The nonlinear diffusion equation in cylindrical coordinates

Nonlinear corrections to some classical solutions of the linear diffusion equation in cylindrical coordinates are studied within quadratic approximation. When cylindrical coordinates are used, we try to find a nonlinear correction using quadratic polynomials of Bessel functions whose coefficients are Laurent polynomials of radius. This usual perturbation technique inevitably leads to a series of overdetermined systems of linear algebraic equations for the unknown coefficients (in contrast with the Cartesian coordinates). Using a computer algebra system we show that all these overdetermined systems become compatible if we formally add one function on radius $W(r)$. Solutions can be constructed as linear combinations of these quadratic polynomials of the Bessel functions and the functions $W(r)$ and $W'(r)$. This gives a series of solutions to the nonlinear diffusion equation; these are found with the same accuracy as the equation is derived.