Fractional Derivatives for Physicists and Engineers

Учайкин, В. В.

doi:10.1007/978-3-642-33911-0

Cited by 631 publications

(422 citation statements)

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“…All these equations form a basis of fractional nonlocal statistical mechanics. can be considered as the conjugate Riesz derivative [6] with respect to x j . Therefore, the operator (79) can be called a generalized conjugate derivative of the Riesz type.…”

Section: Resultsmentioning

confidence: 99%

“…Fractional calculus [3,4,5,6,7,8,9] has a lot of applications in physics [10,11,12,13,14,15,16] and it allows us to take into account fractional power-law nonlocality of continuously distributed systems. Using the fractional calculus, we can consider fractional differential equations for conservation of probability in generalized phase spaces.…”

Section: Introductionmentioning

confidence: 99%

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Fractional Liouville equation on lattice phase-space

Tarasov

2015

Physica A: Statistical Mechanics and its Applications

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In this paper we propose a lattice analog of phase-space fractional Liouville equation. The Liouville equation for phase-space lattice with long-range jumps of power-law types is suggested. We prove that the continuum limit transforms this lattice equation into Liouville equation with conjugate Riesz fractional derivatives of non-integer orders with respect to coordinates of continuum phase-space. An application of the fractional Liouville equation with these Riesz fractional derivatives to describe properties of plasma-like nonlocal media is considered.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Fractional Liouville equation on lattice phase-space

Tarasov

2015

Physica A: Statistical Mechanics and its Applications

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“…The fractional diffusionwave equation [19] is the linear fractional differential equation obtained from the classical diffusion or wave equations by replacing the first-or second-order time derivatives by a fractional derivative (in the Caputo sense) [41][42][43] of order  with 02  ,…”

Section: -2 Fractional Dynamicsmentioning

confidence: 99%

Fractional Dynamics in Bioscience and Biomedicine and the Physics of Cancer

Nasrolahpour

2017

Preprint

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Almost all phenomena and structures in nature exhibit some degrees of fractionality or fractality. Fractional calculus and fractal theory are two interrelated concepts. In this article we study the memory effects in nature and particularly in biological structures. Based on this fact that natural way to incorporate memory effects in the modeling of various phenomena and dealing with complexities is using of fractional calculus, in this article we present different examples in various branch of science from cosmology to biology and we investigate this idea that are we able to describe all of such these phenomena using the well-know and powerful tool of fractional calculus. In particular we focus on fractional calculus approach as an effective tool for better understanding of physics of living systems and organism and especially physics of cancer.Keywords: Fractional Dynamics; Memory Effect; Biological Structures; Physics of Cancer IntroductionThe concept of memory effect plays an important role in a large number of phenomena in different contexts and systems from biological structures to cosmological phenomena. For instance: memory effects in nanoscale systems [1,2], optical memory effect [3], gravitational and cosmological memory effect (which means: the induction of a permanent change in the relative separation of the test particles when gravitational radiation passes through a configuration of test particles that make up a gravitational wave detector) [4], memory effects in gene regulatory networks [5], memory effects in economics [6] have been investigated. In addition other kinds of memory effect including: shape memory effect [7][8][9][10], magnetic shape memory effect [11] and temperature memory effect [12] have been also considered in recent years. In all above mentioned references authors have used the standard calculus, however nowadays it is well-known that a natural way to incorporate memory effects in the modeling of various phenomena is using of fractional calculus. On the other hand almost all phenomena and structures in nature exhibit some degrees and levels of fractionality or fractality (low or high level or something between them), also nowadays it is well-known that there is a close relation between fractality and fractionality. In this work we investigate this idea that are we able to describe all of such these phenomena using the well-know and powerful tool of fractional calculus. Therefore for this purpose in the following, concepts of fractality and fractional dynamics are briefly reviewed respectively in Sec. 2.Then in Sec. 3 we introduce fractional calculus as a powerful tool for modeling of memory effects in different context and we present some important applications. At last, in Sec. 4, we will present some conclusions.

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“…The "hot spot" is formed in the end of the 80-th of the last century when many researches working in different application fields understood that this new tool suggested by the mathematics of the fractional calculus can open new features and generalizations of the previous phenomena associated with fractal geometry studied. For beginners one can recommend some monographs [14,1,12,15,13] and reviews [6,7] included extended old and recent historical survey, where the foundations of this "hot spot" are explained. The interest to relationship between fractals and fractional calculus is renewed again.…”

Section: Introductionmentioning

confidence: 99%

Accurate relationships between fractals and fractional integrals: New approaches and evaluations

Nigmatullin

Zhang

Gubaidullin

2017

FCAA

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In this paper the accurate relationships between the averaging procedure of a smooth function over 1D-fractal sets and the fractional integral of the RL-type are established. The numerical verifications are realized for confirmation of the analytical results and the physical meaning of these obtained formulas is discussed. Besides, the generalizations of the results for a combination of fractal circuits having a discrete set of fractal dimensions were obtained. We suppose that these new results help understand deeper the intimate links between fractals and fractional integrals of different types. These results can be used in different branches of the interdisciplinary physics, where the different equations describing the different physical phenomena and containing the fractional derivatives and integrals are used.MSC 2010 : Primary 28A80, 26A33; Secondary: 60G18, 26A30, 28A78 Key Words and Phrases: fractal object, self-similar object, spatial fractional integral, averaging of smooth functions on spatial fractal sets, Cantor set c

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Fractional Derivatives for Physicists and Engineers

Cited by 631 publications

References 0 publications

Fractional Liouville equation on lattice phase-space

Fractional Liouville equation on lattice phase-space

Fractional Dynamics in Bioscience and Biomedicine and the Physics of Cancer

Accurate relationships between fractals and fractional integrals: New approaches and evaluations

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