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Regularity of the solution to 1-D fractional order diffusion equations

Authors: V. J. Ervin, N. Heuer and J. P. Roop
Journal: Math. Comp.
MSC (2010): Primary 65N30, 35B65, 41A10, 33C45
Published electronically: January 26, 2018
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Abstract: In this article we investigate the solution of the steady-state fractional diffusion equation on a bounded domain in $ \mathbb{R}^{1}$. The diffusion operator investigated, motivated by physical considerations, is neither the Riemann-Liouville nor the Caputo fractional diffusion operator. We determine a closed form expression for the kernel of the fractional diffusion operator which, in most cases, determines the regularity of the solution. Next we establish that the Jacobi polynomials are pseudo eigenfunctions for the fractional diffusion operator. A spectral type approximation method for the solution of the steady-state fractional diffusion equation is then proposed and studied.

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Additional Information

V. J. Ervin
Affiliation: Department of Mathematical Sciences, Clemson University, Clemson, South Carolina 29634-0975

N. Heuer
Affiliation: Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Macul, Santiago, Chile

J. P. Roop
Affiliation: Department of Mathematics, North Carolina A & T State University, Greensboro, North Carolina 27411

Keywords: Fractional diffusion equation, Jacobi polynomials, spectral method
Received by editor(s): August 14, 2016
Received by editor(s) in revised form: January 7, 2017, March 15, 2017, and May 4, 2017
Published electronically: January 26, 2018
Additional Notes: The first author was partially support by CONICYT through FONDECYT project 1150056
The second author was partially support by CONICYT through FONDECYT projects 1150056, and Anillo ACT1118 (ANANUM)
Article copyright: © Copyright 2018 American Mathematical Society

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