scholarly journals General Fractional Dynamics

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1464
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Fractional Calculus ◽  
Exact Solutions ◽  
Arbitrary Order ◽  
Nonlinear Dynamical Systems ◽  
Fractional Integrals ◽  
Fractional Dynamics ◽  
Fractional Systems ◽  
Nonlinear Dynamical ◽  
Fractional Differential ◽  
Solutions Of Equations

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.

INTRODUCTION

2012 ◽  
Vol 22 (04) ◽  
pp. 1202002 ◽  
Author(s):  
CHANGPIN LI ◽  
YANG QUAN CHEN ◽  
BLAS M. VINAGRE ◽  
IGOR PODLUBNY
Keyword(s):  
Fractional Calculus ◽  
Scaling Laws ◽  
Power Laws ◽  
Fractional Dynamics ◽  
Long Range Interactions ◽  
Potential Applications ◽  
Dynamics And Control ◽  
Long Memory Processes ◽  
Fractional Differential ◽  
And Control

Fractional Dynamics and Control is emerging as a new hot topic of research which draws tremendous attention and great interest. Although the fractional calculus appeared almost in the same era when the classical (or integer-order) calculus was born, it has recently been found that it can better characterize long-memory processes and materials, anomalous diffusion, long-range interactions, long-term behaviors, power laws, allometric scaling laws, and so on. Complex dynamical evolutions of these fractional differential equation models, as well as their controls, are becoming more and more important due to their potential applications in the real world. This special issue includes one review article and twenty-three regular papers, covering fundamental theories of fractional calculus, dynamics and control of fractional differential systems, and numerical calculation of fractional differential equations.

Fractional calculus and Sinc methods

2011 ◽  
Vol 14 (4) ◽  
Author(s):  
Gerd Baumann ◽  
Frank Stenger
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Integral Equations ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Singular Integral Equations ◽  
Fractional Integrals ◽  
Sinc Methods ◽  
Weakly Singular Integral Equations ◽  
Fractional Differential

AbstractFractional integrals, fractional derivatives, fractional integral equations, and fractional differential equations are numerically solved by Sinc methods. Sinc methods are able to deal with singularities of the weakly singular integral equations of Riemann-Liouville and Caputo type. The convergence of the numerical method is numerically examined and shows exponential behavior. Different examples are used to demonstrate the effective derivation of numerical solutions for different types of fractional differential and integral equations, linear and non-linear ones. Equations of mixed ordinary and fractional derivatives, integro-differential equations are solved using Sinc methods. We demonstrate that the numerical calculation needed in fractional calculus can be gained with high accuracy using Sinc methods.

scholarly journals Special Functions of Fractional Calculus in the Form of Convolution Series and Their Applications

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2132
Author(s):  
Yuri Luchko
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Special Functions ◽  
Analytic Solutions ◽  
Fractional Integrals ◽  
Function Type ◽  
Fractional Differential ◽  
Derivatives Of

In this paper, we first discuss the convolution series that are generated by Sonine kernels from a class of functions continuous on a real positive semi-axis that have an integrable singularity of power function type at point zero. These convolution series are closely related to the general fractional integrals and derivatives with Sonine kernels and represent a new class of special functions of fractional calculus. The Mittag-Leffler functions as solutions to the fractional differential equations with the fractional derivatives of both Riemann-Liouville and Caputo types are particular cases of the convolution series generated by the Sonine kernel κ(t)=tα−1/Γ(α),0<α<1. The main result of the paper is the derivation of analytic solutions to the single- and multi-term fractional differential equations with the general fractional derivatives of the Riemann-Liouville type that have not yet been studied in the fractional calculus literature.

scholarly journals Wavelet Methods for Solving Fractional Order Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
A. K. Gupta ◽  
S. Saha Ray
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Fractional Differential Equations ◽  
Arbitrary Order ◽  
Applied Mathematics ◽  
Science And Engineering ◽  
The Past ◽  
Biological Population ◽  
Fractional Differential ◽  
Wavelet Methods

Fractional calculus is a field of applied mathematics which deals with derivatives and integrals of arbitrary orders. The fractional calculus has gained considerable importance during the past decades mainly due to its application in diverse fields of science and engineering such as viscoelasticity, diffusion of biological population, signal processing, electromagnetism, fluid mechanics, electrochemistry, and many more. In this paper, we review different wavelet methods for solving both linear and nonlinear fractional differential equations. Our goal is to analyze the selected wavelet methods and assess their accuracy and efficiency with regard to solving fractional differential equations. We discuss challenges faced by researchers in this field, and we emphasize the importance of interdisciplinary effort for advancing the study on various wavelets in order to solve differential equations of arbitrary order.

scholarly journals Mittag-Leffler Stability Theorem for Fractional Nonlinear Systems with Delay

2010 ◽  
Vol 2010 ◽  
pp. 1-7 ◽  
Author(s):  
S. J. Sadati ◽  
D. Baleanu ◽  
A. Ranjbar ◽  
R. Ghaderi ◽  
T. Abdeljawad (Maraaba)
Keyword(s):  
Time Delay ◽  
Fractional Calculus ◽  
Nonlinear Dynamical Systems ◽  
Stability Theorem ◽  
Delay Systems ◽  
Time Delay Systems ◽  
Science And Engineering ◽  
Nonlinear Dynamical ◽  
Caputo’S Derivative ◽  
The Stability

Fractional calculus started to play an important role for analysis of the evolution of the nonlinear dynamical systems which are important in various branches of science and engineering. In this line of taught in this paper we studied the stability of fractional order nonlinear time-delay systems for Caputo's derivative, and we proved two theorems for Mittag-Leffler stability of the fractional nonlinear time delay systems.

Editorial: Fractional differential and integral operators with non-singular and non-local kernel with application to nonlinear dynamical systems

2020 ◽  
Vol 132 ◽  
pp. 109493 ◽  
Author(s):  
Abdon Atangana ◽  
Jose Francisco Gomez Aguilar ◽  
Matthew Owolabi Kolade ◽  
Jordan Yankov Hristov
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Integral Operators ◽  
Nonlinear Dynamical ◽  
Non Local ◽  
Fractional Differential

scholarly journals General Fractional Calculus: Multi-Kernel Approach

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1501
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivatives ◽  
Arbitrary Order ◽  
Fractional Integrals ◽  
Extended Formulation ◽  
Different Types ◽  
Kernel Approach ◽  
Definition Of ◽  
Fundamental Theorems ◽  
First Time

For the first time, a general fractional calculus of arbitrary order was proposed by Yuri Luchko in 2021. In Luchko works, the proposed approaches to formulate this calculus are based either on the power of one Sonin kernel or the convolution of one Sonin 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.

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