Fractional Partial Differential Equations

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Time-Fractional Differential Equations

Adam Kubica,Katarzyna Ryszewska,Masahiro Yamamoto
Time Fractional Differential Equations Book PDF

Author : Adam Kubica,Katarzyna Ryszewska,Masahiro Yamamoto
Publisher : Springer Nature
Page : 134 pages
File Size : 46,7 Mb
Release : 2020-11-29
Category : Mathematics
ISBN : 9789811590665

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Time-Fractional Differential Equations by Adam Kubica,Katarzyna Ryszewska,Masahiro Yamamoto Pdf

This book aims to establish a foundation for fractional derivatives and fractional differential equations. The theory of fractional derivatives enables considering any positive order of differentiation. The history of research in this field is very long, with its origins dating back to Leibniz. Since then, many great mathematicians, such as Abel, have made contributions that cover not only theoretical aspects but also physical applications of fractional calculus. The fractional partial differential equations govern phenomena depending both on spatial and time variables and require more subtle treatments. Moreover, fractional partial differential equations are highly demanded model equations for solving real-world problems such as the anomalous diffusion in heterogeneous media. The studies of fractional partial differential equations have continued to expand explosively. However we observe that available mathematical theory for fractional partial differential equations is not still complete. In particular, operator-theoretical approaches are indispensable for some generalized categories of solutions such as weak solutions, but feasible operator-theoretic foundations for wide applications are not available in monographs. To make this monograph more readable, we are restricting it to a few fundamental types of time-fractional partial differential equations, forgoing many other important and exciting topics such as stability for nonlinear problems. However, we believe that this book works well as an introduction to mathematical research in such vast fields.

Fractional Partial Differential Equations and Their Numerical Solutions

Boling Guo,Xueke Pu,Fenghui Huang
Fractional Partial Differential Equations and Their Numerical Solutions Book PDF

Author : Boling Guo,Xueke Pu,Fenghui Huang
Publisher : World Scientific
Page : 348 pages
File Size : 42,8 Mb
Release : 2015-03-09
Category : Mathematics
ISBN : 9789814667067

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Fractional Partial Differential Equations and Their Numerical Solutions by Boling Guo,Xueke Pu,Fenghui Huang Pdf

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

Fractional Partial Differential Equations

Yong Zhou
Fractional Partial Differential Equations Book PDF

Author : Yong Zhou
Publisher : World Scientific
Page : 319 pages
File Size : 42,5 Mb
Release : 2024-03-12
Category : Mathematics
ISBN : 9789811290428

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Fractional Partial Differential Equations by Yong Zhou Pdf

This monograph offers a comprehensive exposition of the theory surrounding time-fractional partial differential equations, featuring recent advancements in fundamental techniques and results. The topics covered encompass crucial aspects of the theory, such as well-posedness, regularity, approximation, and optimal control. The book delves into the intricacies of fractional Navier-Stokes equations, fractional Rayleigh-Stokes equations, fractional Fokker-Planck equations, and fractional Schrödinger equations, providing a thorough exploration of these subjects. Numerous real-world applications associated with these equations are meticulously examined, enhancing the practical relevance of the presented concepts.The content in this monograph is based on the research works carried out by the author and other excellent experts during the past five years. Rooted in the latest advancements, it not only serves as a valuable resource for understanding the theoretical foundations but also lays the groundwork for delving deeper into the subject and navigating the extensive research landscape. Geared towards researchers, graduate students, and PhD scholars specializing in differential equations, applied analysis, and related research domains, this monograph facilitates a nuanced understanding of time-fractional partial differential equations and their broader implications.

Inverse Problems for Fractional Partial Differential Equations

Barbara Kaltenbacher,William Rundell
Inverse Problems for Fractional Partial Differential Equations Book PDF

Author : Barbara Kaltenbacher,William Rundell
Publisher : American Mathematical Society
Page : 522 pages
File Size : 55,5 Mb
Release : 2023-07-13
Category : Mathematics
ISBN : 9781470472771

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Inverse Problems for Fractional Partial Differential Equations by Barbara Kaltenbacher,William Rundell Pdf

As the title of the book indicates, this is primarily a book on partial differential equations (PDEs) with two definite slants: toward inverse problems and to the inclusion of fractional derivatives. The standard paradigm, or direct problem, is to take a PDE, including all coefficients and initial/boundary conditions, and to determine the solution. The inverse problem reverses this approach asking what information about coefficients of the model can be obtained from partial information on the solution. Answering this question requires knowledge of the underlying physical model, including the exact dependence on material parameters. The last feature of the approach taken by the authors is the inclusion of fractional derivatives. This is driven by direct physical applications: a fractional derivative model often allows greater adherence to physical observations than the traditional integer order case. The book also has an extensive historical section and the material that can be called "fractional calculus" and ordinary differential equations with fractional derivatives. This part is accessible to advanced undergraduates with basic knowledge on real and complex analysis. At the other end of the spectrum, lie nonlinear fractional PDEs that require a standard graduate level course on PDEs.

