The nonlinear diffusion equation in cylindrical coordinates

A. M. Shermenev

Fundamentalʹnaâ i prikladnaâ matematika, Tome 13 (2007) no. 1, pp. 235-245

Résumé

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.