$$-K=\nu \frac{1}{r}\frac{\partial u}{\partial r}-a\varepsilon^{2}\frac{\partial u}{\partial r}\left ( 2\frac{\partial^2 u}{\partial r^2}+\frac{1}{r}\frac{\partial u }{\partial r} \right )$$
$$u(r=R)=0; \frac{\partial u}{\partial r}\left [ r=0 \right ] = 0 $$
$$\varepsilon \ is \ the\ mixing\ length \ and \ is \ of \ the \ order \ of \ R$$ $$\nu \ is \ the\ kinematic\ viscosity$$ u represents the velocity and is a function of r.
Everything other than r and u is a known constant.
This is Prandtl-Mixing Length for turbulent flows in a circular pipe with radius R. I really need help solving this ODE. I've tried these ways without any success: Finite difference, shooting Method, bvp4c and bvp5c on Matlab. I need to write a code such that I can adjust K in order to achieve exactly what I want.
Any help will be welcome.
$$-K=\nu \frac{1}{r}\frac{d u}{d r}-a\varepsilon^{2}\frac{d u}{d r}\left ( 2\frac{d^2 u}{d r^2}+\frac{1}{r}\frac{d u }{d r} \right )$$ You can simplify the second order ODE to a first order ODE thanks to the change of unknown function : $$U(r)=\frac{d u}{d r}\quad;\quad u(r)=\int U(r)dr+\text{constant}$$ $$-K=\nu \frac{1}{r}U-a\varepsilon^{2}U\left ( 2\frac{d U}{d r}+\frac{1}{r}U \right )$$
$$ U\frac{d U}{d r}=-\frac{1}{2r}U^2+\frac{\nu}{2a\varepsilon^{2}r} U+\frac{K}{2a\varepsilon^{2}}$$ This is an Abel's differential equation of the second kind.
Analytical solving is not always possible in terms of a finite number of standard functions. Generally very difficult even in particular cases. For example see : https://arxiv.org/ftp/arxiv/paper/1503/1503.05929.pdf
Even more, in the general case, it seems doubtfull that one can found an antiderivative of a probably complicated function $U(r)$ in order to obtain $u(r)$.