FDA Express

This volume in the “Encyclopedia of Complexity and Systems Science, Second Edition,” offers a detailed account of fractional calculus tools, signatures of complex systems, hidden connections to fractional calculus, and applications and case studies involving fractional calculus in complex signal analysis and complex system modelling, analysis and control (MAD). The authors document both the foundational concepts of fractional calculus in complexity science as well as their applications to, and role in the optimization of, complex engineered systems. Fractional calculus is about differentiation and integration of non-integer orders. Convenience has driven the use of integer-order models and controllers for complex natural or man-made systems, but these traditional models and tools for the control of dynamic systems may result in suboptimum performance and even “anomalous” phenomena. In contrast, the growing literature documents “more optimal” performance when fractional order calculus tools are applied. From an engineering point of view, new and beneficial uses of this versatile mathematical tool are both possible and important, and may become an enabler of new science discoveries.

· Presents the first comprehensive coverage of the fractional calculus role in complex systems.

· Discusses major existing signatures such as power law of complex systems with emphasis on the connections to the fractional calculus

· Includes a wide variety of real world case studies in signal analysis and complex system modeling and control

Topic Areas (Table of Contents in preparation):

· Fractional calculus:

o definitions, history, basic properties

o Fractional order dynamic systems

o Fractional noises

· Signatures of complex systems and its fractional calculus connection

o Power law

o Long range dependence

o Long memory

o Long range interaction

o Heavytailedness

o etc.

· Complex signal analysis using fractional calculus

· Complex system modeling using fractional calculus

· Complex system control using fractional calculus

Key parameters: Minimum number of chapters 30 to 50 (no upper limit); 9,000–12,000 words (plus figures and references) per chapter (or 10-12 published pages); delivery date 12/31/2018; submission via online system only at https://meteor.springer.com

If you are interested in contributing a chapter in this new volume “Fractional Calculus for Complex Systems”, please visit http://mechatronics.ucmerced.edu/fccs for Guidelines and background information. Send me a chapter proposal (one page) with title/authors/affiliations/contact info/synopsis/keywords to email: yqchen@ieee.org with FCCS on email subject title for easy search. Thank you! 6/24/2017

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Doctoral thesis

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Numerical study on unsteady convection, heat and mass transfer of fractional viscoelastic fluid

Author: Zhao Jinhu

Supervisor： Zheng Liancun
School of Energy and Environmental Engineering
University of Science and Technology Beijing
30 Xueyuan Road， Haidian District
Beijing 100083， P.R.CHINA

Abstract

Recently, the fractional derivatives were found to be quite flexible in the description of complex dynamics (such as constitutive relation and heat conduction law) in viscoelastic fluids due to its nonlocal property or global dependency of fractional operators. In general, the constitutive relationships for generalized viscoelastic fluids are modified from well-known fluid models by replacing the time derivative of an integer order by the so-called fractional calculus operator. However, in classical study on viscoelastic fluid complex flow and heat transfer, the authors commonly ignored the effects of nonlinear convection and dealt only with the cases when the governing equations are linear. Their methods, i.e., integral transforms, are difficult to solve the problems when the equations are nonlinear. Very little efforts have so far been made to discuss nonlinear convection terms with fractional derivatives.

This paper presents a research on unsteady boundary layer convection heat and mass transfer of viscoelastic fluid. Fractional shear stress and heat flux models are introduced and the fractional boundary layer governing equations are firstly formulated and derived. From such derivation, the model constitutes nonlinear coupled equations with mixed time-space derivatives in the convection and diffusion terms, which are solved by a newly developed finite difference method combined with an L1-algorithm. Moreover, the fractional Marangoni boundary condition is firstly established via the balance between the surface tension and shear stress. Modified Darcy’s law is employed to investigate flow and heat transfer characteristics of fractional viscoelastic fluid through a porous medium. Some novel phenomena are found that both the velocity and temperature boundary layers manifest short-term memory and basic relaxation time characteristics, shear stress shows slow response to external body force. The Marangoni surface tension and local Nusselt number show different variation tendencies dependent on temperature. The effects of fractional derivative parameters and other involved parameters on velocity, temperature and concentration fields are presented graphically and analyzed in detail to characterize the complexity of heat and mass transfer of viscoelastic fluid. The Marangoni number plays a connecting role between the velocity and temperature gradients on the boundary surface and only has slight influence on the thickness of the boundary layers. The temperature distributions decline with the increase of porosity, which demonstrate a loss of the thickness of the thermal boundary layer. Better permeability in porous medium not only promotes momentum transmission of the fluid, but also reduces the heat conduction loss in the convection flow. Brownian motion number accelerates the convection flow and improves the efficiency of mass transfer, while thermophoresis number has strong effect on nanoparticle diffusion during the convection flow. Key Words： Viscoelastic fluid, Fractional model, Marangoni convection, Natural convection, Heat and mass transfer

