Solving nonlinear singular differential equations

[wasi-uz-zaman]

TL;DR Summary

how to tackle singularity in the differential equations

hi, i am going through differential equations which are nonlinear and singular - like Lane-Emden equation etc.
my question is how to tackle singularity - while coding.
regards

 

[pasmith]

What method are you using to solve these equations?

If you solve $$\frac{1}{r^{n-1}} \frac{d}{dr}\left(r^{n-1}\frac{d\theta}{dr}\right) = -\theta^m$$ for $n = 2$ or $3$ subject to $\theta(0) =1$ and $\theta'(0) = 0$ then there is no difficulty: The singular point is at the origin, but you aren't enforcing the ODE there because $\theta$ and $\theta'$ are fixed by the boundary conditions.

 

[fresh_42]

I like to think of the solutions of an ODE as a vector field. @pasmith's answer means to avoid singularities by choosing initial values, i.e. trajectories which lead you through the field without encountering singularities.

If you still consider the entire field, then there are certain types of singularities: attractors, repellers, or intersecting trajectories, maybe even isolated points. Such a classification allows handling different types of singularities differently. It's always the same: what is the situation and what is the goal?

If you approach it numerically, then you have to consider how robust a solution is. This means if you vary the initial values, how much do the trajectories vary? Again a matter of the given situation and goal.

 

[PAllen]

Yes, I remember the idea of solutions as a vector field being a revelation for me. I was fortunate that this notion was introduced in my initial ODE course.