Introduction To Fractional And Pseudo Differential Equations With Singular Symbols

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Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

Sabir Umarov
Introduction to Fractional and Pseudo Differential Equations with Singular Symbols Book PDF

Author : Sabir Umarov
Publisher : Springer
Page : 434 pages
File Size : 47,9 Mb
Release : 2015-08-18
Category : Mathematics
ISBN : 9783319207711

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Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols by Sabir Umarov Pdf

The book systematically presents the theories of pseudo-differential operators with symbols singular in dual variables, fractional order derivatives, distributed and variable order fractional derivatives, random walk approximants, and applications of these theories to various initial and multi-point boundary value problems for pseudo-differential equations. Fractional Fokker-Planck-Kolmogorov equations associated with a large class of stochastic processes are presented. A complex version of the theory of pseudo-differential operators with meromorphic symbols based on the recently introduced complex Fourier transform is developed and applied for initial and boundary value problems for systems of complex differential and pseudo-differential equations.

Introduction to Fractional Differential Equations

Constantin Milici,Gheorghe Drăgănescu,J. Tenreiro Machado
Introduction to Fractional Differential Equations Book PDF

Author : Constantin Milici,Gheorghe Drăgănescu,J. Tenreiro Machado
Publisher : Springer
Page : 188 pages
File Size : 51,5 Mb
Release : 2018-10-28
Category : Technology & Engineering
ISBN : 9783030008956

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Introduction to Fractional Differential Equations by Constantin Milici,Gheorghe Drăgănescu,J. Tenreiro Machado Pdf

This book introduces a series of problems and methods insufficiently discussed in the field of Fractional Calculus – a major, emerging tool relevant to all areas of scientific inquiry. The authors present examples based on symbolic computation, written in Maple and Mathematica, and address both mathematical and computational areas in the context of mathematical modeling and the generalization of classical integer-order methods. Distinct from most books, the present volume fills the gap between mathematics and computer fields, and the transition from integer- to fractional-order methods.

Regional Analysis of Time-Fractional Diffusion Processes

Fudong Ge,YangQuan Chen,Chunhai Kou
Regional Analysis of Time Fractional Diffusion Processes Book PDF

Author : Fudong Ge,YangQuan Chen,Chunhai Kou
Publisher : Springer
Page : 250 pages
File Size : 40,7 Mb
Release : 2018-01-08
Category : Technology & Engineering
ISBN : 9783319728964

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Regional Analysis of Time-Fractional Diffusion Processes by Fudong Ge,YangQuan Chen,Chunhai Kou Pdf

This monograph provides an accessible introduction to the regional analysis of fractional diffusion processes. It begins with background coverage of fractional calculus, functional analysis, distributed parameter systems and relevant basic control theory. New research problems are then defined in terms of their actuation and sensing policies within the regional analysis framework. The results presented provide insight into the control-theoretic analysis of fractional-order systems for use in real-life applications such as hard-disk drives, sleep stage identification and classification, and unmanned aerial vehicle control. The results can also be extended to complex fractional-order distributed-parameter systems and various open questions with potential for further investigation are discussed. For instance, the problem of fractional order distributed-parameter systems with mobile actuators/sensors, optimal parameter identification, optimal locations/trajectory of actuators/sensors and regional actuation/sensing configurations are of great interest. The book’s use of illustrations and consistent examples throughout helps readers to understand the significance of the proposed fractional models and methodologies and to enhance their comprehension. The applications treated in the book run the gamut from environmental science to national security. Academics and graduate students working with cyber-physical and distributed systems or interested in the applications of fractional calculus will find this book to be an instructive source of state-of-the-art results and inspiration for further research.

Transmutation Operators and Applications

Vladislav V. Kravchenko,Sergei M. Sitnik
Transmutation Operators and Applications Book PDF

Author : Vladislav V. Kravchenko,Sergei M. Sitnik
Publisher : Springer Nature
Page : 685 pages
File Size : 47,7 Mb
Release : 2020-04-11
Category : Mathematics
ISBN : 9783030359140

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Transmutation Operators and Applications by Vladislav V. Kravchenko,Sergei M. Sitnik Pdf

Transmutation operators in differential equations and spectral theory can be used to reveal the relations between different problems, and often make it possible to transform difficult problems into easier ones. Accordingly, they represent an important mathematical tool in the theory of inverse and scattering problems, of ordinary and partial differential equations, integral transforms and equations, special functions, harmonic analysis, potential theory, and generalized analytic functions. This volume explores recent advances in the construction and applications of transmutation operators, while also sharing some interesting historical notes on the subject.

