On Mittag-Leffler Kernel-Dependent Fractional Operators with Variable Order

2019 ◽  
pp. 41-58
Author(s):  
G. M. Bahaa ◽  
T. Abdeljawad ◽  
F. Jarad
Keyword(s):  
Variable Order ◽  
Fractional Operators

scholarly journals Applications of Distributed-Order Fractional Operators: A Review

Entropy ◽  
2021 ◽  
Vol 23 (1) ◽  
pp. 110
Author(s):  
Wei Ding ◽  
Sansit Patnaik ◽  
Sai Sidhardh ◽  
Fabio Semperlotti
Keyword(s):  
Fractional Calculus ◽  
Real World ◽  
Transport Processes ◽  
Model Systems ◽  
Variable Order ◽  
Fractional Operators ◽  
Research Activity ◽  
Road Map ◽  
Real World Applications ◽  
Distributed Order

Distributed-order fractional calculus (DOFC) is a rapidly emerging branch of the broader area of fractional calculus that has important and far-reaching applications for the modeling of complex systems. DOFC generalizes the intrinsic multiscale nature of constant and variable-order fractional operators opening significant opportunities to model systems whose behavior stems from the complex interplay and superposition of nonlocal and memory effects occurring over a multitude of scales. In recent years, a significant amount of studies focusing on mathematical aspects and real-world applications of DOFC have been produced. However, a systematic review of the available literature and of the state-of-the-art of DOFC as it pertains, specifically, to real-world applications is still lacking. This review article is intended to provide the reader a road map to understand the early development of DOFC and the progressive evolution and application to the modeling of complex real-world problems. The review starts by offering a brief introduction to the mathematics of DOFC, including analytical and numerical methods, and it continues providing an extensive overview of the applications of DOFC to fields like viscoelasticity, transport processes, and control theory that have seen most of the research activity to date.

Synthesis of multifractional Gaussian noises based on variable-order fractional operators

2011 ◽  
Vol 91 (7) ◽  
pp. 1645-1650 ◽  
Author(s):  
Hu Sheng ◽  
Hongguang Sun ◽  
YangQuan Chen ◽  
TianShuang Qiu
Keyword(s):  
Variable Order ◽  
Fractional Operators

Modeling Nonlinear Oscillators via Variable-Order Fractional Operators

2019 ◽  
Author(s):  
Sansit Patnaik ◽  
Fabio Semperlotti
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Models ◽  
Contact Problems ◽  
Variable Order ◽  
Mathematical Tool ◽  
Fractional Operators ◽  
State Variables ◽  
Governing Equations ◽  
Seamless Transition ◽  
Physical Phenomena

Abstract Fractional derivatives and integrals are intrinsically multiscale operators that can act on both space and time dependent variables. Contrarily to their integer-order counterpart, fractional operators can have either fixed or variable order (VO) where, in the latter case, the order can also be function of either independent or state variables. When using VO differential governing equations to describe the response of dynamical systems, the order can evolve as a function of the response itself therefore allowing a natural and seamless transition between largely dissimilar dynamics (e.g. linear, nonlinear, and even contact problems). Such an intriguing characteristic allows defining governing equations for dynamical systems that are evolutionary in nature. In this study, we present the possible application of VO operators to a class of nonlinear lumped parameter models that has great practical relevance in mechanics and dynamics. Specific examples include hysteresis and contact problems for discrete oscillators. Within this context, we present a methodology to define VO operators capable of capturing such complex physical phenomena. Despite using simplified lumped parameters nonlinear models to present the application of VO operators to mechanics and dynamics, we provide a more qualitative discussion of the possible applications of this mathematical tool in the broader context of continuous multiscale systems.

scholarly journals Applications of variable-order fractional operators: a review

2020 ◽  
Vol 476 (2234) ◽  
pp. 20190498 ◽  
Author(s):  
Sansit Patnaik ◽  
John P. Hollkamp ◽  
Fabio Semperlotti
Keyword(s):  
Transport Processes ◽  
Point Of View ◽  
Variable Order ◽  
Fractional Operators ◽  
Mathematical Concepts ◽  
Physical Systems ◽  
Practical Applications ◽  
Starting Point ◽  
Made In ◽  
Real World Problems

Variable-order fractional operators were conceived and mathematically formalized only in recent years. The possibility of formulating evolutionary governing equations has led to the successful application of these operators to the modelling of complex real-world problems ranging from mechanics, to transport processes, to control theory, to biology. Variable-order fractional calculus (VO-FC) is a relatively less known branch of calculus that offers remarkable opportunities to simulate interdisciplinary processes. Recognizing this untapped potential, the scientific community has been intensively exploring applications of VO-FC to the modelling of engineering and physical systems. This review is intended to serve as a starting point for the reader interested in approaching this fascinating field. We provide a concise and comprehensive summary of the progress made in the development of VO-FC analytical and computational methods with application to the simulation of complex physical systems. More specifically, following a short introduction of the fundamental mathematical concepts, we present the topic of VO-FC from the point of view of practical applications in the context of scientific modelling.

An original perspective on variable-order fractional operators for viscoelastic materials

Meccanica ◽  
2021 ◽  
Vol 56 (4) ◽  
pp. 769-784
Author(s):  
Andrea Burlon ◽  
Gioacchino Alotta ◽  
Mario Di Paola ◽  
Giuseppe Failla
Keyword(s):  
Variable Order ◽  
Fractional Operators ◽  
Viscoelastic Materials

Synchronization of chaotic systems involving fractional operators of Liouville–Caputo type with variable-order

2017 ◽  
Vol 487 ◽  
pp. 1-21 ◽  
Author(s):  
A. Coronel-Escamilla ◽  
J.F. Gómez-Aguilar ◽  
L. Torres ◽  
R.F. Escobar-Jiménez ◽  
M. Valtierra-Rodríguez
Keyword(s):  
Chaotic Systems ◽  
Variable Order ◽  
Fractional Operators

scholarly journals Modeling Contacts and Hysteretic Behavior in Discrete Systems Via Variable-Order Fractional Operators

2020 ◽  
Vol 15 (9) ◽  
Author(s):  
Sansit Patnaik ◽  
Fabio Semperlotti
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Contact Problems ◽  
Discrete Systems ◽  
Hysteretic Behavior ◽  
Variable Order ◽  
Fractional Operators ◽  
State Variables ◽  
Nonlinear Dynamical ◽  
Integer Order

Abstract The modeling of nonlinear dynamical systems subject to strong and evolving nonsmooth nonlinearities is typically approached via integer-order differential equations. In this study, we present the possible application of variable-order (VO) fractional operators to a class of nonlinear lumped parameter models that have great practical relevance in mechanics and dynamics. Fractional operators are intrinsically multiscale operators that can act on both space- and time-dependent variables. Contrarily to their integer-order counterpart, fractional operators can have either fixed or VO. In the latter case, the order can be function of either independent or state variables. We show that when using VO equations to describe the response of dynamical systems, the order can evolve as a function of the response itself; therefore, allowing a natural and seamless transition between widely dissimilar dynamics. Such an intriguing characteristic allows defining governing equations for dynamical systems that are evolutionary in nature. Within this context, we present a physics-driven strategy to define VO operators capable of capturing complex and evolutionary phenomena. Specific examples include hysteresis in discrete oscillators and contact problems. Despite using simplified models to illustrate the applications of VO operators, we show numerical evidence of their unique modeling capabilities as well as their connection to more complex dynamical systems.

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