The Fractional Laplacian

Author: C. Pozrikidis

Publisher: CRC Press

ISBN: 1315359936

Category: Mathematics

Page: 278

View: 6488

The fractional Laplacian, also called the Riesz fractional derivative, describes an unusual diffusion process associated with random excursions. The Fractional Laplacian explores applications of the fractional Laplacian in science, engineering, and other areas where long-range interactions and conceptual or physical particle jumps resulting in an irregular diffusive or conductive flux are encountered. Presents the material at a level suitable for a broad audience of scientists and engineers with rudimentary background in ordinary differential equations and integral calculus Clarifies the concept of the fractional Laplacian for functions in one, two, three, or an arbitrary number of dimensions defined over the entire space, satisfying periodicity conditions, or restricted to a finite domain Covers physical and mathematical concepts as well as detailed mathematical derivations Develops a numerical framework for solving differential equations involving the fractional Laplacian and presents specific algorithms accompanied by numerical results in one, two, and three dimensions Discusses viscous flow and physical examples from scientific and engineering disciplines Written by a prolific author well known for his contributions in fluid mechanics, biomechanics, applied mathematics, scientific computing, and computer science, the book emphasizes fundamental ideas and practical numerical computation. It includes original material and novel numerical methods.
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Fractional Order Systems

Optimization, Control, Circuit Realizations and Applications

Author: Ahmad Taher Azar,Ahmed G. Radwan,Sundarapandian Vaidyanathan

Publisher: Academic Press

ISBN: 0128163089

Category: Technology & Engineering

Page: 741

View: 7679

Fractional Order Systems: Optimization, Control, Circuit Realizations and Applications consists of 21 contributed chapters by subject experts. Chapters offer practical solutions and novel methods for recent research problems in the multidisciplinary applications of fractional order systems, such as FPGA, circuits, memristors, control algorithms, photovoltaic systems, robot manipulators, oscillators, etc. This book is ideal for researchers working in the modeling and applications of both continuous-time and discrete-time dynamics and chaotic systems. Researchers from academia and industry who are working in research areas such as control engineering, electrical engineering, mechanical engineering, computer science, and information technology will find the book most informative. Discusses multi-disciplinary applications with new fundamentals, modeling, analysis, design, realization and experimental results Includes new circuits and systems based on the new nonlinear elements Covers most of the linear and nonlinear fractional-order theorems that will solve many scientific issues for researchers Closes the gap between theoretical approaches and real-world applications Provides MATLAB® and Simulink code for many of the applications in the book
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Fractional Partial Differential Equations and Their Numerical Solutions

Author: Boling Guo,Xueke Pu,Fenghui Huang

Publisher: World Scientific

ISBN: 9814667064

Category: Mathematics

Page: 348

View: 8978

This book aims to introduce some new trends and results on the study of the fractional differential equations, and to provide a good understanding of this field to beginners who are interested in this field, which is the authors' beautiful hope. This book describes theoretical and numerical aspects of the fractional partial differential equations, including the authors' researches in this field, such as the fractional Nonlinear Schrödinger equations, fractional Landau–Lifshitz equations and fractional Ginzburg–Landau equations. It also covers enough fundamental knowledge on the fractional derivatives and fractional integrals, and enough background of the fractional PDEs. Contents:Physics BackgroundFractional Calculus and Fractional Differential EquationsFractional Partial Differential EquationsNumerical Approximations in Fractional CalculusNumerical Methods for the Fractional Ordinary Differential EquationsNumerical Methods for Fractional Partial Differential Equations Readership: Graduate students and researchers in mathematical physics, numerical analysis and computational mathematics. Key Features:This book covers the fundamentals of this field, especially for the beginnersThe book covers new trends and results in this fieldThe book covers numerical results, which will be of broad interests to researchersKeywords:Fractional Partial Differential Equations;Numerical Solutions
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Partial Differential Equations and Geometric Measure Theory

