I am new to solving numerically ODES and thus it is difficult for me to judge the reliability/trustworthiness of the results that I have produced for the following problem:
I am dealing with a 2nd order ODE (I can add a non-linear term to it, too) on the interval $[0,1]$ with regular singularity at $0$ and $1$. I converted the ODE into a 1st order one and interpreted it as a IVP by setting an initial condition at $x=1$ motivated by symmetry arguments and let ODE45 integrate from $x=1$ to $x=0$. In this way I can produce solutions which is nice.
But here comes the problem: I have to use these solutions later on, e.g., I can also compute the $L^2$-norm of the solutions on the interval $[0,1]$ with respect to some measure and what I get as a result seems to depend sensitively on
(1.) the cut-off at $x=0$ (I let ODE45 run until "eps", the machine precision but I could also take smaller and bigger values)
(2.) the smallness of AbsTol and RelTol (both around eps)
(3.) the tspan size, i.e., the number of slices into which I cut the interval, like $10^6$.
Depending on how I vary the values in (1.)-(3.) I get results which differ from one another and not to a small extent. I believe that the code is correctly implemented, rather I guess the problem comes from the regular singularity which leads me to the question how trustworthy ODE45 is for such a problem. Does anybody have experiences with such a problem and can recommend what I could do to get more reliable results? Thanks!
Update:
The equation reads $2x~(1-x)~f''(x)+(3-4x)~f'(x)+a~f(x)+b~f^n(x)=0;~~ a,b\in\mathbb{R}, n\in\mathbb{N}.$
"not to a small extent" means a change in the result for the normalization from e.g $10^{-2}$ to $10^{20}$ if I solve until $x=eps*100$ vs. $x=eps$.
