Diffusion Wave Initiation Problem for a Nonlinear Parabolic System in the Case of Spherical and Cylindrical Symmetry
A. L. Kazakov1, L. F. Spevak2
1Matrosov Institute of System Dynamics and Control Theory, Siberian Branch, Russian Academy of Sciences, Irkutsk, Russia
2Institute of Engineering Science, Ural Branch, Russian Academy of Sciences, Yekaterinburg, Russia
Keywords: reaction-diffusion system, diffusion wave, existence and uniqueness theorem, exact solution, numerical method
Abstract
For a nonlinear parabolic reaction - diffusion system, solutions are constructed and investigated that have the form of a diffusion wave propagating in a medium at rest with a finite velocity. For the first time, for cases of spherical and cylindrical symmetry, the problem of initiating a diffusion wave by boundary conditions specified on a sphere (circular cylindrical surface) is considered. A theorem of existence and uniqueness of a solution in the class of analytic functions is proved. An exact solution is constructed, which is presented in the form of explicit formulas. A step-by-step iterative algorithm is proposed, based on the collocation method and expansion in radial basis functions. Numerical calculations are performed, for the verification of the results of which the exact solution is used.
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