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.

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Last modified: December 21, 2006