Fractional Sobolev Spaces via Riemann-Liouville Derivatives

“…In a similar way as in Idczak and Walczak, 15 we introduce the fractional Sobolev space W ,p a + (Ω), specifically adapted to our problem, with the norm ‖·‖ W ,p a + (Ω) given by…”

Section: Fractional Teodorescu and Cauchy-bitsadze Operatorsmentioning

confidence: 99%

“…Another field of application consists of addressing differential equations related to flows with permeable boundaries, such as for instance dam‐fill problems which provides a further important motivation to develop three‐dimensional generalizations of harmonic and Clifford analysis tools for the fractional setting. Preceding work pointing in this direction can be found in Idczak and Walczak and Hossein et al where the fractional p ‐Laplace equation has been treated.…”

Section: Introductionmentioning

confidence: 99%

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A higher dimensional fractional Borel‐Pompeiu formula and a related hypercomplex fractional operator calculus

Ferreira

Kraußhar

Rodrigues

et al. 2019

Math Methods in App Sciences

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In this paper, we develop a fractional integro‐differential operator calculus for Clifford algebra‐valued functions. To do that, we introduce fractional analogues of the Teodorescu and Cauchy‐Bitsadze operators, and we investigate some of their mapping properties. As a main result, we prove a fractional Borel‐Pompeiu formula based on a fractional Stokes formula. This tool in hand allows us to present a Hodge‐type decomposition for the fractional Dirac operator. Our results exhibit an amazing duality relation between left and right operators and between Caputo and Riemann‐Liouville fractional derivatives. We round off this paper by presenting a direct application to the resolution of boundary value problems related to Laplace operators of fractional order.

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“…Moreover, the function p(x, t) := p 0,0 (x, t) is the first fundamental solution of (14). From now on we consider c = 1 in (12) and (13), and α = 1 in (14), (15) and (16), which implies that p(x, t) = G β n (x, t).…”

Section: Estimates Of the Fundamental Solution Of The Time-fractionalmentioning

confidence: 99%

A Time-Fractional Borel–Pompeiu Formula and a Related Hypercomplex Operator Calculus

Ferreira

Rodrigues

Vieira

2019

Complex Anal. Oper. Theory

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In this paper we develop a time-fractional operator calculus in fractional Clifford analysis. Initially we study the Lp-integrability of the fundamental solutions of the multi-dimensional time-fractional diffusion operator and the associated time-fractional parabolic Dirac operator. Then we introduce the time-fractional analogues of the Teodorescu and Cauchy-Bitsadze operators in a cylindrical domain, and we investigate their main mapping properties. As a main result, we prove a time-fractional version of the Borel-Pompeiu formula based on a time-fractional Stokes' formula. This tool in hand allows us to present a Hodge-type decomposition for the forward time-fractional parabolic Dirac operator with left Caputo fractional derivative in the time coordinate. The obtained results exhibit an interesting duality relation between forward and backward parabolic Dirac operators and Caputo and Riemann-Liouville time-fractional derivatives. We round off this paper by giving a direct application of the obtained results for solving time-fractional boundary value problems.

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Fractional Sobolev Spaces via Riemann-Liouville Derivatives

Cited by 37 publications

References 3 publications

Hartman-Wintner-Type Inequality for a Fractional Boundary Value Problem via a Fractional Derivative with respect to Another Function

Hartman-Wintner-Type Inequality for a Fractional Boundary Value Problem via a Fractional Derivative with respect to Another Function

A higher dimensional fractional Borel‐Pompeiu formula and a related hypercomplex fractional operator calculus

A Time-Fractional Borel–Pompeiu Formula and a Related Hypercomplex Operator Calculus

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