Local Fractional Integral Transforms and Their Applications

Xiao Jun Yang,Dumitru Baleanu,H. M. Srivastava
Local Fractional Integral Transforms and Their Applications Book PDF

Author : Xiao Jun Yang,Dumitru Baleanu,H. M. Srivastava
Publisher : Academic Press
Page : 262 pages
File Size : 42,9 Mb
Release : 2015-10-22
Category : Mathematics
ISBN : 9780128040324

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Local Fractional Integral Transforms and Their Applications by Xiao Jun Yang,Dumitru Baleanu,H. M. Srivastava Pdf

Local Fractional Integral Transforms and Their Applications provides information on how local fractional calculus has been successfully applied to describe the numerous widespread real-world phenomena in the fields of physical sciences and engineering sciences that involve non-differentiable behaviors. The methods of integral transforms via local fractional calculus have been used to solve various local fractional ordinary and local fractional partial differential equations and also to figure out the presence of the fractal phenomenon. The book presents the basics of the local fractional derivative operators and investigates some new results in the area of local integral transforms. Provides applications of local fractional Fourier Series Discusses definitions for local fractional Laplace transforms Explains local fractional Laplace transforms coupled with analytical methods

Fractional Differential Equations

Bangti Jin
Fractional Differential Equations Book PDF

Author : Bangti Jin
Publisher : Springer Nature
Page : 377 pages
File Size : 42,7 Mb
Release : 2021-07-22
Category : Mathematics
ISBN : 9783030760434

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Fractional Differential Equations by Bangti Jin Pdf

This graduate textbook provides a self-contained introduction to modern mathematical theory on fractional differential equations. It addresses both ordinary and partial differential equations with a focus on detailed solution theory, especially regularity theory under realistic assumptions on the problem data. The text includes an extensive bibliography, application-driven modeling, extensive exercises, and graphic illustrations throughout to complement its comprehensive presentation of the field. It is recommended for graduate students and researchers in applied and computational mathematics, particularly applied analysis, numerical analysis and inverse problems.

Basic Theory Of Fractional Differential Equations (Third Edition)

Yong Zhou
Basic Theory Of Fractional Differential Equations Third Edition Book PDF

Author : Yong Zhou
Publisher : World Scientific
Page : 516 pages
File Size : 51,5 Mb
Release : 2023-10-06
Category : Mathematics
ISBN : 9789811271700

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Basic Theory Of Fractional Differential Equations (Third Edition) by Yong Zhou Pdf

This accessible monograph is devoted to a rapidly developing area on the research of qualitative theory of fractional ordinary differential equations and evolution equations. It is self-contained and unified in presentation, and provides the readers the necessary background material required to go further into the subject and explore the rich research literature. The tools used include many classical and modern nonlinear analysis methods such as fixed point theory, measure of noncompactness method, topological degree method, Picard operators technique, critical point theory and semigroups theory. This book is based on the research work done so far by the author and other experts, and contains comprehensive up-to-date materials on the topic.In this third edition, four new topics have been added: Hilfer fractional evolution equations and infinite interval problems, oscillations and nonoscillations, fractional Hamiltonian systems, fractional Rayleigh-Stokes equations, and wave equations. The bibliography has also been updated and expanded.This book is useful to researchers, graduate or PhD students dealing with fractional calculus and applied analysis, differential equations, and related areas of research.

Fractional Differential Equations

Juan J. Nieto,Rosana Rodríguez-López
Fractional Differential Equations Book PDF

Author : Juan J. Nieto,Rosana Rodríguez-López
Publisher : MDPI
Page : 172 pages
File Size : 41,6 Mb
Release : 2019-11-19
Category : Mathematics
ISBN : 9783039217328

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Fractional Differential Equations by Juan J. Nieto,Rosana Rodríguez-López Pdf

Fractional calculus provides the possibility of introducing integrals and derivatives of an arbitrary order in the mathematical modelling of physical processes, and it has become a relevant subject with applications to various fields, such as anomalous diffusion, propagation in different media, and propogation in relation to materials with different properties. However, many aspects from theoretical and practical points of view have still to be developed in relation to models based on fractional operators. This Special Issue is related to new developments on different aspects of fractional differential equations, both from a theoretical point of view and in terms of applications in different fields such as physics, chemistry, or control theory, for instance. The topics of the Issue include fractional calculus, the mathematical analysis of the properties of the solutions to fractional equations, the extension of classical approaches, or applications of fractional equations to several fields.

