General Fractional Dynamics

Vasily E. Tarasov

doi:10.3390/math9131464

Abstract

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

Highlights

Fractional dynamics [1,2,3,4] is an interdisciplinary science in which the nonlocal properties of dynamical systems are studied by using methods of fractional calculus [5,6,7,8,9,10,11,12], integro-differential equations of non-integer orders and discrete nonlocal mappings
(1) General fractional dynamics with continuous time is described by the equations with general fractional derivatives (GFD) and general fractional integrals (GFI) with the kernels belonging to the Sonin set S−1
This paper proposes a new direction of research that can be called general fractional dynamics (GFDynamics)

Summary

Introduction

Fractional dynamics [1,2,3,4] is an interdisciplinary science in which the nonlocal properties of dynamical systems are studied by using methods of fractional calculus [5,6,7,8,9,10,11,12], integro-differential equations of non-integer orders and discrete nonlocal mappings. Such discrete nonlocal mappings must be described by kernels of the operators that are used in general fractional calculus These mappings can be derived from equations with GFD and GFI without approximations. These general nonlocal mappings are the exact solutions of fractional differential equations at discrete points. In this paper, using general fractional calculus, we considered fractional systems with nonlocality in time which are described by equations with general fractional derivatives, integrals and periodic kicks. The exact solutions for these nonlinear fractional differential and integral equations with kicks were obtained These exact solutions for discrete timepoints used to derive mappings with nonlocality in time were described without approximations. The nonlocality of general nonlocal mappings was determined by the kernels that belong to the Sonin and Luchko sets of kernels which were used in GFI and GFD of the initial equations

Equations of General Fractional Dynamical Systems

General Fractional Dynamics

Examples of Nonlocality in the Form of Kernels from the Sonin Set

General Fractional Calculus of Arbitrary Order

General Momenta of Arbitrary Order

Conclusions