Special Functions for Applied Scientists

Mathai, A. M.; Haubold, H. J.

doi:10.1007/978-0-387-75894-7

Cited by 371 publications

(269 citation statements)

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“…The kernel function of fractional derivative is called memory function [45]. Recently it is showed that the fractional model perfectly fits the test data of memory phenomena in different disciplines [46] they have found that a possible physical meaning of the fractional order is an index of memory.…”

Section: Fractional Calculus In Bioscience and Biomedicinementioning

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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Section: Fractional Calculus In Bioscience and Biomedicinementioning

confidence: 99%

Fractional Dynamics in Bioscience and Biomedicine and the Physics of Cancer

Nasrolahpour

2017

Preprint

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“…Both functions of the above type (i.e., Equations (9) and (10)) are essentially used as an explicit representation of solutions to integral and differential equations of the fractional order (see [6,30]). …”

Section: History Of the Mittag-leffler Functionmentioning

confidence: 99%

The Role of the Mittag-Leffler Function in Fractional Modeling

Rogosin

2015

Mathematics

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“…(25) A relation between the lower-part incomplete Beta function and the lower-part incomplete Gamma function is defined [31] as ( ) …”

Section: Appendixmentioning

confidence: 99%

The rate coefficients of unimolecular reactions in the systems with power-law distributions

Yin

Guo

2014

Physica A: Statistical Mechanics and its Applications

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Abstract:The rate coefficient formulae of unimolecular reactions are generalized to the systems with the power-law distributions based on nonextensive statistics, and the power-law rate coefficients are derived in the high and low pressure limits, respectively. The numerical analyses are made of the rate coefficients as functions of the ν-parameter, the threshold energy, the temperature and the number of degrees of freedom. We show that the new rate coefficients depend strongly on the ν-parameter different from one (thus from a Boltzmann-Gibbs distribution). Two unimolecular reactions, CH 3 CO→CH 3 +CO and CH 3 NC→CH 3 CN, are taken as application examples to calculate their power-law rate coefficients, which obtained with the ν-parameters slightly different from one can be exactly in agreement with all the experimental studies on these two reactions in the given temperature ranges.

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Special Functions for Applied Scientists

Cited by 371 publications

References 0 publications

Fractional Dynamics in Bioscience and Biomedicine and the Physics of Cancer

Fractional Dynamics in Bioscience and Biomedicine and the Physics of Cancer

The Role of the Mittag-Leffler Function in Fractional Modeling

The rate coefficients of unimolecular reactions in the systems with power-law distributions

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