arXiv subjects

Vasily E. Tarasov

Publications and source records attributed to Vasily E. Tarasov.

At least 19 recordsLinked to original sources

arXiv2025

Predator-prey models with memory and kicks: Exact solution and discrete maps with memory

In this paper, we proposed new predator-prey models that take into account memory and kicks. Memory is understood as the dependence of current behavior on the history of past behavior. The equations of these proposed models are generalizations of the Lotka-Volterra and Kolmogorov equations by using the Caputo fractional derivative of non-integer order and periodic kicks. This fractional derivative allows us to take into account memory with power-law fading. The periodic kicks, which are described by Dirac delta-functions, take into account short duration of interaction between predators and prey. For the proposed equations, which are fractional differential equations with kicks, we obtain exact solutions that describe behaviors of predator and prey with power-law fading memory. Using these exact solutions, we derive, without using any approximations, new discrete maps with memory that represent the proposed predator-prey models with memory.

q-bio.PE↗

arXiv2025

General Fractional Dynamics

General fractional dynamics (GFDynamics) can be viewed as an interdisciplinary science, in which the non-local properties of linear and nonlinear dynamical systems are studied by using of general fractional calculus, equations with general fractional integrals (GFI) and derivatives (GFD), or general nonlocal mappings with discrete time. The GFDynamics implies research and obtaining results concerning of general form of nonlocality, which can be described by general form operator kernels, and not its particular implementations and representations. In this paper, it is proposed the concept of "general nonlocal maps" that are exact solutions of equations with GFI and GFD at discrete points. In these maps, the non-locality is determined by the kernels that are associated to the Sonin and Luchko kernels of general fractional integrals and derivatives, which are used in initial equations. Using general fractional calculus, we consider fractional systems with general non-locality in time, which are described by equations with general fractional operators and periodic kicks. Equations with GFI and GFD of arbitrary order are also used to derive general nonlocal maps. Exact solutions for these general fractional differential and integral equations with kicks are obtained. These exact solutions with discrete time points are used to derive general nonlocal maps without approximations. Some examples of non-locality in time are described.

nlin.CD↗

arXiv2025

From Fractional Differential Equations with Hilfer Derivatives: To Discrete Maps with Memory

In this article, we proposed new discrete maps with memory (DMM). These maps are derived from fractional differential equations (FDE) with the Hilfer fractional derivatives of non-integer orders and periodic sequence of kicks. The suggested DMM are obtained from these equations without any approximation, and they are a discrete form of the exact solutions. DMM are proposed for arbitrary positive orders of equations with the Hilfer fractional derivatives. As an example, the suggested maps are described for the orders lying in the intervals (0,1) and (1,2). The maps, which are derived from the equations with the Hilfer operators, allow us to consider a whole range of the maps derived from the FDE with the Caputo and Riemann-Liouville fractional derivatives.

nlin.CD↗

arXiv2025

Fractional Dynamics with Non-Local Scaling

Nonlinear fractional dynamics with scale invariance in continuous and discrete time approaches are described. We use non-integer-order integro-differential operators that can be interpreted as generalizations of scaling (dilation) differential operator for the case of non-locality. Nonlinear integro-differential equations with Hadamard type operators of non-integer orders with respect to time and periodic sequence of kicks are considered. Exact solutions of these equations are derived without using approximations. Using these solutions for discrete time points, we derive mappings with non-local scaling in time from proposed equations without approximation. Non-local mappings are obtained in general for arbitrary orders of the Hadamard type fractional operators. An example of these mappings with non-local scaling in time is given for arbitrary positive orders of integro-differential equations with kicks. The proposed mappings are independent of the period of kicks at zero initial conditions. The proposed approach can be used to describe the nonlinear fractional dynamics that is characterized by scale invariance.

nlin.PS↗

arXiv2025

Parametric General Fractional Calculus: Nonlocal Operators Acting on Function with Respect to Another Function

In this paper, we present the definitions and some properties of the general fractional integrals (GFIs) and general fractional derivatives (GFDs) of a function f(x) with respect to another function g(x). Examples of special cases of parametric GF operators are described. Some properties of the parametric GFIs and the parametric GFDs are proved. Representation of the parametric GFIs and parametric GFDs via the GF integrals on the finite interval (a,b). Fundamental theorems of parametric GFC are proved. Applied mathematical interpretations of parametric general fractional derivatives for processes with memory are given.

