Journal
FRACTIONAL CALCULUS AND APPLIED ANALYSIS
Volume 22, Issue 3, Pages 767-794Publisher
WALTER DE GRUYTER GMBH
DOI: 10.1515/fca-2019-0042
Keywords
fractional Laplacian; Caputo derivative; evolution problems
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This work introduces and analyzes a finite element scheme for evolution problems involving fractional-in-time and in-space differentiation operators up to order two. The left-sided fractional-order derivative in time we consider is employed to represent memory effects, while a nonlocal differentiation operator in space accounts for long-range dispersion processes. We discuss well-posedness and obtain regularity estimates for the evolution problems under consideration. The discrete scheme we develop is based on piecewise linear elements for the space variable and a convolution quadrature for the time component. We illustrate the method's performance with numerical experiments in one-and two-dimensional domains.
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