General Fractional Dynamics

Tarasov, Vasily

doi:10.3390/math9131464

Cited by 56 publications

(46 citation statements)

References 68 publications

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“…Therefore, these results are valid for any operator kernels from the Sonin set. This approach to non-Markovian quantum theory is directly connected with the concept of general fractional dynamics suggested in [72].…”

Section: Discussionmentioning

confidence: 99%

“…Operational calculus for equations with general fractional derivatives was proposed in [60]. The general fractional calculus was also developed in works [61][62][63][64][65][66][67][68][69][70][71][72] devoted to mathematical aspects and some applications in classical physics.…”

Section: Introductionmentioning

confidence: 99%

“…The general fractional calculus is suggested as a mathematical basis to formulate the non-Markovian quantum dynamics. In the framework of the general approach, it was assumed and implied to obtain, first of all, general results that do not depend on specific types (particular implementations) of nonlocality and kernels [72]. In the general approach to non-Markovian quantum dynamics, all research and results should be related to the general form of nonlocality, operator kernels of almost all types (a wide set of operator kernels).…”

Section: Introductionmentioning

confidence: 99%

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General Non-Markovian Quantum Dynamics

Tarasov

2021

Entropy

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A general approach to the construction of non-Markovian quantum theory is proposed. Non-Markovian equations for quantum observables and states are suggested by using general fractional calculus. In the proposed approach, the non-locality in time is represented by operator kernels of the Sonin type. A wide class of the exactly solvable models of non-Markovian quantum dynamics is suggested. These models describe open (non-Hamiltonian) quantum systems with general form of nonlocality in time. To describe these systems, the Lindblad equations for quantum observable and states are generalized by taking into account a general form of nonlocality. The non-Markovian quantum dynamics is described by using integro-differential equations with general fractional derivatives and integrals with respect to time. The exact solutions of these equations are derived by using the operational calculus that is proposed by Yu. Luchko for general fractional differential equations. Properties of bi-positivity, complete positivity, dissipativity, and generalized dissipativity in general non-Markovian quantum dynamics are discussed. Examples of a quantum oscillator and two-level quantum system with a general form of nonlocality in time are suggested.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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General Non-Markovian Quantum Dynamics

Tarasov

2021

Entropy

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“…It is important to have a general fractional calculus that allows us to describe nonlocality in the most general form. We proposed [44,45] to use general fractional calculus (GFC) to describe systems, media and fields with general form of nonlocality in space.…”

Section: Introductionmentioning

confidence: 99%

“…Operational calculus for equations with general fractional derivatives is proposed in [43]. The GFC is also developed and applied in physics in works [44,45,[53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68].…”

Section: Introductionmentioning

confidence: 99%

General Fractional Vector Calculus

Tarasov

2021

Mathematics

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A generalization of fractional vector calculus (FVC) as a self-consistent mathematical theory is proposed to take into account a general form of non-locality in kernels of fractional vector differential and integral operators. Self-consistency involves proving generalizations of all fundamental theorems of vector calculus for generalized kernels of operators. In the generalization of FVC from power-law nonlocality to the general form of nonlocality in space, we use the general fractional calculus (GFC) in the Luchko approach, which was published in 2021. This paper proposed the following: (I) Self-consistent definitions of general fractional differential vector operators: the regional and line general fractional gradients, the regional and surface general fractional curl operators, the general fractional divergence are proposed. (II) Self-consistent definitions of general fractional integral vector operators: the general fractional circulation, general fractional flux and general fractional volume integral are proposed. (III) The general fractional gradient, Green’s, Stokes’ and Gauss’s theorems as fundamental theorems of general fractional vector calculus are proved for simple and complex regions. The fundamental theorems (Gradient, Green, Stokes, Gauss theorems) of the proposed general FVC are proved for a wider class of domains, surfaces and curves. All these three parts allow us to state that we proposed a calculus, which is a general fractional vector calculus (General FVC). The difficulties and problems of defining general fractional integral and differential vector operators are discussed to the nonlocal case, caused by the violation of standard product rule (Leibniz rule), chain rule (rule of differentiation of function composition) and semigroup property. General FVC for orthogonal curvilinear coordinates, which includes general fractional vector operators for the spherical and cylindrical coordinates, is also proposed.

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Variational Method for Solving the Time-Fractal Heat Conduction Problem in the Claydite-Block Construction

Shymanskyi

Sokolovskyy

et al. 2022

Advances in Computer Science for Engineering and Education

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General Fractional Dynamics

Cited by 56 publications

References 68 publications

General Non-Markovian Quantum Dynamics

General Non-Markovian Quantum Dynamics

General Fractional Vector Calculus

Variational Method for Solving the Time-Fractal Heat Conduction Problem in the Claydite-Block Construction

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