Nomenclature The Challenge of Complexity Little Science, Big Science Complexity Chapter Overviews After Thoughts Yesterday's Science Simple Linearity Complicated Linearity Nonlinear Dynamics After Thoughts Appendix Chapter 2 New Ways of Thinking Why Now? Through the Looking Glass Non-differentiablity can be Physical The size effect After Thoughts Simple Fractional Operators Random Walks Fractional Derivatives Fractional Rate Equations After Thoughts Tomorrow's Dynamics What We Think We Know Linear Systems 126 5.2 Fractional Linear System Applications of FLE Control of Complexity Fractional Logistic Equation Fractional Leibniz Rule After Thoughts Appendix Chapter 5 Fractional Cooperation HRV and Levy Statistics Fractional Wave Equations Turbulence Fractional Magnetization Equations Fractional Search Hypothesis The Network Effect After Thoughts Strange Statistics Fractional Hamiltonian Formalism Anomalous Transport Fractional Fokker-Planck Equation Fractional Sturm-Liouville Theory Fractional Kinetics Equation Physiology and Complexity Loss Diffusion Entropy After Thoughts Appendices Chapter 7 What have we learned? Why the fractional calculus? Conformation Bias and Scientific Truth Final Thoughts

Fractional calculus was born in 1695 on September 30 due to a very deep question raised in a letter of L鈥橦ospital to Leibniz. The prophetical answer of Leibniz to that deep question encapsulated a huge inspiration for all generations of scientists and is continuing to stimulate the minds of contemporary researchers. During 325 years of existence, fractional calculus has kept the attention of top level mathematicians, and during the last period of time it has become a very useful tool for tacklin...

#2Mamikon Gulian(SNL: Sandia National Laboratories)H-Index: 8

Last. Marta D'Elia(SNL: Sandia National Laboratories)H-Index: 16

view all 4 authors...

Modeling of phenomena such as anomalous transport via fractional-order differential equations has been established as an effective alternative to partial differential equations, due to the inherent ability to describe large-scale behavior with greater efficiency than fully-resolved classical models. In this review article, we first provide a broad overview of fractional-order derivatives with a clear emphasis on the stochastic processes that underlie their use. We then survey three exemplary app...

Last. Huanying Xu(SDU: Shandong University)H-Index: 9

view all 4 authors...

Abstract null null Laser technology has been widely used in biomedical therapies and external surgeries. To promote its applications, a good thermal model is required to analyze the temperature distribution within the living tissues. In this paper, we develop a time-space fractional hyperbolic bioheat transfer model to study the non-Fourier bioheat transfer process within the living biological tissues during laser irradiation. Based on the the null null null null L null 1 null null null null app...

Abstract null null In this study, a stochastic differential equation capable of describing (using the motion function) the automatic manufacturing process of a lithium battery with a sleeve shell is introduced. The boundary-condition modeling method for this type of motion is an ordinary differential equation. The nonlinear equation is found using a dynamic method. The equations of the motions for the assembly process are derived by reducing the order of terms and separating the variables. Both ...

We address the problem of telegraphic transport in several dimensions. We review the derivation of two and three dimensional telegrapher鈥檚 equations鈥攁s well as their fractional generalizations鈥攆rom microscopic random walk models for transport (normal and anomalous). We also present new results on solutions of the higher dimensional fractional equations.

#1Haoyu Niu(UCM: University of California, Merced)H-Index: 5

#2YangQuan Chen(UCM: University of California, Merced)H-Index: 94

Last. Bruce J. West(RTP: Research Triangle Park)H-Index: 73

view all 3 authors...

Fractional-order calculus is about the differentiation and integration of non-integer orders. Fractional calculus (FC) is based on fractional-order thinking (FOT) and has been shown to help us to understand complex systems better, improve the processing of complex signals, enhance the control of complex systems, increase the performance of optimization, and even extend the enabling of the potential for creativity. In this article, the authors discuss the fractional dynamics, FOT and rich fractio...

The Wright function is a generalization of the exponential function and the Bessel functions. Integral relations between the Mittag鈥揕effler functions and the Wright function are presented. The applications of the Wright function and the Mainardi function to description of diffusion, heat conduction, thermal and diffusive stresses, and nonlocal elasticity in the framework of fractional calculus are discussed.

#1Antonio G. Goulart(UFRGS: Universidade Federal do Rio Grande do Sul)H-Index: 12

#2Matheus J. Lazo(UFRGS: Universidade Federal do Rio Grande do Sul)H-Index: 14

Last. J.M.S. Suarez(UFRGS: Universidade Federal do Rio Grande do Sul)H-Index: 2

view all 3 authors...

Abstract In the present work, we propose an advection-diffusion equation with Hausdorff deformed derivatives to stud the turbulent diffusion of contaminants in the atmosphere. We compare the performance of our model to fit experimental data against models with classical and Caputo fractional derivatives. We found that the Hausdorff equation gives better results than the tradition advection-diffusion equation when fitting experimental data. Most importantly, we show that our model and the Caputo ...

Financial time series have a fractal nature that poses challenges for their dynamical characterization. The Dow Jones Industrial Average (DJIA) is one of the most influential financial indices, and due to its importance, it is adopted as a test bed for this study. The paper explores an alternative strategy to the standard time analysis, by joining the multidimensional scaling (MDS) computational tool and the concepts of distance, entropy, fractal dimension, and fractional calculus. First, severa...

In this paper, a generalized fractional-order SEIR model is proposed, denoted by SEIQRP model, which divided the population into susceptible, exposed, infectious, quarantined, recovered and insusceptible individuals and has a basic guiding significance for the prediction of the possible outbreak of infectious diseases like the coronavirus disease in 2019 (COVID-19) and other insect diseases in the future. Firstly, some qualitative properties of the model are analyzed. The basic reproduction numb...

We use cookies to improve your online experience. By continuing to use our website we assume you agree to the placement of these cookies. To learn more, you can find in our Privacy Policy.