Time Fractional 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 : 49,8 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 Differential Equations

Bangti Jin
Fractional Differential Equations Book PDF

Author : Bangti Jin
Publisher : Springer Nature
Page : 377 pages
File Size : 54,8 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.

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 : 52,9 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.

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 : 50,9 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.

Fractional Partial Differential Equations

Yong Zhou
Fractional Partial Differential Equations Book PDF

Author : Yong Zhou
Publisher : World Scientific
Page : 319 pages
File Size : 55,8 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.

Time-Fractional Order Biological Systems with Uncertain Parameters

Snehashish Chakraverty,Rajarama Mohan Jena,Subrat Kumar Jena
Time Fractional Order Biological Systems with Uncertain Parameters Book PDF

Author : Snehashish Chakraverty,Rajarama Mohan Jena,Subrat Kumar Jena
Publisher : Springer Nature
Page : 144 pages
File Size : 43,7 Mb
Release : 2022-06-01
Category : Mathematics
ISBN : 9783031024238

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Time-Fractional Order Biological Systems with Uncertain Parameters by Snehashish Chakraverty,Rajarama Mohan Jena,Subrat Kumar Jena Pdf

The subject of fractional calculus has gained considerable popularity and importance during the past three decades, mainly due to its validated applications in various fields of science and engineering. It is a generalization of ordinary differentiation and integration to arbitrary (non-integer) order. The fractional derivative has been used in various physical problems, such as frequency-dependent damping behavior of structures, biological systems, motion of a plate in a Newtonian fluid, λμ controller for the control of dynamical systems, and so on. It is challenging to obtain the solution (both analytical and numerical) of related nonlinear partial differential equations of fractional order. So for the last few decades, a great deal of attention has been directed towards the solution for these kind of problems. Different methods have been developed by other researchers to analyze the above problems with respect to crisp (exact) parameters. However, in real-life applications such as for biological problems, it is not always possible to get exact values of the associated parameters due to errors in measurements/experiments, observations, and many other errors. Therefore, the associated parameters and variables may be considered uncertain. Here, the uncertainties are considered interval/fuzzy. Therefore, the development of appropriate efficient methods and their use in solving the mentioned uncertain problems are the recent challenge. In view of the above, this book is a new attempt to rigorously present a variety of fuzzy (and interval) time-fractional dynamical models with respect to different biological systems using computationally efficient method. The authors believe this book will be helpful to undergraduates, graduates, researchers, industry, faculties, and others throughout the globe.

New Trends in Fractional Differential Equations with Real-World Applications in Physics

Jagdev Singh,Jordan Yankov Hristov,Zakia Hammouch
New Trends in Fractional Differential Equations with Real World Applications in Physics Book PDF

Author : Jagdev Singh,Jordan Yankov Hristov,Zakia Hammouch
Publisher : Frontiers Media SA
Page : 172 pages
File Size : 43,5 Mb
Release : 2020-12-30
Category : Science
ISBN : 9782889663040

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New Trends in Fractional Differential Equations with Real-World Applications in Physics by Jagdev Singh,Jordan Yankov Hristov,Zakia Hammouch Pdf

This eBook is a collection of articles from a Frontiers Research Topic. Frontiers Research Topics are very popular trademarks of the Frontiers Journals Series: they are collections of at least ten articles, all centered on a particular subject. With their unique mix of varied contributions from Original Research to Review Articles, Frontiers Research Topics unify the most influential researchers, the latest key findings and historical advances in a hot research area! Find out more on how to host your own Frontiers Research Topic or contribute to one as an author by contacting the Frontiers Editorial Office: frontiersin.org/about/contact.

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 : 50,5 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.

Numerical Methods for Fractional Differentiation

Kolade M. Owolabi,Abdon Atangana
Numerical Methods for Fractional Differentiation Book PDF

Author : Kolade M. Owolabi,Abdon Atangana
Publisher : Springer Nature
Page : 328 pages
File Size : 48,6 Mb
Release : 2019-10-14
Category : Mathematics
ISBN : 9789811500985

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Numerical Methods for Fractional Differentiation by Kolade M. Owolabi,Abdon Atangana Pdf

This book discusses numerical methods for solving partial differential and integral equations, as well as ordinary differential and integral equations, involving fractional differential and integral operators. Differential and integral operators presented in the book include those with exponential decay law, known as Caputo–Fabrizio differential and integral operators, those with power law, known as Riemann–Liouville fractional operators, and those for the generalized Mittag–Leffler function, known as the Atangana–Baleanu fractional operators. The book reviews existing numerical schemes associated with fractional operators including those with power law, while also highlighting new trends in numerical schemes for recently introduced differential and integral operators. In addition, the initial chapters address useful properties of each differential and integral fractional operator. Methods discussed in the book are subsequently used to solved problems arising in many fields of science, technology, and engineering, including epidemiology, chaos, solitons, fractals, diffusion, groundwater, and fluid mechanics. Given its scope, the book offers a valuable resource for graduate students of mathematics and engineering, and researchers in virtually all fields of science, technology, and engineering, as well as an excellent addition to libraries.

