Fractional Integrals and Derivatives: Mapping Properties

Rafeiro, Humberto

doi:10.1515/fca-2016-0032

Cited by 28 publications

(4 citation statements)

References 92 publications

(82 reference statements)

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“…According to the definition of the fractional derivative [28], Equation (24) can be written as Q = ∂ 1/2 C ∂t 1/2 . If no radioactive decay occurs (i.e., Λ = 0) but diffusion is described by the generalized Fick's law (3) (i.e., 0 < α < 1 (0 < β < 1/2)), the mass fluxes can be given as…”

Section: Analytical Solutionmentioning

confidence: 99%

Mathematical Modeling of Non-Fickian Diffusional Mass Exchange of Radioactive Contaminants in Geological Disposal Formations

Suzuki

Fomin

Чугунов

et al. 2018

Water

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Abstract:Deep geological repositories for nuclear wastes consist of both engineered and natural geologic barriers to isolate the radioactive material from the human environment. Inappropriate repositories of nuclear waste would cause severe contamination to nearby aquifers. In this complex environment, mass transport of radioactive contaminants displays anomalous behaviors and often produces power-law tails in breakthrough curves due to spatial heterogeneities in fractured rocks, velocity dispersion, adsorption, and decay of contaminants, which requires more sophisticated models beyond the typical advection-dispersion equation. In this paper, accounting for the mass exchange between a fracture and a porous matrix of complex geometry, the universal equation of mass transport within a fracture is derived. This equation represents the generalization of the previously used models and accounts for anomalous mass exchange between a fracture and porous blocks through the introduction of the integral term of convolution type and fractional derivatives. This equation can be applied for the variety of processes taking place in the complex fractured porous medium, including the transport of radioactive elements. The Laplace transform method was used to obtain the solution of the fractional diffusion equation with a time-dependent source of radioactive contaminant.

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Section: Analytical Solutionmentioning

confidence: 99%

Mathematical Modeling of Non-Fickian Diffusional Mass Exchange of Radioactive Contaminants in Geological Disposal Formations

Suzuki

Fomin

Чугунов

et al. 2018

Water

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“…Riesz potentials pervade the wider mathematical context of equilibrium measures, with many results in Riesz potential theory going back to the original study of these constructions by the eponymous M. Riesz [56,57]. Some standard surveys which include modern and historical aspects are [55,34,66,1].…”

Section: Introductionmentioning

confidence: 99%

Computation of Power Law Equilibrium Measures on Balls of Arbitrary Dimension

Gutleb¹,

Carrillo²,

Olver³

2021

Preprint

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We present a numerical approach for computing attractive-repulsive power law equilibrium measures in arbitrary dimension. We prove new recurrence relationships for radial Jacobi polynomials on d-dimensional ball domains, providing a substantial generalization of the work started in [27] for the one-dimensional case based on recurrence relationships of Riesz potentials on arbitrary dimensional balls. Among the attractive features of the numerical method are good efficiency due to recursively generated banded and approximately banded Riesz potential operators and computational complexity independent of the dimension d, in stark contrast to the widely used particle swarm simulation approaches for these problems which scale catastrophically with the dimension. We present several numerical experiments to showcase the accuracy and applicability of the method and discuss how our method compares with alternative numerical approaches and conjectured analytical solutions which exist for certain special cases. Finally, we discuss how our method can be used to explore the analytically poorly understood gap formation boundary to spherical shell support.

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“…1203-1224 , DOI: 10.1515/fca-2019-0064 such spaces. We refer, for instance, to the books [15], [26] and [28] and papers [3], [4], [5], [6], [9], [12], [16], [20], [22] and references therein. In this paper we consider weighted generalized Morrey spaces on underlying quasi-metric measure sets.…”

Section: Introductionmentioning

confidence: 99%

Embeddings of Weighted Generalized Morrey Spaces Into Lebesgue Spaces on Fractal Sets

Samko

2019

FCAA

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We study embeddings of weighted local and consequently global generalized Morrey spaces defined on a quasi-metric measure set (X, d, μ) of general nature which may be unbounded, into Lebesgue spaces Ls(X), 1 ≤ s ≤ p < ∞. The main motivation for obtaining such an embedding is to have an embedding of non-separable Morrey space into a separable space. In the general setting of quasi-metric measure spaces and arbitrary weights we give a sufficient condition for such an embedding. In the case of radial weights related to the center of local Morrey space, we obtain an effective sufficient condition in terms of (fractional in general) upper Ahlfors dimensions of the set X. In the case of radial weights we also obtain necessary conditions for such embeddings of local and global Morrey spaces, with the use of (fractional in general) lower and upper Ahlfors dimensions. In the case of power-logarithmic-type weights we obtain a criterion for such embeddings when these dimensions coincide.

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Fractional Integrals and Derivatives: Mapping Properties

Cited by 28 publications

References 92 publications

Mathematical Modeling of Non-Fickian Diffusional Mass Exchange of Radioactive Contaminants in Geological Disposal Formations

Mathematical Modeling of Non-Fickian Diffusional Mass Exchange of Radioactive Contaminants in Geological Disposal Formations

Computation of Power Law Equilibrium Measures on Balls of Arbitrary Dimension

Embeddings of Weighted Generalized Morrey Spaces Into Lebesgue Spaces on Fractal Sets

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