Fractional Differential Equations

Bees are subject to numerous pressures in the modern world. The abundance and diversity of flowers has declined, bees are chronically exposed to cocktails of agrochemicals, and they are simultaneously exposed to novel parasites accidentally spread by humans. Climate change is likely to exacerbate these problems in the future. Stressors do not act in isolation; for example pesticide exposure can impair both detoxification mechanisms and immune responses, rendering bees more susceptible to parasites. It seems certain that chronic exposure to multiple, interacting stressors is driving honey bee colony losses and declines of wild pollinators, but such interactions are not addressed by current regulatory procedures and studying these interactions experimentally poses a major challenge. In the meantime, taking steps to reduce stress on bees would seem prudent; incorporating flower-rich habitat into farmland, reducing pesticide use through adopting more sustainable farming methods, and enforcing effective quarantine measures on bee movements are all practical measures that should be adopted. Effective monitoring of wild pollinator populations is urgently needed to inform management strategies into the future.

https://science.sciencemag.org/content/sci/early/2015/02/25/science.1255957.full.pdf

Bee declines driven by combined stress from parasites, pesticides, and lack of flowers

1. Introduction 2. Reference kinematics 3. Analytical techniques 4. Mechanics of deformable bodies 5. Floating frame of reference formulation 6. Finite element formulation 7. Large deformation problem Appendix: Linear algebra References Index.

Dynamics of multibody systems

In the paper, we present a new definition of fractional deriva tive with a smooth kernel which takes on two different representations for the temporal and spatial variable. The first works on the time variables; thus it is suitable to use th e Laplace transform. The second definition is related to the spatial va riables, by a non-local fractional derivative, for which it is more convenient to work with the Fourier transform. The interest for this new approach with a regular kernel was born from the prospect that there is a class of non-local systems, which have the ability to descri be the material heterogeneities and the fluctuations of diff erent scales, which cannot be well described by classical local theories or by fractional models with singular kernel.

http://www.naturalspublishing.com/files/published/0gb83k287mo759.pdf

A new Definition of Fractional Derivative without Singular Kernel

This chapter introduces the finite element method (FEM) as a tool for solution of classical electromagnetic problems. Although we discuss the main points in the application of the finite element method to electromagnetic design, including formulation and implementation, those who seek deeper understanding of the finite element method should consult some of the works listed in the bibliography section.

Introduction to the Finite Element Method

ductor, where the conductivity is sufficiently high, the displacement current density can be neglected. In this case, the conduction current density is given by the product of the electric field and the conductance. One of the aspects of with SE is it attenuates the higher frequency components of a signal. The SE was first verified by Kelvin in 1887. Since then many researchers developed work on the subject and presently a comprehensive physical model, based on the Maxwell equations, is well established. The Maxwell formalism plays a fundamental role in the electromagnetic theory. These equations lead to the derivation of mathematical descriptions useful in many applications in physics and engineering. Maxwell is generally The Maxwell equations involve only the integer-order calculus and, therefore, it is natural that the resulting classical models adopted in electrical engineering reflect this perspective. Recently, a closer look of some phenomof precise models, seem to point out the requirement for a fractional calculus approach. Bearing these ideas in mind, in this study we address the SE and we re-evaluate the results demonstrating its fractional-order nature. Department of Electrical Engineering, Institute of Engineering of Porto Rua Dr. Antonio Bernardino de Almeida, 4200-072 Porto, Portugal; E-mail: jtm,isj,amf@isep.ipp.pt Engineering Systems, Vila-Real, Portugal; E-mail: jboavent@utad.pt Institute of Intelligent Engineering Systems, Budapest Tech, John von Neumann Faculty of Informatics, Budapest, Hungary; E-mail: tar@nik.bmf.hu the high-frequency effects is the skin effect (SE ). The fundamental problem regarded as the 19th century scientist who had the greatest influence on 20th century physics, making contributions to the fundamental models of nature. enas present in electrical systems and the motivation towards the development

Advances in Fractional Calculus

In the last two decades, fractional (or non integer) differentiation has played a very important role in various fields such as mechanics, electricity, chemistry, biology, economics, control theory and signal and image processing. For example, in the last three fields, some important considerations such as modelling, curve fitting, filtering, pattern recognition, edge detection, identification, stability, controllability, observability and robustness are now linked to long-range dependence phenomena. Similar progress has been made in other fields listed here. The scope of the book is thus to present the state of the art in the study of fractional systems and the application of fractional differentiation. As this volume covers recent applications of fractional calculus, it will be of interest to engineers, scientists, and applied mathematicians.

Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering

Formulation of Euler–Lagrange equations for fractional variational problems

Accurate modeling of many dynamic systems leads to a set of Fractional Differential Equations (FDEs). This paper presents a general formulation and a solution scheme for a class of Fractional Optimal Control Problems (FOCPs) for those systems. The fractional derivative is described in the Riemann–Liouville sense. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of FDEs. The Calculus of Variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler–Lagrange equations for the FOCP. The formulation presented and the resulting equations are very similar to those that appear in the classical optimal control theory. Thus, the present formulation essentially extends the classical control theory to fractional dynamic system. The formulation is used to derive the control equations for a quadratic linear fractional control problem. An approach similar to a variational virtual work coupled with the Lagrange multiplier technique is presented to find the approximate numerical solution of the resulting equations. Numerical solutions for two fractional systems, a time-invariant and a time-varying, are presented to demonstrate the feasibility of the method. It is shown that (1) the solutions converge as the number of approximating terms increase, and (2) the solutions approach to classical solutions as the order of the fractional derivatives approach to 1. The formulation presented is simple and can be extended to other FOCPs. It is hoped that the simplicity of this formulation will initiate a new interest in the area of optimal control of fractional systems.

A General Formulation and Solution Scheme for Fractional Optimal Control Problems

A general solution is given for a fractional diffusion-wave equation defined in a bounded space domain. The fractional time derivative is described in the Caputo sense. The finite sine transform technique is used to convert a fractional differential equation from a space domain to a wavenumber domain. Laplace transform is used to reduce the resulting equation to an ordinary algebraic equation. Inverse Laplace and inverse finite sine transforms are used to obtain the desired solutions. The response expressions are written in terms of the Mittag–Leffler functions. For the first and the second derivative terms, these expressions reduce to the ordinary diffusion and wave solutions. Two examples are presented to show the application of the present technique. Results show that for fractional time derivatives of order 1/2 and 3/2, the system exhibits, respectively, slow diffusion and mixed diffusion-wave behaviors.

Om P. Agrawal

Papers

Advances in Fractional Calculus

Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering

Formulation of Euler–Lagrange equations for fractional variational problems

A General Formulation and Solution Scheme for Fractional Optimal Control Problems

Solution for a Fractional Diffusion-Wave Equation Defined in a Bounded Domain