Can we approximate the solution to a singular IVP on a domain that extends past a known singularity?
As an example, consider the problem
y' = y^2, y(0) = c > 1,
on the interval [0,1]. The solution is given by
y(x) = c / (1 - cx), x in [0,1] and x =/= 1/c,
and has a singularity at 1/c. Is there a finite method that approximates the solution on [0,1] and handles the singularity in a reasonable way?
what is the need for numerical solutions when you have the analytic solutions??
Thank you for the response.
I understand how to define a finite difference scheme and how to solve it up to the singularity. I am curious if there is a method of solving the finite problem on an interval that contains the singular point.
I do not expect a time-marching approach to be successful.
"what is the need for numerical solutions when you have the analytic solutions??"
The example problem is used to illustrate the primary goal: Find a numerical method that will approximate a singular solution to an IVP on a domain that extends beyond the singularity. The purpose is to apply such a method to problems where an analytic solution is not known.
Any numerical method that is based on Taylor expansion such as Euler and Runge Kutta solvers can not handle your problem since all these methods are derived under the assumption the solution is smooth and of course bounded in the interval.
But there exists some methods based on pade approximations, rational approximation of the solution. These kinds of methods can represent the singularity and go beyond the singularity. I recommend the paper " A numerical methodology for the painleve equation, journal of computational physics, 2011" by Fornberg and Weideman
See my paper though in Russian О частных случаях динамики вращения твердого тела вокруг неглавной центральной оси инерции при сухом трении в опорах On some particular cases of the rotational dynamics of a rigid body around central but non-principal axis of inertia under action of dry friction. I solved there numerically the IVPs with discontinuous right side different using Maple procedures. Surely some methods could not extrapolate the solution some did. Please try different scehemes rosenbrock, rgk. rk45 an so on
Realise that integration through such a singularity relies on you making additional modelling assumptions over that which would have given the DE in the first place. Be sure that any assumptions you make are appropriate for the application.
However, the most obvious assumption is that you want to 'analytically' continue the solution past the singularity. What does 'analytically' mean? Usually we implicitly mean that it has a natural continuation into the complex plane. So there is your answer: get past the singularity by taking a detour into complex x, go around the singularity, and then back to the real line.
Of course you then have to resolve what you want if you go around the singularity both sides of the singularity and get a different answer back on the real line. Example, solve dy/dx=1/(1-x) with solution log(1-x) is multivalued for x>1.
You can use one of the semi-analytical methods like Homotopy analysis method
You can check the following reference:
Applied Numerical Mathematics
Volume 67, May 2013, Pages 204–219
"A globally adaptive explicit numerical method for exploding systems of ordinary differential equations"
by Nabil R. Nassif, Noha Makhoul-Karam and Jocelyne Erhel
It treats blowing up solutions to systems of ODE that include your example.
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