On Some New Properties of the Fundamental Solution to the Multi-Dimensional Space- and Time-Fractional Diffusion-Wave Equation

Luchko, Yuri

doi:10.3390/math5040076

Cited by 24 publications

(22 citation statements)

References 33 publications

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“…The first fundamental solution of (12) satisfying the initial conditions u(x, 0) = δ(x) if 0 < β < 1, and u(x, 0) = δ(x) and ∂ t u(x, 0) = 0 if 1 < β < 2 was deduced by several authors (see e.g. [10,15,19,20]). In [10] the authors obtained the fundamental solution G β n (x, t) in the form…”

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

confidence: 99%

“…Taking into account (27), (19), and the fact that G β n (x, t) = p 0,0 (x, t), we get the following relation between C G β + and the derivatives of p 0,0 :…”

Section: The Case Of the Time-fractional Parabolic Dirac Operatormentioning

confidence: 99%

“…The multi-dimensional time-fractional diffusion equation case was studied in several papers, e.g. [3,4,10,12,13,[19][20][21]32]. In [32] the fundamental solution of this equation was deduced in terms of H-functions.…”

Section: Introductionmentioning

confidence: 99%

“…In [12,13] fractional diffusion-wave equations. In [3,4,[19][20][21] Luchko and his collaborators used the representation in terms of Mellin-Barnes type-integrals to study some properties of the multi-dimensional space-time-fractional diffusionwave equation. Explicit series representations for the fundamental solution of the diffusion-wave operator and the so-called time-fractional parabolic Dirac operator were obtained in [10], for arbitrary dimension.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

confidence: 99%

“…Taking into account (27), (19), and the fact that G β n (x, t) = p 0,0 (x, t), we get the following relation between C G β + and the derivatives of p 0,0 :…”

Section: The Case Of the Time-fractional Parabolic Dirac Operatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Ferreira

Rodrigues

Vieira

2019

Complex Anal. Oper. Theory

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“…The FDE has been treated by a number of authors (see [9,14,17,19,21]) and, among the several applications that have been studied, Mainardi [20] studied the application to the theory of linear viscoelasticity.…”

Section: Introductionmentioning

confidence: 99%

Hopf lemma for the fractional diffusion operator and its application to a fractional free-boundary problem

Roscani

2016

Journal of Mathematical Analysis and Applications

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This paper deals with a theoretical mathematical analysis of a one-dimensional-moving-boundary problem for the time-fractional diffusion equation, where the time-fractional derivative of order α ∈ (0, 1) is taken in the Caputo's sense. A generalization of the Hopf's lemma is proved, and then this result is used to prove a monotonicity property for the free-boundary when a fractional free-boundary Stefan problem is considered.Keywords: fractional diffusion equation; Caputo's derivative; moving-boundary problem; free-boudary problemwhere the fractional derivative in the Caputo's sense of arbitrary order α > 0 is given bywhere n ∈ N and Γ is the Gamma function defined by Γ(x) = ∞ 0 w x−1 e −w dw.Whereas that the one-dimensional heat equation has become the paradigm for the all-embracing study of parabolic partial differential equations, linear or nonlinear (see Cannon [4]), the FDE plays a similar role in fractional parabolic operators.

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The fundamental solution of the master equation for a jump‐diffusion Ornstein–Uhlenbeck process

Rozanova,

Krutov

2024

Mathematische Nachrichten

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An integro‐differential equation for the probability density of the generalized stochastic Ornstein–Uhlenbeck process with jump diffusion is considered for a special case of the Laplacian distribution of jumps. It is shown that for a certain ratio between the intensity of jumps and the speed of reversion, the fundamental solution can be found explicitly, as a finite sum. Alternatively, the fundamental solution can be represented as converging power series. The properties of this solution are investigated. The fundamental solution makes it possible to obtain explicit formulas for the density at each instant of time, which is important, for example, for testing numerical methods.

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On Some New Properties of the Fundamental Solution to the Multi-Dimensional Space- and Time-Fractional Diffusion-Wave Equation

Cited by 24 publications

References 33 publications

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

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

Hopf lemma for the fractional diffusion operator and its application to a fractional free-boundary problem

The fundamental solution of the master equation for a jump‐diffusion Ornstein–Uhlenbeck process

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