Cetraro, Italy 2014

Author: Alessio Figalli,Enrico Valdinoci,Ireneo Peral

Publisher: Springer

ISBN: 3319740423

Category: Mathematics

Page: 216

View: 5013

This book collects together lectures by some of the leaders in the field of partial differential equations and geometric measure theory. It features a wide variety of research topics in which a crucial role is played by the interaction of fine analytic techniques and deep geometric observations, combining the intuitive and geometric aspects of mathematics with analytical ideas and variational methods. The problems addressed are challenging and complex, and often require the use of several refined techniques to overcome the major difficulties encountered. The lectures, given during the course "Partial Differential Equations and Geometric Measure Theory'' in Cetraro, June 2–7, 2014, should help to encourage further research in the area. The enthusiasm of the speakers and the participants of this CIME course is reflected in the text.
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Potential Analysis of Stable Processes and its Extensions

Author: Krzysztof Bogdan,Tomasz Byczkowski,Tadeusz Kulczycki,Michal Ryznar,Renming Song,Zoran Vondracek

Publisher: Springer Science & Business Media

ISBN: 3642021417

Category: Mathematics

Page: 194

View: 6713

Stable Lévy processes and related stochastic processes play an important role in stochastic modelling in applied sciences, in particular in financial mathematics. This book is about the potential theory of stable stochastic processes. It also deals with related topics, such as the subordinate Brownian motions (including the relativistic process) and Feynman–Kac semigroups generated by certain Schrödinger operators. The authors focus on classes of stable and related processes that contain the Brownian motion as a special case. This is the first book devoted to the probabilistic potential theory of stable stochastic processes, and, from the analytical point of view, of the fractional Laplacian. The introduction is accessible to non-specialists and provides a general presentation of the fundamental objects of the theory. Besides recent and deep scientific results the book also provides a didactic approach to its topic, as all chapters have been tested on a wide audience, including young mathematicians at a CNRS/HARP Workshop, Angers 2006. The reader will gain insight into the modern theory of stable and related processes and their potential analysis with a theoretical motivation for the study of their fine properties.
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Physics of Fractal Operators

Author: Bruce West,Mauro Bologna,Paolo Grigolini

Publisher: Springer Science & Business Media

ISBN: 9780387955544

Category: Mathematics

Page: 354

View: 7955

In Chapter One we review the foundations of statistieal physies and frac­ tal functions. Our purpose is to demonstrate the limitations of Hamilton's equations of motion for providing a dynamical basis for the statistics of complex phenomena. The fractal functions are intended as possible models of certain complex phenomena; physical.systems that have long-time mem­ ory and/or long-range spatial interactions. Since fractal functions are non­ differentiable, those phenomena described by such functions do not have dif­ ferential equations of motion, but may have fractional-differential equations of motion. We argue that the traditional justification of statistieal mechan­ ics relies on aseparation between microscopic and macroscopie time scales. When this separation exists traditional statistieal physics results. When the microscopic time scales diverge and overlap with the macroscopie time scales, classieal statistieal mechanics is not applicable to the phenomenon described. In fact, it is shown that rather than the stochastic differential equations of Langevin describing such things as Brownian motion, we ob­ tain fractional differential equations driven by stochastic processes.
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Dispersion in Heterogeneous Geological Formations

Author: Brian Berkowitz

Publisher: Springer Science & Business Media

ISBN: 9780792367796

Category: Mathematics

Page: 263

View: 9015

In spite of many years of intensive study, our current abilities to quantify and predict contaminant migration in natural geological formations remain severely limited. The heterogeneity of these formations over a wide range of scales necessitates consideration of sophisticated transport theories. The evolution of such theories has escalated to the point that a review of the subject seems timely. While conceptual and mathematical developments were crucial to the introduction of these new approaches, there are now too many publications that contain theoretical abstractions without regard to real systems, or incremental improvements to existing theories which are known not to be applicable. This volume brings together articles representing a broad spectrum of state-of-the-art approaches for characterization and quantification of contaminant dispersion in heterogeneous porous media. Audience: The contributions are intended to be as accessible as possible to a wide readership of academics and professionals with diverse backgrounds such as earth sciences, subsurface hydrology, petroleum engineering, and soil physics.
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Stochastic Partial Differential Equations and Applications - VII