Fractional Differential Equations

Igor Podlubny
Fractional Differential Equations Book PDF

Author : Igor Podlubny
Publisher : Elsevier
Page : 366 pages
File Size : 43,8 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

Computational Fractional Dynamical Systems

Snehashish Chakraverty,Rajarama M. Jena,Subrat K. Jena
Computational Fractional Dynamical Systems Book PDF

Author : Snehashish Chakraverty,Rajarama M. Jena,Subrat K. Jena
Publisher : John Wiley & Sons
Page : 276 pages
File Size : 53,6 Mb
Release : 2022-11-15
Category : Mathematics
ISBN : 9781119696957

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Computational Fractional Dynamical Systems by Snehashish Chakraverty,Rajarama M. Jena,Subrat K. Jena Pdf

A rigorous presentation of different expansion and semi-analytical methods for fractional differential equations Fractional differential equations, differential and integral operators with non-integral powers, are used in various science and engineering applications. Over the past several decades, the popularity of the fractional derivative has increased significantly in diverse areas such as electromagnetics, financial mathematics, image processing, and materials science. Obtaining analytical and numerical solutions of nonlinear partial differential equations of fractional order can be challenging and involve the development and use of different methods of solution. Computational Fractional Dynamical Systems: Fractional Differential Equations and Applications presents a variety of computationally efficient semi-analytical and expansion methods to solve different types of fractional models. Rather than focusing on a single computational method, this comprehensive volume brings together more than 25 methods for solving an array of fractional-order models. The authors employ a rigorous and systematic approach for addressing various physical problems in science and engineering. Covers various aspects of efficient methods regarding fractional-order systems Presents different numerical methods with detailed steps to handle basic and advanced equations in science and engineering Provides a systematic approach for handling fractional-order models arising in science and engineering Incorporates a wide range of methods with corresponding results and validation Computational Fractional Dynamical Systems: Fractional Differential Equations and Applications is an invaluable resource for advanced undergraduate students, graduate students, postdoctoral researchers, university faculty, and other researchers and practitioners working with fractional and integer order differential equations.

Fractional-in-Time Semilinear Parabolic Equations and Applications

Ciprian G. Gal,Mahamadi Warma
Fractional in Time Semilinear Parabolic Equations and Applications Book PDF

Author : Ciprian G. Gal,Mahamadi Warma
Publisher : Springer Nature
Page : 193 pages
File Size : 47,8 Mb
Release : 2020-09-23
Category : Mathematics
ISBN : 9783030450434

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Fractional-in-Time Semilinear Parabolic Equations and Applications by Ciprian G. Gal,Mahamadi Warma Pdf

This book provides a unified analysis and scheme for the existence and uniqueness of strong and mild solutions to certain fractional kinetic equations. This class of equations is characterized by the presence of a nonlinear time-dependent source, generally of arbitrary growth in the unknown function, a time derivative in the sense of Caputo and the presence of a large class of diffusion operators. The global regularity problem is then treated separately and the analysis is extended to some systems of fractional kinetic equations, including prey-predator models of Volterra–Lotka type and chemical reactions models, all of them possibly containing some fractional kinetics. Besides classical examples involving the Laplace operator, subject to standard (namely, Dirichlet, Neumann, Robin, dynamic/Wentzell and Steklov) boundary conditions, the framework also includes non-standard diffusion operators of "fractional" type, subject to appropriate boundary conditions. This book is aimed at graduate students and researchers in mathematics, physics, mathematical engineering and mathematical biology, whose research involves partial differential equations.

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 : 53,8 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

The Analysis of Fractional Differential Equations

Kai Diethelm
The Analysis of Fractional Differential Equations Book PDF

Author : Kai Diethelm
Publisher : Springer
Page : 247 pages
File Size : 40,5 Mb
Release : 2010-08-18
Category : Mathematics
ISBN : 9783642145742

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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.

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 : 47,7 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.