Key Words： Viscoelastic fluid, Fractional model, Marangoni convection, Natural convection, Heat and mass transfer

For more information on this doctoral thesis, readers can contact the author by the following email address:

zhaojin_hu@126.com

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Books

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Stochastic Calculus for Fractional Brownian Motion and Related Processes

Mishura, Yuliya

Book Description

The theory of fractional Brownian motion and other long-memory processes are addressed in this volume. Interesting topics for PhD students and specialists in probability theory, stochastic analysis and financial mathematics demonstrate the modern level of this field. Among these are results about Levy characterization of fractional Brownian motion, maximal moment inequalities for Wiener integrals including the values 0<H<1/2 of Hurst index, the conditions of existence and uniqueness of solutions to SDE involving additive Wiener integrals, and of solutions of the mixed Brownian—fractional Brownian SDE. The author develops optimal filtering of mixed models including linear case, and studies financial applications and statistical inference with hypotheses testing and parameter estimation. She proves that the market with stock guided by the mixed model is arbitrage-free without any restriction on the dependence of the components and deduces different forms of the Black-Scholes equation for fractional market.

More information on this book can be found by the following links:

http://www.springer.com/de/book/9783540758723

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Fractional and Multivariable Calculus: Model Building and Optimization Problems

Mathai, A.M., Haubold, H.J.

Book Description

This textbook presents a rigorous approach to multivariable calculus in the context of model building and optimization problems. This comprehensive overview is based on lectures given at five SERC Schools from 2008 to 2012 and covers a broad range of topics that will enable readers to understand and create deterministic and nondeterministic models. Researchers, advanced undergraduate, and graduate students in mathematics, statistics, physics, engineering, and biological sciences will find this book to be a valuable resource for finding appropriate models to describe real-life situations. The first chapter begins with an introduction to fractional calculus moving on to discuss fractional integrals, fractional derivatives, fractional differential equations and their solutions. Multivariable calculus is covered in the second chapter and introduces the fundamentals of multivariable calculus (multivariable functions, limits and continuity, differentiability, directional derivatives and expansions of multivariable functions). Illustrative examples, input-output process, optimal recovery of functions and approximations are given; each section lists an ample number of exercises to heighten understanding of the material. Chapter three discusses deterministic/mathematical and optimization models evolving from differential equations, difference equations, algebraic models, power function models, input-output models and pathway models. Fractional integral and derivative models are examined. Chapter four covers non-deterministic/stochastic models. The random walk model, branching process model, birth and death process model, time series models, and regression type models are examined. The fifth chapter covers optimal design. General linear models from a statistical point of view are introduced; the Gauss–Markov theorem, quadratic forms, and generalized inverses of matrices are covered. Pathway, symmetric, and asymmetric models are covered in chapter six, the concepts are illustrated with graphs.

More information on this book can be found by the following links:

http://www.springer.com/de/book/9783319599922#aboutBook

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Journals

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Chaos, Solitons & Fractals

(Selected)

Restricted fractional differential transform for solving irrational order fractional differential equations

Ayad R. Khudair, S.A.M. Haddad, Sanaa L. khalaf

A fractional Gauss–Jacobi quadrature rule for approximating fractional integrals and derivatives

S. Jahanshahi, E. Babolian, D.F.M. Torres, A.R. Vahidi

Time fractional quantum mechanics

Nick Laskin

On generalized fractional vibration equation

Hongzhe Dai, Zhibao Zheng, Wei Wang

On disappearance of chaos in fractional systems

Amey S. Deshpande, Varsha Daftardar-Gejji

On Hopf bifurcation in fractional dynamical systems

Amey S. Deshpande, Varsha Daftardar-Gejji, Yogita V. Sukale

A search for a spectral technique to solve nonlinear fractional differential equations