Basic Theory

Anatoly Kochubei,Yuri Luchko
Basic Theory Book PDF

Author : Anatoly Kochubei,Yuri Luchko
Publisher : Walter de Gruyter GmbH & Co KG
Page : 489 pages
File Size : 51,6 Mb
Release : 2019-02-19
Category : Mathematics
ISBN : 9783110571622

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Basic Theory by Anatoly Kochubei,Yuri Luchko Pdf

This multi-volume handbook is the most up-to-date and comprehensive reference work in the field of fractional calculus and its numerous applications. This first volume collects authoritative chapters covering the mathematical theory of fractional calculus, including fractional-order operators, integral transforms and equations, special functions, calculus of variations, and probabilistic and other aspects.

Implicit Fractional Differential and Integral Equations

Saïd Abbas,Mouffak Benchohra,John R. Graef,Johnny Henderson
Implicit Fractional Differential and Integral Equations Book PDF

Author : Saïd Abbas,Mouffak Benchohra,John R. Graef,Johnny Henderson
Publisher : Walter de Gruyter GmbH & Co KG
Page : 359 pages
File Size : 51,5 Mb
Release : 2018-02-05
Category : Mathematics
ISBN : 9783110553185

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Implicit Fractional Differential and Integral Equations by Saïd Abbas,Mouffak Benchohra,John R. Graef,Johnny Henderson Pdf

This book deals with the existence and stability of solutions to initial and boundary value problems for functional differential and integral equations and inclusions involving the Riemann-Liouville, Caputo, and Hadamard fractional derivatives and integrals. A wide variety of topics is covered in a mathematically rigorous manner making this work a valuable source of information for graduate students and researchers working with problems in fractional calculus. Contents Preliminary Background Nonlinear Implicit Fractional Differential Equations Impulsive Nonlinear Implicit Fractional Differential Equations Boundary Value Problems for Nonlinear Implicit Fractional Differential Equations Boundary Value Problems for Impulsive NIFDE Integrable Solutions for Implicit Fractional Differential Equations Partial Hadamard Fractional Integral Equations and Inclusions Stability Results for Partial Hadamard Fractional Integral Equations and Inclusions Hadamard–Stieltjes Fractional Integral Equations Ulam Stabilities for Random Hadamard Fractional Integral Equations

Fractional Differential Equations

Anatoly Kochubei,Yuri Luchko
Fractional Differential Equations Book PDF

Author : Anatoly Kochubei,Yuri Luchko
Publisher : Walter de Gruyter GmbH & Co KG
Page : 528 pages
File Size : 53,9 Mb
Release : 2019-02-19
Category : Mathematics
ISBN : 9783110571660

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Fractional Differential Equations by Anatoly Kochubei,Yuri Luchko Pdf

This multi-volume handbook is the most up-to-date and comprehensive reference work in the field of fractional calculus and its numerous applications. This second volume collects authoritative chapters covering the mathematical theory of fractional calculus, including ordinary and partial differential equations of fractional order, inverse problems, and evolution equations.

Differential Equations on Measures and Functional Spaces

Vassili Kolokoltsov
Differential Equations on Measures and Functional Spaces Book PDF

Author : Vassili Kolokoltsov
Publisher : Springer
Page : 525 pages
File Size : 55,6 Mb
Release : 2019-06-20
Category : Mathematics
ISBN : 9783030033774

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Differential Equations on Measures and Functional Spaces by Vassili Kolokoltsov Pdf

This advanced book focuses on ordinary differential equations (ODEs) in Banach and more general locally convex spaces, most notably the ODEs on measures and various function spaces. It briefly discusses the fundamentals before moving on to the cutting edge research in linear and nonlinear partial and pseudo-differential equations, general kinetic equations and fractional evolutions. The level of generality chosen is suitable for the study of the most important nonlinear equations of mathematical physics, such as Boltzmann, Smoluchovskii, Vlasov, Landau-Fokker-Planck, Cahn-Hilliard, Hamilton-Jacobi-Bellman, nonlinear Schroedinger, McKean-Vlasov diffusions and their nonlocal extensions, mass-action-law kinetics from chemistry. It also covers nonlinear evolutions arising in evolutionary biology and mean-field games, optimization theory, epidemics and system biology, in general models of interacting particles or agents describing splitting and merging, collisions and breakage, mutations and the preferential-attachment growth on networks. The book is intended mainly for upper undergraduate and graduate students, but is also of use to researchers in differential equations and their applications. It particularly highlights the interconnections between various topics revealing where and how a particular result is used in other chapters or may be used in other contexts, and also clarifies the links between the languages of pseudo-differential operators, generalized functions, operator theory, abstract linear spaces, fractional calculus and path integrals.

