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Finite difference methods and their physical constraints for the fractional klein‐kramers equation
Author(s) -
Deng Weihua,
Li Can
Publication year - 2011
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.20596
Subject(s) - mathematics , formalism (music) , klein–gordon equation , stability (learning theory) , partial differential equation , finite difference , convergence (economics) , space (punctuation) , mathematical analysis , physics , quantum mechanics , nonlinear system , computer science , art , musical , machine learning , operating system , economics , visual arts , economic growth
Incorporating subdiffusive mechanisms into the Klein‐Kramers formalism leads to the fractional Klein‐Kramers equation. Then, the equation can effectively describe subdiffusion in the presence of an external force field in the phase space. This article presents the finite difference methods for numerically solving the fractional Klein‐Kramers equation and does the detailed stability and error analyses. The stability condition, mv R 2 β ≤ 16, shows the ratio between the kinetic energy of the particle and the temperature of the fluid can not be too large, which well agrees with the physical property of the subdiffusive particle, we call it “physical constraint.” The numerical examples are provided to verify the theoretical results on rate of convergence. Moreover, we simulate the fractional Klein‐Kramers dynamics and the simulation results further confirm the effectiveness of our numerical schemes. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1561–1583, 2010