienna*References*: <85esu5$pca@smc.vnet.net> <85mn1s$27t@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Paul Abbott <paul at physics.uwa.edu.au> writes: > Klaus Wallmann wrote: > > > Using MATHEMATICA Version 4.0 on a Pentium III PC with Windows NT Version > > 4, I tried to solve the following ordinary homogeneous non-linear second > > order differential equation with variable coefficients: > > > > y''[t] + p[t] y'[t] + q[t] y[t]/(y[t]+K) = 0 > > > > with boundary conditions > > > > y[10] = 29 > > > > y'[6000] = 0 > > > > and > > > > K = 1 > > > > p[t] = ((-1.15301 (-3 + ln[(0.769 + 0.102 exp(-0.036 t))^(2)]) + 0.2307 > > exp(+0.036 t) (-1 + ln[(0.769 + 0.102 exp(-0.036 t))^(2)])^(2))/(314 (0.871 > > + 0.769 (-1 + exp(+0.036 t) (-1 + ln[(0.769 + 0.102 exp(-0.036 t))^(2)])) > > > > q[t] = -[0.00203318 (0.231 - 0.102 exp(-0.036 t)) (0.102 (0.697676 - > > exp(-0.036 t))+ 0.008316 (93.2 + t))^(-1.142) (1 - ln[(0.769 + 0.102 > > exp(-0.036 t))^(2)])]/(0.769 + 0.102 exp (-0.036 t)) > > > > I tried NDSolve to solve this equation but I did not succeed. > > > > Is it possible to solve this equation or similar differential equations > > using the numerical procedures implemented in MATHEMATICA? > > The following Notebook outlines one possible approach. > > Cheers, > Paul > > ............. > In other words, for the boundary values you have supplied, this \ > problem is not well conditioned. \ > \>", "Text"] > } > ] Hello, I just have to be a little bit nit-picking: I would guess that the boundary value problem itself is well conditioned, (If you change the BCs to y(10) = 29+eps1 y'(6000) = eps2, then the solution should change only by the order of eps. ) But the solution method by shooting is very ill-conditioned. This behaviour is to be expected close to saddle points. I think finite difference codes or collocation methods like Colnew shouldn't encounter severe problems. A second thought about this problem: I would guess (Mr. Wallmann didn't yet react to my query) that the point x=6000 originally means x=\infty. Maybe the real task is to find the stable solution to the problem with the initial condition y(10)=29. In this case it seems one could use a much smaller domain of integration and replace the far boundary condition by some asymptotic boundary condition, which requires the endpoint to lie on the stable manifold. In any case I would use Colnew to solve the problem. Best regards Alois