FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2007, VOLUME 13, NUMBER 1, PAGES 235-245

**The nonlinear diffusion equation in cylindrical coordinates**

A. M. Shermenev

Abstract

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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 $W\text{'}(r)$.
This gives a series of solutions to the nonlinear diffusion
equation; these are found with the same accuracy as the equation is
derived.

Location: http://mech.math.msu.su/~fpm/eng/k07/k071/k07115h.htm

Last modified: December 21, 2006