Malgorzata Turalska, Bruce J. West

Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system

Abdon Atangana

Conditions for continuity of fractional velocity and existence of fractional Taylor expansions

Dimiter Prodanov

Time fractional equations and probabilistic representation

Zhen-Qing Chen

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ISA Transactions

(Selected)

Modeling and analysis of fractional order DC-DC converter

Ahmed G. Radwan, Ahmed A. Emira, Amr M. AbdelAty, Ahmed T. Azar

Convergence of fractional adaptive systems using gradient approach

Javier A. Gallegos, Manuel A. Duarte-Mermoud

Impulsive stabilization of fractional differential systems

Liguang Xu, Jiankang Li, Shuzhi Sam Ge

Robust fast controller design via nonlinear fractional differential equations

Xi Zhou, Yiheng Wei, Shu Liang, Yong Wang

On the fragility of fractional-order PID controllers for FOPDT processes

Fabrizio Padula, Antonio Visioli

Diffusion control for a tempered anomalous diffusion system using fractional-order PI controllers

Juan Chen, Bo Zhuang, YangQuan Chen, Baotong Cui

Practical stability analysis of fractional-order impulsive control systems

Ivanka Stamova, Johnny Henderson

Performance analysis of fractional order extremum seeking control

Hadi Malek, Sara Dadras, YangQuan Chen

Admissibility and robust stabilization of continuous linear singular fractional order systems with the fractional order α: The 0<α<1 case

Xuefeng Zhang, YangQuan Chen

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Paper Highlight
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A finite deformation fractional viscoplastic model for the glass transition behavior of amorphous polymers

Xiao, Rui; Sun, HongGuang; Chen, Wen

Publication information: INTERNATIONAL JOURNAL OF NON-LINEAR MECHANICS Volume: 93 Pages: 7-14 Published: JUL 2017

http://www.sciencedirect.com/science/article/pii/S0020746216303493

Abstract

The stress response of amorphous polymers exhibits tremendous change during the glass transition region, from soft viscoelastic response to stiff viscoplastic response. In order to describe the temperature-dependent and rate-dependent stress response of amorphous polymers, we extend the one-dimensional small strain fractional Zener model to the three-dimensional finite deformation model. The Eyring model is adopted to represent the stress-activated viscous flow. A phenomenological evolution equation of yield strength is used to describe the strain softening behaviors. We demonstrate that the stress response predicted by the three-dimensional model is consistent with that of one-dimensional model under uniaxial deformation, which confirms the validity of the extension. The model is then applied to describe the stress response of an amorphous thermoset at various temperatures and strain rates, which shows good agreement between experiments and simulation. We further perform a parameter study to investigate the influence of the model parameters on the stress response. The results show that a smaller fractional order results in a larger yield strain while has little effect on the yield stress when the temperature is below the glass transition temperature. For the stress relaxation tests, a smaller fractional order leads to a slower relaxation rate.

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Uncertain viscoelastic models with fractional order: A new spectral tau method to study the numerical simulations of the solution

Ahmadian, A.; Ismail, F.; Salahshour, S.; et al.

Publication information: COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION Volume: 53 Pages: 44-64 Published: DEC 2017

http://www.sciencedirect.com/science/article/pii/S1007570417300874

Abstract

The analysis of the behaviors of physical phenomena is important to discover significant features of the character and the structure of mathematical models. Frequently the unknown parameters involve in the models are assumed to be unvarying over time. In reality, some of them are uncertain and implicitly depend on several factors. In this study, to consider such uncertainty in variables of the models, they are characterized based on the fuzzy notion. We propose here a new model based on fractional calculus to deal with the Kelvin–Voigt (KV) equation and non-Newtonian fluid behavior model with fuzzy parameters. A new and accurate numerical algorithm using a spectral tau technique based on the generalized fractional Legendre polynomials (GFLPs) is developed to solve those problems under uncertainty. Numerical simulations are carried out and the analysis of the results highlights the significant features of the new technique in comparison with the previous findings. A detailed error analysis is also carried out and discussed.

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The End of This Issue

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