MaplePrimes - answers and comments on Question, singularitys

By series

hirnyk — Fri, 07 May 2010 07:27:15 Z

Solve the system by series, replacing

> sol4 := dsolve([eqn12, eqn13, f(0) = 1, e(0) = 1], numeric, maxfun = 1000000000);

>sol4 := dsolve({eqn12, eqn13}union {f(0) = 1, e(0) = 1},{e(z),f(z)},type='series');

It works. Next, you might follow

http://www.mapleprimes.com/forum/odewpowerseriesplotting in this forum.

O

Robert Israel — Fri, 07 May 2010 09:39:00 Z

First of all, you shouldn't use O (which has a special meaning in terms of series, and causes a "too many levels of recursion" error). Try o instead.

Now it looks to me like there is a real singularity there: f(z)^4*(e(z)^2-.27*(1+z)^3)^2 hits 0.0081 at approximately z = .76663139, after which you'll be taking the square root of a negative quantity. If you want any chance of a sensible continuation past that singularity, perhaps you'll want to insert an absolute value. Thus

> eqn10 := diff(e(z)^2, z) = ((3/2)*e(z)^2-3*(n^2/f(z)^2)^2/(2*(e(z)^2-o*(1+z)^(3*g)))
     +(1/2)*((g-1)*3)*o*(1+z)^(3*g))/(2*(1+z));
  eqn11 := diff(f(z), z) = sqrt(abs(1-(n^2/f(z)^2)^2/(e(z)^2-o*(1+z)^(3*g))^2))/(e(z)*(1+z));
  eqn12 := subs(o = .27, g = 1, n = .3, eqn10);
  eqn13 := subs(o = .27, g = 1, n = .3, eqn11);
  sol4 := dsolve([eqn12, eqn13, f(0) = 1, e(0) = 1], numeric,method=classical[rk4],stepsize=0.00001,maxfun=10^6);
  plots[odeplot](sol4,[[z,e(z),colour=red],[z,f(z),colour=blue]],z=0 .. 2,numpoints=1000);

As you can see, something strange happens to f(z).
It turns out that (e(z)^2-.27*(1+z)^3) hits 0, resulting in df/dz going to infinity, at around z = .7867866833. Because of the square root, I think the singularity should still be integrable, so f won't go to infinity, but it's going to be very hard for numerical methods to accurately reflect what does happen.

Series and singularities

alec — Fri, 07 May 2010 17:21:36 Z

If there is a singularity, the series solution will have the radius of convergence equal or less than the distance from the center of the series to the singularity (perhaps, there are exceptions, but that's what usually happen), so it can not be used for continuation after the singularity. That can be cleary seen both here and in the example that you cited.

Alec