Fractional partial differential equations with boundary conditions |
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Authors: | Boris Baeumer Mihály Kovács Harish Sankaranarayanan |
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Institution: | 1. University of Otago, New Zealand;2. Chalmers University of Technology, Sweden;3. Michigan State University, USA |
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Abstract: | We identify the stochastic processes associated with one-sided fractional partial differential equations on a bounded domain with various boundary conditions. This is essential for modelling using spatial fractional derivatives. We show well-posedness of the associated Cauchy problems in and . In order to do so we develop a new method of embedding finite state Markov processes into Feller processes on bounded domains and then show convergence of the respective Feller processes. This also gives a numerical approximation of the solution. The proof of well-posedness closes a gap in many numerical algorithm articles approximating solutions to fractional differential equations that use the Lax–Richtmyer Equivalence Theorem to prove convergence without checking well-posedness. |
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Keywords: | Nonlocal operators Fractional differential equations Stable processes Reflected stable processes Feller processes |
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