Beyond The Triangle: Brownian Motion, Ito Calculus, And Fokker-planck Equation - Fractional Generalizations

Sabir Umarov,Marjorie Hahn,Kei Kobayashi
Beyond The Triangle Brownian Motion Ito Calculus And Fokker planck Equation Fractional Generalizations Book PDF

Author : Sabir Umarov,Marjorie Hahn,Kei Kobayashi
Publisher : World Scientific
Page : 192 pages
File Size : 50,9 Mb
Release : 2018-02-13
Category : Mathematics
ISBN : 9789813230996

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Beyond The Triangle: Brownian Motion, Ito Calculus, And Fokker-planck Equation - Fractional Generalizations by Sabir Umarov,Marjorie Hahn,Kei Kobayashi Pdf

The book is devoted to the fundamental relationship between three objects: a stochastic process, stochastic differential equations driven by that process and their associated Fokker-Planck-Kolmogorov equations. This book discusses wide fractional generalizations of this fundamental triple relationship, where the driving process represents a time-changed stochastic process; the Fokker-Planck-Kolmogorov equation involves time-fractional order derivatives and spatial pseudo-differential operators; and the associated stochastic differential equation describes the stochastic behavior of the solution process. It contains recent results obtained in this direction.This book is important since the latest developments in the field, including the role of driving processes and their scaling limits, the forms of corresponding stochastic differential equations, and associated FPK equations, are systematically presented. Examples and important applications to various scientific, engineering, and economics problems make the book attractive for all interested researchers, educators, and graduate students.

The Analysis of Fractional Differential Equations

Kai Diethelm
The Analysis of Fractional Differential Equations Book PDF

Author : Kai Diethelm
Publisher : Springer Science & Business Media
Page : 251 pages
File Size : 45,7 Mb
Release : 2010-09-03
Category : Mathematics
ISBN : 9783642145735

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The Analysis of Fractional Differential Equations by Kai Diethelm Pdf

Fractional calculus was first developed by pure mathematicians in the middle of the 19th century. Some 100 years later, engineers and physicists have found applications for these concepts in their areas. However there has traditionally been little interaction between these two communities. In particular, typical mathematical works provide extensive findings on aspects with comparatively little significance in applications, and the engineering literature often lacks mathematical detail and precision. This book bridges the gap between the two communities. It concentrates on the class of fractional derivatives most important in applications, the Caputo operators, and provides a self-contained, thorough and mathematically rigorous study of their properties and of the corresponding differential equations. The text is a useful tool for mathematicians and researchers from the applied sciences alike. It can also be used as a basis for teaching graduate courses on fractional differential equations.

Fractional Calculus: Models And Numerical Methods

Baleanu Dumitru,Diethelm Kai,Scalas Enrico
Fractional Calculus Models And Numerical Methods Book PDF

Author : Baleanu Dumitru,Diethelm Kai,Scalas Enrico
Publisher : World Scientific
Page : 428 pages
File Size : 41,8 Mb
Release : 2012-01-27
Category : Mathematics
ISBN : 9789814458634

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Fractional Calculus: Models And Numerical Methods by Baleanu Dumitru,Diethelm Kai,Scalas Enrico Pdf

The subject of fractional calculus and its applications (that is, convolution-type pseudo-differential operators including integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past three decades or so, mainly due to its applications in diverse fields of science and engineering. These operators have been used to model problems with anomalous dynamics, however, they also are an effective tool as filters and controllers, and they can be applied to write complicated functions in terms of fractional integrals or derivatives of elementary functions, and so on.This book will give readers the possibility of finding very important mathematical tools for working with fractional models and solving fractional differential equations, such as a generalization of Stirling numbers in the framework of fractional calculus and a set of efficient numerical methods. Moreover, we will introduce some applied topics, in particular fractional variational methods which are used in physics, engineering or economics. We will also discuss the relationship between semi-Markov continuous-time random walks and the space-time fractional diffusion equation, which generalizes the usual theory relating random walks to the diffusion equation. These methods can be applied in finance, to model tick-by-tick (log)-price fluctuations, in insurance theory, to study ruin, as well as in macroeconomics as prototypical growth models.All these topics are complementary to what is dealt with in existing books on fractional calculus and its applications. This book was written with a trade-off in mind between full mathematical rigor and the needs of readers coming from different applied areas of science and engineering. In particular, the numerical methods listed in the book are presented in a readily accessible way that immediately allows the readers to implement them on a computer in a programming language of their choice. Numerical code is also provided.

Functional Calculus of Pseudodifferential Boundary Problems

Gerd Grubb
Functional Calculus of Pseudodifferential Boundary Problems Book PDF

Author : Gerd Grubb
Publisher : Springer Science & Business Media
Page : 536 pages
File Size : 44,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461207696

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Functional Calculus of Pseudodifferential Boundary Problems by Gerd Grubb Pdf

Pseudodifferential methods are central to the study of partial differential equations, because they permit an "algebraization." The main purpose of this book is to set up an operational calculus for operators defined from differential and pseudodifferential boundary values problems via a resolvent construction. A secondary purposed is to give a complete treatment of the properties of the calculus of pseudodifferential boundary problems with transmission, both the first version by Boutet de Monvel (brought completely up to date in this edition) and in version containing a parameter running in an unbounded set. And finally, the book presents some applications to evolution problems, index theory, fractional powers, spectral theory and singular perturbation theory. Thus the book’s improved proofs and modern points of view will be useful to research mathematicians and to graduate students studying partial differential equations and pseudodifferential operators.

