On the Laplace integral representation of multivariate Mittag‐Leffler functions in anomalous relaxation

Nigmatullin, Raoul R.; Khamzin, A. A.; Băleanu, Dumitru

doi:10.1002/mma.3746

Cited by 17 publications

(18 citation statements)

References 22 publications

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“…Subordination identities (10) and (11) can be also deduced as particular cases of the generalized multiplication theorem of Efros. 25 This theorem is used, eg, in Nigmatullin et al, 26 to obtain integral representation of multivariate Mittag-Leffler functions.…”

Section: (K) -Relaxation Equationmentioning

confidence: 99%

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Estimates for a general fractional relaxation equation and application to an inverse source problem

Bazhlekova

2018

Math Methods in App Sciences

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A general fractional relaxation equation is considered with a convolutional derivative in time introduced by A. Kochubei (Integr. Equ. Oper. Theory 71 (2011), 583-600). This equation generalizes the single-term, multiterm, and distributed-order fractional relaxation equations. The fundamental and the impulse-response solutions are studied in detail. Properties such as analyticity and subordination identities are established and employed in the proof of an upper and a lower bound. The obtained results extend some known properties of the Mittag-Leffler functions. As an application of the estimates, uniqueness and conditional stability are established for an inverse source problem for the general time-fractional diffusion equation on a bounded domain. KEYWORDS completely monotone function, general fractional derivative, inverse source problem, Mittag-Leffler function, subordination principle 9018

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Section: (K) -Relaxation Equationmentioning

confidence: 99%

“…where f n = (f, n ). Introducing the notations h n = (h, n ) and Q n (t) = ∫ t 0 v n (t − )q( ) d , (26) gives…”

Section: Inverse Problem: Uniqueness and A Conditional Stability Estimentioning

confidence: 99%

Estimates for a general fractional relaxation equation and application to an inverse source problem

Bazhlekova

2018

Math Methods in App Sciences

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“…Mainly, the model emphasizes the concept of fractal capacity (fractance) -implicitly Choquet non-additive measure and integrals -whose charge is ruled by the non-integer differential equation i ∼ d α U/dt α with α = 1/d [15,16], where U is the experimental potential. In the simplest case of the first order local transfer, hence, for canonical transfer, the Fourier transform must be expressed through Cole and Cole type of impedance: [17][18][19][20] which is a generalization of the exponential transfer turned by convolving with the d-fractal geometry. Many other interesting expressions and forms can be found, but being basically related to exponential operator, the canonic form appears as seminal.…”

Section: Zeta Function and "α-Expona Ntiation"mentioning

confidence: 99%

From the arrow of time in Badiali's quantum approach to the dynamic meaning of Riemann's hypothesis

Riot¹,

Méhauté²

2017

Condens. Matter Phys.

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The novelty of the Jean Pierre Badiali last scientific works stems to a quantum approach based on both (i) a return to the notion of trajectories (Feynman paths) and (ii) an irreversibility of the quantum transitions. These iconoclastic choices find again the Hilbertian and the von Neumann algebraic point of view by dealing statistics over loops. This approach confers an external thermodynamic origin to the notion of a quantum unit of time (Rovelli Connes' thermal time). This notion, basis for quantization, appears herein as a mere criterion of parting between the quantum regime and the thermodynamic regime. The purpose of this note is to unfold the content of the last five years of scientific exchanges aiming to link in a coherent scheme the Jean Pierre's choices and works, and the works of the authors of this note based on hyperbolic geodesics and the associated role of Riemann zeta functions. While these options do not unveil any contradictions, nevertheless they give birth to an intrinsic arrow of time different from the thermal time. The question of the physical meaning of Riemann hypothesis as the basis of quantum mechanics, which was at the heart of our last exchanges, is the backbone of this note. From algebraic analysis of quantum mechanics to "irreversible" Feynman paths integralDespite the unstoppable success of the technosciences based on both quantum mechanics, standard particle model and cosmological model, at least two questions must be investigated among many issues that the theories leave open [1, 2]: (i) the question of the ontological status of the time and (ii) the obsessive interrogation concerning the existence or the absence of an intrinsic "arrow of time". The origin of these questions comes from the equivocal equivalence of the status of time in any types of mechanical formalisms. For example, within Newtonian vision, the observable f can be analysed algebraically using action-integral through the Lagrangian L while Poisson brackets gives time differential representations d f /dt = {H, f }. According to Noether theorem, the energy, referred to the Hamiltonian H, is no other than the tag of a time-shift independence of physical laws, namely a compact commutativity. The statistical knowledge of the high dimensions system requires (i) the definition of a Liouville measure µ L based on the symplectic structure of the phase space and (ii) the value of the configuration distribution Z C , therefore dµ ∼ (1/Z C ) e −βH , with β = 1/k B T related to the inverse of the temperature. This point of view is discretized in quantum mechanics (QM).

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“…The provided numerical simulations suggest that the obtained solutions and expressions are convergent and valid. Due to the vast applications of these special functions, their calculations as well as approximations have always intrigued scientists . As a sample, one may refer to Pade approximations of Mittag‐Leffler functions in Zeng and Chen and Starovoitov and Starovoitova .…”

Section: Introductionmentioning

confidence: 99%

“…Due to the vast applications of these special functions, their calculations as well as approximations have always intrigued scientists. 21,22 As a sample, one may refer to Pade approximations of Mittag-Leffler functions in Zeng and Chen 23 and Starovoitov and Starovoitova. 24 Therefore, new efficient approximations are also presented here by using partition polynomials.…”

Section: Introductionmentioning

confidence: 99%

The use of partition polynomial series in Laplace inversion of composite functions with applications in fractional calculus

Taghavian

2019

Math Methods in App Sciences

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This paper presents an analytical method towards Laplace transform inversion of composite functions with the aid of Bell polynomial series. The presented results are used to derive the exact solution of fractional distributed order relaxation processes as well as time‐domain impulse response of fractional distributed order operators in new series forms. Evaluation of the obtained series expansions through computer simulations is also given. The results are then used to present novel series expansions for some special functions, including the one‐parameter Mittag‐Leffler function. It is shown that truncating these series expansions when combined with using potential partition polynomials provides efficient approximations for these functions. At the end, the results are shown to be also useful in studying asymptotical behavior of partial Bell polynomials. Numerical simulations as well as analytical examples are provided to verify the results of this paper.

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On the Laplace integral representation of multivariate Mittag‐Leffler functions in anomalous relaxation

Cited by 17 publications

References 22 publications

Estimates for a general fractional relaxation equation and application to an inverse source problem

Estimates for a general fractional relaxation equation and application to an inverse source problem

From the arrow of time in Badiali's quantum approach to the dynamic meaning of Riemann's hypothesis

The use of partition polynomial series in Laplace inversion of composite functions with applications in fractional calculus

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