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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
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