Fractional Differential Equations

Igor Podlubny
Fractional Differential Equations Book PDF

Author : Igor Podlubny
Publisher : Elsevier
Page : 366 pages
File Size : 53,7 Mb
Release : 1998-10-27
Category : Mathematics
ISBN : 9780080531984

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Fractional Differential Equations by Igor Podlubny Pdf

This book is a landmark title in the continuous move from integer to non-integer in mathematics: from integer numbers to real numbers, from factorials to the gamma function, from integer-order models to models of an arbitrary order. For historical reasons, the word 'fractional' is used instead of the word 'arbitrary'. This book is written for readers who are new to the fields of fractional derivatives and fractional-order mathematical models, and feel that they need them for developing more adequate mathematical models. In this book, not only applied scientists, but also pure mathematicians will find fresh motivation for developing new methods and approaches in their fields of research. A reader will find in this book everything necessary for the initial study and immediate application of fractional derivatives fractional differential equations, including several necessary special functions, basic theory of fractional differentiation, uniqueness and existence theorems, analytical numerical methods of solution of fractional differential equations, and many inspiring examples of applications. A unique survey of many applications of fractional calculus Presents basic theory Includes a unified presentation of selected classical results, which are important for applications Provides many examples Contains a separate chapter of fractional order control systems, which opens new perspectives in control theory The first systematic consideration of Caputo's fractional derivative in comparison with other selected approaches Includes tables of fractional derivatives, which can be used for evaluation of all considered types of fractional derivatives

Introduction to the Fractional Calculus of Variations

Agnieszka B Malinowska,Delfim F M Torres
Introduction to the Fractional Calculus of Variations Book PDF

Author : Agnieszka B Malinowska,Delfim F M Torres
Publisher : World Scientific Publishing Company
Page : 292 pages
File Size : 55,7 Mb
Release : 2012-09-14
Category : Mathematics
ISBN : 9781848169685

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Introduction to the Fractional Calculus of Variations by Agnieszka B Malinowska,Delfim F M Torres Pdf

This invaluable book provides a broad introduction to the fascinating and beautiful subject of Fractional Calculus of Variations (FCV). In 1996, FVC evolved in order to better describe non-conservative systems in mechanics. The inclusion of non-conservatism is extremely important from the point of view of applications. Forces that do not store energy are always present in real systems. They remove energy from the systems and, as a consequence, Noether's conservation laws cease to be valid. However, it is still possible to obtain the validity of Noether's principle using FCV. The new theory provides a more realistic approach to physics, allowing us to consider non-conservative systems in a natural way. The authors prove the necessary Euler–Lagrange conditions and corresponding Noether theorems for several types of fractional variational problems, with and without constraints, using Lagrangian and Hamiltonian formalisms. Sufficient optimality conditions are also obtained under convexity, and Leitmann's direct method is discussed within the framework of FCV. The book is self-contained and unified in presentation. It may be used as an advanced textbook by graduate students and ambitious undergraduates in mathematics and mechanics. It provides an opportunity for an introduction to FCV for experienced researchers. The explanations in the book are detailed, in order to capture the interest of the curious reader, and the book provides the necessary background material required to go further into the subject and explore the rich research literature.

Partial Differential Equations II

Yu.V. Egorov,A.I. Komech,M.A. Shubin
Partial Differential Equations II Book PDF

Author : Yu.V. Egorov,A.I. Komech,M.A. Shubin
Publisher : Springer Science & Business Media
Page : 269 pages
File Size : 53,5 Mb
Release : 2013-12-01
Category : Mathematics
ISBN : 9783642578762

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Partial Differential Equations II by Yu.V. Egorov,A.I. Komech,M.A. Shubin Pdf

This book, the first printing of which was published as Volume 31 of the Encyclopaedia of Mathematical Sciences, contains a survey of the modern theory of general linear partial differential equations and a detailed review of equations with constant coefficients. Readers will be interested in an introduction to microlocal analysis and its applications including singular integral operators, pseudodifferential operators, Fourier integral operators and wavefronts, a survey of the most important results about the mixed problem for hyperbolic equations, a review of asymptotic methods including short wave asymptotics, the Maslov canonical operator and spectral asymptotics, a detailed description of the applications of distribution theory to partial differential equations with constant coefficients including numerous interesting special topics.