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Author: Changpin Li
Publisher: CRC Press
ISBN: 148225381X
Category : Mathematics
Languages : en
Pages : 299
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Book Description
Numerical Methods for Fractional Calculus presents numerical methods for fractional integrals and fractional derivatives, finite difference methods for fractional ordinary differential equations (FODEs) and fractional partial differential equations (FPDEs), and finite element methods for FPDEs.The book introduces the basic definitions and propertie
Author: Changpin Li
Publisher: CRC Press
ISBN: 148225381X
Category : Mathematics
Languages : en
Pages : 299
Get Book Here
Book Description
Numerical Methods for Fractional Calculus presents numerical methods for fractional integrals and fractional derivatives, finite difference methods for fractional ordinary differential equations (FODEs) and fractional partial differential equations (FPDEs), and finite element methods for FPDEs.The book introduces the basic definitions and propertie
Author: Changpin Li
Publisher: SIAM
ISBN: 1611975883
Category : Mathematics
Languages : en
Pages : 327
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Book Description
Due to its ubiquity across a variety of fields in science and engineering, fractional calculus has gained momentum in industry and academia. While a number of books and papers introduce either fractional calculus or numerical approximations, no current literature provides a comprehensive collection of both topics. This monograph introduces fundamental information on fractional calculus, provides a detailed treatment of existing numerical approximations, and presents an inclusive review of fractional calculus in terms of theory and numerical methods and systematically examines almost all existing numerical approximations for fractional integrals and derivatives. The authors consider the relationship between the fractional Laplacian and the Riesz derivative, a key component absent from other related texts, and highlight recent developments, including their own research and results. The core audience spans several fractional communities, including those interested in fractional partial differential equations, the fractional Laplacian, and applied and computational mathematics. Advanced undergraduate and graduate students will find the material suitable as a primary or supplementary resource for their studies.
Author: Dumitru Baleanu
Publisher: World Scientific
ISBN: 9814355208
Category : Mathematics
Languages : en
Pages : 426
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Book Description
This title 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.
Author: Matthew Harker
Publisher: Springer
ISBN: 9783030323769
Category : Technology & Engineering
Languages : en
Pages : 466
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Book Description
This book provides a comprehensive set of practical tools for exploring and discovering the world of fractional calculus and its applications, and thereby a means of bridging the theory of fractional differential equations (FDE) with real-world facts. These tools seamlessly blend centuries old numerical methods such as Gaussian quadrature that have stood the test of time with pioneering concepts such as hypermatrix equations to harness the emerging capabilities of modern scientific computing environments. This unique fusion of old and new leads to a unified approach that intuitively parallels the classic theory of differential equations, and results in methods that are unprecedented in computational speed and numerical accuracy. The opening chapter is an introduction to fractional calculus that is geared towards scientists and engineers. The following chapter introduces the reader to the key concepts of approximation theory with an emphasis on the tools of numerical linear algebra. The third chapter provides the keystone for the remainder of the book with a comprehensive set of methods for the approximation of fractional order integrals and derivatives. The fourth chapter describes the numerical solution of initial and boundary value problems for FDE of a single variable, both linear and nonlinear. Moving to two, three, and four dimensions, the ensuing chapter is devoted to a novel approach to the numerical solution of partial FDE that leverages the little-known one-to-one relation between partial differential equations and matrix and hypermatrix equations. The emphasis on applications culminates in the final chapter by addressing inverse problems for ordinary and partial FDE, such as smoothing for data analytics, and the all-important system identification problem. Over a century ago, scientists such as Ludwig Boltzmann and Vito Volterra formulated mathematical models of real materials that -- based on physical evidence -- integrated the history of the system. The present book will be invaluable to students and researchers in fields where analogous phenomena arise, such as viscoelasticity, rheology, polymer dynamics, non-Newtonian fluids, bioengineering, electrochemistry, non-conservative mechanics, groundwater hydrology, NMR and computed tomography, mathematical economics, thermomechanics, anomalous diffusion and transport, control theory, supercapacitors, and genetic algorithms, to name but a few. These investigators will be well-equipped with reproducible numerical methods to explore and discover their particular field of application of FDE.
Author: J. Sabatier
Publisher: Springer Science & Business Media
ISBN: 1402060424
Category : Technology & Engineering
Languages : en
Pages : 550
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Book Description
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.
Author: Igor Podlubny
Publisher: Elsevier
ISBN: 0080531989
Category : Mathematics
Languages : en
Pages : 366
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Book Description
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
Author: Varsha Daftardar-Gejji
Publisher: Springer
ISBN: 9811392277
Category : Mathematics
Languages : en
Pages : 187
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Book Description
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.
Author: Xiao-Jun Yang
Publisher: CRC Press
ISBN: 0429811527
Category : Mathematics
Languages : en
Pages : 391
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Book Description
General Fractional Derivatives: Theory, Methods and Applications provides knowledge of the special functions with respect to another function, and the integro-differential operators where the integrals are of the convolution type and exist the singular, weakly singular and nonsingular kernels, which exhibit the fractional derivatives, fractional integrals, general fractional derivatives, and general fractional integrals of the constant and variable order without and with respect to another function due to the appearance of the power-law and complex herbivores to figure out the modern developments in theoretical and applied science. Features: Give some new results for fractional calculus of constant and variable orders. Discuss some new definitions for fractional calculus with respect to another function. Provide definitions for general fractional calculus of constant and variable orders. Report new results of general fractional calculus with respect to another function. Propose news special functions with respect to another function and their applications. Present new models for the anomalous relaxation and rheological behaviors. This book serves as a reference book and textbook for scientists and engineers in the fields of mathematics, physics, chemistry and engineering, senior undergraduate and graduate students. Dr. Xiao-Jun Yang is a full professor of Applied Mathematics and Mechanics, at China University of Mining and Technology, China. He is currently an editor of several scientific journals, such as Fractals, Applied Numerical Mathematics, Mathematical Modelling and Analysis, International Journal of Numerical Methods for Heat & Fluid Flow, and Thermal Science.
Author: Constantin Milici
Publisher: Springer
ISBN: 3030008959
Category : Technology & Engineering
Languages : en
Pages : 199
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Book Description
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.
Author: Boling Guo
Publisher: World Scientific
ISBN: 9814667064
Category : Mathematics
Languages : en
Pages : 347
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Book Description
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.
