Fractional Dynamics and Discrete Maps with Memory

In this paper, we use integral equations of non-integer orders to derive discrete maps with memory. Note that discrete maps with memory were not previously derived from fractional integral equations of non-integer orders. Such a derivation of discrete maps with memory is proposed for the first time in this work. In this paper, we derived discrete maps with nonlocality in time and memory from exact solutions of fractional integral equations with the Riemann–Liouville and Hadamard type fractional integrals of non-integer orders and periodic sequence of kicks that are described by Dirac delta-functions. The suggested discrete maps with nonlocality in time are derived from these fractional integral equations without any approximation and can be considered as exact discrete analogs of these equations. The discrete maps with memory, which are derived from integral equations with the Hadamard type fractional integrals, do not depend on the period of kicks.

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Mass Generation and Non-Euclidean Metric from Fractional Dynamics

10.20944/preprints202011.0350.v1 ◽

2020 ◽

Author(s):

Ervin Goldfain

Keyword(s):

Field Theory ◽

Quantum Field Theory ◽

Fractional Dynamics ◽

Quantum Field ◽

Fractional Differentiation ◽

Quantum Processes ◽

Mass Generation ◽

Euclidean Metric ◽

Standard Index ◽

With Memory

Fractional-time Schrödinger equation (FTSE) describes the evolution of quantum processes endowed with memory effects. FTSE manifestly breaks all consistency requirements of quantum field theory (unitarity, locality and compliance with the clustering theorem), unless the order of fractional differentiation and integration ( ) falls close to the standard index . Working in the context of the minimal fractal manifold (where , ), we confirm here that FTSE approximates the attributes of gravitational metric and provides an unforeseen generation mechanism for massive fields.

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REVIEW OF SOME PROMISING FRACTIONAL PHYSICAL MODELS

International Journal of Modern Physics B ◽

10.1142/s0217979213300053 ◽

2013 ◽

Vol 27 (09) ◽

pp. 1330005 ◽

Cited By ~ 75

Author(s):

VASILY E. TARASOV

Keyword(s):

Fractional Calculus ◽

Power Law ◽

Particle Interaction ◽

Discrete Systems ◽

Physical Models ◽

Long Term Memory ◽

Fractional Dynamics ◽

Fractal Media ◽

Systems With Memory ◽

With Memory

Fractional dynamics is a field of study in physics and mechanics investigating the behavior of objects and systems that are characterized by power-law nonlocality, power-law long-term memory or fractal properties by using integrations and differentiation of non-integer orders, i.e., by methods in the fractional calculus. This paper is a review of physical models that look very promising for future development of fractional dynamics. We suggest a short introduction to fractional calculus as a theory of integration and differentiation of noninteger order. Some applications of integro-differentiations of fractional orders in physics are discussed. Models of discrete systems with memory, lattice with long-range inter-particle interaction, dynamics of fractal media are presented. Quantum analogs of fractional derivatives and model of open nano-system systems with memory are also discussed.

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