Author: Giuseppe Da Prato,Luciano Tubaro

Publisher: CRC Press

ISBN: 9781420028720

Category: Mathematics

Page: 347

View: 3963

Stochastic Partial Differential Equations and Applications gives an overview of current state-of-the-art stochastic PDEs in several fields, such as filtering theory, stochastic quantization, quantum probability, and mathematical finance. Featuring contributions from leading expert participants at an international conference on the subject, this book presents valuable information for PhD students in probability and PDEs as well as for researchers in pure and applied mathematics. Coverage includes Navier-Stokes equations, Ornstein-Uhlenbeck semigroups, quantum stochastic differential equations, applications of SPDE, 3D stochastic Navier-Stokes equations, and nonlinear filtering.
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Spherical and Plane Integral Operators for PDEs

Construction, Analysis, and Applications

Author: Karl K. Sabelfeld,Irina A. Shalimova

Publisher: Walter de Gruyter

ISBN: 3110315335

Category: Mathematics

Page: 338

View: 558

The book presents integral formulations for partial differential equations, with the focus on spherical and plane integral operators. The integral relations are obtained for different elliptic and parabolic equations, and both direct and inverse mean value relations are studied. The derived integral equations are used to construct new numerical methods for solving relevant boundary value problems, both deterministic and stochastic based on probabilistic interpretation of the spherical and plane integral operators.
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Semigroups of Operators -Theory and Applications

Będlewo, Poland, October 2013

Author: Jacek Banasiak,Adam Bobrowski,Mirosław Lachowicz

Publisher: Springer

ISBN: 3319121456

Category: Mathematics

Page: 337

View: 5894

Many results, both from semi group theory itself and from the applied sciences, are phrased in discipline-specific languages and hence are hardly known to a broader community. This volume contains a selection of lectures presented at a conference that was organised as a forum for all mathematicians using semi group theory to learn what is happening outside their own field of research. The collection will help to establish a number of new links between various sub-disciplines of semigroup theory, stochastic processes, differential equations and the applied fields. The theory of semigroups of operators is a well-developed branch of functional analysis. Its foundations were laid at the beginning of the 20th century, while the fundamental generation theorem of Hille and Yosida dates back to the forties. The theory was, from the very beginning, designed as a universal language for partial differential equations and stochastic processes, but at the same time it started to live as an independent branch of operator theory. Nowadays, it still has the same distinctive flavour: it develops rapidly by posing new ‘internal’ questions and in answering them, discovering new methods that can be used in applications. On the other hand, it is influenced by questions from PDEs and stochastic processes as well as from applied sciences such as mathematical biology and optimal control, and thus it continually gathers a new momentum. Researchers and postgraduate students working in operator theory, partial differential equations, probability and stochastic processes, analytical methods in biology and other natural sciences, optimization and optimal control will find this volume useful.
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Nonlocal Diffusion and Applications

Author: Claudia Bucur,Enrico Valdinoci

Publisher: Springer

ISBN: 3319287397

Category: Mathematics

Page: 155

View: 3565

Working in the fractional Laplace framework, this book provides models and theorems related to nonlocal diffusion phenomena. In addition to a simple probabilistic interpretation, some applications to water waves, crystal dislocations, nonlocal phase transitions, nonlocal minimal surfaces and Schrödinger equations are given. Furthermore, an example of an s-harmonic function, its harmonic extension and some insight into a fractional version of a classical conjecture due to De Giorgi are presented. Although the aim is primarily to gather some introductory material concerning applications of the fractional Laplacian, some of the proofs and results are new. The work is entirely self-contained, and readers who wish to pursue related subjects of interest are invited to consult the rich bibliography for guidance.
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