math.GM↗

arXiv2021

General Fractional Vector Calculus

A generalization of fractional vector calculus 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 in the Luchko approach. This paper proposed the following: (1) 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. (2) Self-consistent definitions of general fractional integral vector operators: the general fractional circulation, general fractional flux and general fractional volume integral are proposed. (3) The general fractional gradient, Green, Stokes and Gauss 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. Therefore we can state that we proposed a calculus, which is a general fractional vector calculus. 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 (Leibniz) rule, chain rule, and semigroup property. General FVC for orthogonal curvilinear coordinates, which includes general fractional vector operators for the spherical and cylindrical coordinates, is also proposed.

math.CA↗

arXiv2021

General fractional calculus: Multi-kernel approach

For the first time, a general fractional calculus of arbitrary order was proposed by Yuri Luchko in the works Mathematics 9(6) (2021) 594 and Symmetry 13(5) (2021) 755. In these works, the proposed approaches to formulate this calculus are based either on the power of one Sonine kernel, or the convolution of one Sonine kernel with the kernels of the integer-order integrals. To apply general fractional calculus, it is useful to have a wider range of operators, for example, by using the Laplace convolution of different types of kernels. In this paper, an extended formulation of the general fractional calculus of arbitrary order is proposed. Extension is achieved by using different types (subsets) of pairs of operator kernels in definitions general fractional integrals and derivatives. For this, the definition of the Luchko pair of kernels is somewhat broadened, which leads to the symmetry of the definition of the Luchko pair. The proposed set of kernel pairs are subsets of the Luchko set of kernel pairs. The fundamental theorems for the proposed general fractional derivatives and integrals are proved.

math.GM↗

arXiv2021

Trends, Directions for Further Research, and Some Open Problems of Fractional Calculus

The area of fractional calculus (FC) has been fast developing and is presently being applied in all scientific fields. Therefore, it is of key relevance to assess the present state of development and to foresee, if possible, the future evolution, or, at least, the challenges identified in the scope of advanced research works. This paper gives a vision about the directions for further research as well as some open problems of FC. A number of topics in mathematics, numerical algorithms and physics are analyzed, giving a systematic perspective for future research.

math.CA↗

arXiv2020

Dirac Particle with Memory: Proper Time Non-Locality

A generalization of the standard model of Dirac particle in external electromagnetic field is proposed. In the generalization we take into account interactions of this particle with environment, which is described by the memory function. This function takes into account that the behavior of the particle at proper time can depend not only at the present time, but also on the history of changes on finite time interval. In this case the Dirac particle can be considered an open quantum system with non-Markovian dynamics. The violation of the semigroup property of dynamic maps is a characteristic property of dynamics with memory. We use the Fock-Schwinger proper time method and derivatives of non-integer orders with respect to proper time. The fractional differential equation, which describes the Dirac particle with memory, and the expression of its exact solution are suggested. The asymptotic behavior of the proposed solutions is described.

quant-ph↗

arXiv2019

Economic Growth Model with Constant Pace and Dynamic Memory

The article discusses a generalization of model of economic growth with constant pace, which takes into account the effects of dynamic memory. Memory means that endogenous or exogenous variable at a given time depends not only on their value at that time, but also on their values at previous times. To describe the dynamic memory we use derivatives of non-integer orders. We obtain the solutions of fractional differential equations with derivatives of non-integral order, which describe the dynamics of the output caused by the changes of the net investments and effects of power-law fading memory.

econ.GN↗

arXiv2018

On Fractional and Fractal Formulations of Gradient Linear and Nonlinear Elasticity

In this paper we consider extensions of the gradient elasticity models proposed earlier by the second author to describe materials with fractional non-locality and fractality using the techniques developed recently by the first author. We derive a generalization of three-dimensional continuum gradient elasticity theory, starting from integral relations and assuming a weak non-locality of power-law type that gives constitutive relations with fractional Laplacian terms, by utilizing the fractional Taylor series in wave-vector space. In the sequel we consider non-linear field equations with fractional derivatives of non-integer order to describe nonlinear elastic effects for gradient materials with power-law long-range interactions in the framework of weak non-locality approximation. The special constitutive relationship that we elaborate on, can form the basis for developing a fractional extension of deformation theory of gradient plasticity. Using the perturbation method, we obtain corrections to the constitutive relations of linear fractional gradient elasticity, when the perturbations are caused by weak deviations from linear elasticity or by fractional gradient non-locality. Finally we discuss fractal materials described by continuum models with non-integer dimensional spaces. Using the recently suggested vector calculus for non-integer dimensional spaces, we consider problems of fractal gradient elasticity.

physics.class-ph↗

arXiv2018

Fractional Derivative Regularization in QFT

In this paper, we propose new regularization, where integer-order differential operators are replaced by fractional-order operators. Regularization for quantum field theories based on application of the Riesz fractional derivatives of non-integer orders is suggested. The regularized loop integrals depend on parameter that is the order alpha>0 of the fractional derivative. The regularization procedure is demonstrated for scalar massless fields in phi^4-theory on n-dimensional pseudo-Euclidean space-time.

