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A numerical method for generalized Fokker-Planck equations. (English) Zbl 1281.82022

Li, Jichun (ed.) et al., Recent advances in scientific computing and applications. Eighth international conference on scientific computing and applications, University of Nevada, Las Vegas, NV, USA, April 1–4, 2012. Proceedings. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-8737-0/hbk; 978-0-8218-9501-6/ebook). Contemporary Mathematics 586, 171-179 (2013).
Summary: Generalized Fokker-Planck (GFP) equations have been employed to approximate the radiative transfer equation in applications of highly forward peaked biological media. In this paper, we discuss a numerical method for solving GFP equations. The numerical method is based on a variational formulation involving even and odd components of the solution. We show the well-posedness of the variational formulation and develop a Galerkin method, where spherical harmonics are used for the angular approximation and finite elements are used for spatial discretization. An iteration procedure is introduced to solve the problem and its convergence is shown.
For the entire collection see [Zbl 1264.65002].

MSC:

82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35Q84 Fokker-Planck equations
35R09 Integro-partial differential equations
45K05 Integro-partial differential equations
92C55 Biomedical imaging and signal processing
35Q92 PDEs in connection with biology, chemistry and other natural sciences
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