Topics in Fractional Differential Equations

Saïd Abbas,Mouffak Benchohra,Gaston M. N'Guérékata
Topics in Fractional Differential Equations Book PDF

Author : Saïd Abbas,Mouffak Benchohra,Gaston M. N'Guérékata
Publisher : Springer Science & Business Media
Page : 403 pages
File Size : 42,8 Mb
Release : 2012-08-17
Category : Mathematics
ISBN : 9781461440369

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Topics in Fractional Differential Equations by Saïd Abbas,Mouffak Benchohra,Gaston M. N'Guérékata Pdf

​​​ Topics in Fractional Differential Equations is devoted to the existence and uniqueness of solutions for various classes of Darboux problems for hyperbolic differential equations or inclusions involving the Caputo fractional derivative. ​​Fractional calculus generalizes the integrals and derivatives to non-integer orders. During the last decade, fractional calculus was found to play a fundamental role in the modeling of a considerable number of phenomena; in particular the modeling of memory-dependent and complex media such as porous media. It has emerged as an important tool for the study of dynamical systems where classical methods reveal strong limitations. Some equations present delays which may be finite, infinite, or state-dependent. Others are subject to an impulsive effect. The above problems are studied using the fixed point approach, the method of upper and lower solution, and the Kuratowski measure of noncompactness. This book is addressed to a wide audience of specialists such as mathematicians, engineers, biologists, and physicists. ​

Fractional Calculus and Fractional Differential Equations

Varsha Daftardar-Gejji
Fractional Calculus and Fractional Differential Equations Book PDF

Author : Varsha Daftardar-Gejji
Publisher : Springer
Page : 180 pages
File Size : 54,6 Mb
Release : 2019-08-10
Category : Mathematics
ISBN : 9789811392276

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Fractional Calculus and Fractional Differential Equations by Varsha Daftardar-Gejji Pdf

This book provides a broad overview of the latest developments in fractional calculus and fractional differential equations (FDEs) with an aim to motivate the readers to venture into these areas. It also presents original research describing the fractional operators of variable order, fractional-order delay differential equations, chaos and related phenomena in detail. Selected results on the stability of solutions of nonlinear dynamical systems of the non-commensurate fractional order have also been included. Furthermore, artificial neural network and fractional differential equations are elaborated on; and new transform methods (for example, Sumudu methods) and how they can be employed to solve fractional partial differential equations are discussed. The book covers the latest research on a variety of topics, including: comparison of various numerical methods for solving FDEs, the Adomian decomposition method and its applications to fractional versions of the classical Poisson processes, variable-order fractional operators, fractional variational principles, fractional delay differential equations, fractional-order dynamical systems and stability analysis, inequalities and comparison theorems in FDEs, artificial neural network approximation for fractional operators, and new transform methods for solving partial FDEs. Given its scope and level of detail, the book will be an invaluable asset for researchers working in these areas.

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 : 44,8 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.

Fractional Differential Equations

Igor Podlubny
Fractional Differential Equations Book PDF

Author : Igor Podlubny
Publisher : Elsevier
Page : 366 pages
File Size : 52,9 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

Theory and Applications of Fractional Differential Equations

A.A. Kilbas,H. M. Srivastava,J.J. Trujillo
Theory and Applications of Fractional Differential Equations Book PDF

Author : A.A. Kilbas,H. M. Srivastava,J.J. Trujillo
Publisher : Elsevier
Page : 550 pages
File Size : 47,5 Mb
Release : 2006-02-16
Category : Mathematics
ISBN : 0444518320

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Theory and Applications of Fractional Differential Equations by A.A. Kilbas,H. M. Srivastava,J.J. Trujillo Pdf

This work aims to present, in a systematic manner, results including the existence and uniqueness of solutions for the Cauchy Type and Cauchy problems involving nonlinear ordinary fractional differential equations.

Fractional Differential Equations

Zhi-Zhong Sun,Guang-hua Gao
Fractional Differential Equations Book PDF

Author : Zhi-Zhong Sun,Guang-hua Gao
Publisher : Walter de Gruyter GmbH & Co KG
Page : 226 pages
File Size : 49,9 Mb
Release : 2020-08-24
Category : Mathematics
ISBN : 9783110615302

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Fractional Differential Equations by Zhi-Zhong Sun,Guang-hua Gao Pdf

Starting with an introduction to fractional derivatives and numerical approximations, this book presents finite difference methods for fractional differential equations, including time-fractional sub-diffusion equations, time-fractional wave equations, and space-fractional differential equations, among others. Approximation methods for fractional derivatives are developed and approximate accuracies are analyzed in detail.

Local Density of Solutions to Fractional Equations

Alessandro Carbotti,Serena Dipierro,Enrico Valdinoci
Local Density of Solutions to Fractional Equations Book PDF

Author : Alessandro Carbotti,Serena Dipierro,Enrico Valdinoci
Publisher : Walter de Gruyter GmbH & Co KG
Page : 143 pages
File Size : 55,8 Mb
Release : 2019-08-19
Category : Mathematics
ISBN : 9783110664355

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Local Density of Solutions to Fractional Equations by Alessandro Carbotti,Serena Dipierro,Enrico Valdinoci Pdf

This book presents in a detailed and self-contained way a new and important density result in the analysis of fractional partial differential equations, while also covering several fundamental facts about space- and time-fractional equations.