Geodesic
The nonlinear diffusion equation in cylindrical coordinates
Fundamentalʹnaâ i prikladnaâ matematika, Tome 13 (2007) no. 1, pp. 235-245.

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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 $W'(r)$. 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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A. M. Shermenev. The nonlinear diffusion equation in cylindrical coordinates. Fundamentalʹnaâ i prikladnaâ matematika, Tome 13 (2007) no. 1, pp. 235-245. http://geodesic.mathdoc.fr/item/FPM_2007_13_1_a14/

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