hep-th↗

arXiv2018

No Nonlocality. No Fractional Derivative

The paper discusses the characteristic properties of fractional derivatives of non-integer order. It is known that derivatives of integer orders are determined by properties of differentiable functions only in an infinitely small neighborhood of the considered point. Therefore differential equation, which is considered for this point and contains a finite number of integer-order derivatives, cannot describe nonlocality in space and time. This allows us to propose a principle of nonlocality for fractional derivatives. We state that if the differential equation with fractional derivative can be presented as a differential equation with a finite number of integer-order derivatives, then this fractional derivative cannot be considered as a derivative of non-integer order. This means that all results obtained for this type of fractional derivatives can be derived by using differential operators with integer orders. To illustrate the application of the nonlocality principle, we prove that the conformable fractional derivative, the M-fractional derivative, the alternative fractional derivative, the local fractional derivative and the Caputo-Fabrizio fractional derivatives with exponential kernels cannot be considered as fractional derivatives of non-integer orders.

math.CA↗

arXiv2018

Local Fractional Derivatives of Differentiable Functions are Integer-order Derivatives or Zero

In this paper we prove that local fractional derivatives of differentiable functions are integer-order derivative or zero operator. We demonstrate that the local fractional derivatives are limits of the left-sided Caputo fractional derivatives. The Caputo derivative of fractional order alpha of function f(x) is defined as a fractional integration of order (n-alpha) of the derivative f^(n)(x) of integer order n. The requirement of the existence of integer-order derivatives allows us to conclude that the local fractional derivative cannot be considered as the best method to describe nowhere differentiable functions and fractal objects. We also prove that unviolated Leibniz rule cannot hold for derivatives of orders alpha, which are not equal to one.

math.CA↗

arXiv2017

Economic interpretation of fractional derivatives

An economic interpretation of the Caputo derivatives of non-integer orders is proposed. The suggested economic interpretation of the fractional derivatives is based on a generalization of average and marginal values of economic indicators. We formulate an economic interpretation by using the concept of the T-indicator that allows us to describe economic processes with memory. The standard average and marginal values of indicator are special cases of the proposed T-indicator, when the order is equal to zero and one, respectively. The fractional derivatives are interpreted as economic characteristics (indicators) that are intermediate between the standard average and marginal values of indicators.

econ.GN↗

arXiv2017

Accelerators in macroeconomics: Comparison of discrete and continuous approaches

We prove that the standard discrete-time accelerator equation cannot be considered as an exact discrete analog of the continuous-time accelerator equation. This leads to fact that the standard discrete-time macroeconomic models cannot be considered as exact discretization of the corresponding continuous-time models. As a result, the equations of the continuous and standard discrete models have different solutions and can predict the different behavior of the economy. In this paper, we propose a self-consistent discrete-time description of the economic accelerators that is based on the exact finite differences. For discrete-time approach, the model equations with exact differences have the same solutions as the corresponding continuous-time models and these discrete and continuous models describe the same behavior of the economy. Using the Harrod-Domar growth model as an example, we show that equations of the continuous-time model and the suggested exact discrete model have the same solutions and these models predict the same behavior of the economy.

econ.GN↗

arXiv2017

Logistic map with memory from economic model

A generalization of the economic model of logistic growth, which takes into account the effects of memory and crises, is suggested. Memory effect means that the economic factors and parameters at any given time depend not only on their values at that time, but also on their values at previous times. For the mathematical description of the memory effects, we use the theory of derivatives of non-integer order. Crises are considered as sharp splashes (bursts) of the price, which are mathematically described by the delta-functions. Using the equivalence of fractional differential equations and the Volterra integral equations, we obtain discrete maps with memory that are exact discrete analogs of fractional differential equations of economic processes. We derive logistic map with memory, its generalizations, and "economic" discrete maps with memory from the fractional differential equations, which describe the economic natural growth with competition, power-law memory and crises.

econ.GN↗

arXiv2017

Dynamic intersectoral models with power-law memory

Intersectoral dynamic models with power-law memory are proposed. The equations of open and closed intersectoral models, in which the memory effects are described by the Caputo derivatives of non-integer orders, are derived. We suggest solutions of these equations, which have the form of linear combinations of the Mittag-Leffler functions and which are characterized by different effective growth rates. Examples of intersectoral dynamics with power-law memory are suggested for two sectoral cases. We formulate two principles of intersectoral dynamics with memory: the principle of changing of technological growth rates and the principle of domination change. It has been shown that in the input-output economic dynamics the effects of fading memory can change the economic growth rate and dominant behavior of economic sectors.

econ.GN↗