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The nonlinear diffusion equation in cylindrical coordinates. (English. Russian original) Zbl 1158.35380
J. Math. Sci., New York 152, No. 4, 608-615 (2008); translation from Fundam. Prikl. Mat. 13, No. 1, 235-245 (2007).
Summary: 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 $$W^{\prime}(r)$$. This gives a series of solutions to the nonlinear diffusion equation; these are found with the same accuracy as the equation is derived.
##### MSC:
 35K57 Reaction-diffusion equations 35C10 Series solutions to PDEs 35C20 Asymptotic expansions of solutions to PDEs
##### Keywords:
Bessel functions; Laurent polynomials; computer algebra
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##### References:
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