Fractional Partial Differential Equations And Their Numerical Solutions

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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 : 40,5 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

Numerical Integration of Space Fractional Partial Differential Equations

Younes Salehi,William E. Schiesser
Numerical Integration of Space Fractional Partial Differential Equations Book PDF

Author : Younes Salehi,William E. Schiesser
Publisher : Springer Nature
Page : 192 pages
File Size : 45,9 Mb
Release : 2022-06-01
Category : Mathematics
ISBN : 9783031024122

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Numerical Integration of Space Fractional Partial Differential Equations by Younes Salehi,William E. Schiesser Pdf

​ Partial differential equations (PDEs) are one of the most used widely forms of mathematics in science and engineering. PDEs can have partial derivatives with respect to (1) an initial value variable, typically time, and (2) boundary value variables, typically spatial variables. Therefore, two fractional PDEs can be considered, (1) fractional in time (TFPDEs), and (2) fractional in space (SFPDEs). The two volumes are directed to the development and use of SFPDEs, with the discussion divided as: Vol 1: Introduction to Algorithms and Computer Coding in R Vol 2: Applications from Classical Integer PDEs. Various definitions of space fractional derivatives have been proposed. We focus on the Caputo derivative, with occasional reference to the Riemann-Liouville derivative. In the second volume, the emphasis is on applications of SFPDEs developed mainly through the extension of classical integer PDEs to SFPDEs. The example applications are: Fractional diffusion equation with Dirichlet, Neumann and Robin boundary conditions Fisher-Kolmogorov SFPDE Burgers SFPDE Fokker-Planck SFPDE Burgers-Huxley SFPDE Fitzhugh-Nagumo SFPDE /div These SFPDEs were selected because they are integer first order in time and integer second order in space. The variation in the spatial derivative from order two (parabolic) to order one (first order hyperbolic) demonstrates the effect of the spatial fractional order with 1 ≤ ≤ 2. All of the example SFPDEs are one dimensional in Cartesian coordinates. Extensions to higher dimensions and other coordinate systems, in principle, follow from the examples in this second volume. The examples start with a statement of the integer PDEs that are then extended to SFPDEs. The format of each chapter is the same as in the first volume. The R routines can be downloaded and executed on a modest computer (R is readily available from the Internet).

Fractional Differential Equations

Igor Podlubny
Fractional Differential Equations Book PDF

Author : Igor Podlubny
Publisher : Elsevier
Page : 366 pages
File Size : 45,9 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

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 : 54,8 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.

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 : 51,8 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.

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 : 45,5 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 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 : 46,5 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.

Numerical Integration of Space Fractional Partial Differential Equations

Younes Salehi,William E. Schiesser
Numerical Integration of Space Fractional Partial Differential Equations Book PDF

Author : Younes Salehi,William E. Schiesser
Publisher : Springer Nature
Page : 188 pages
File Size : 55,9 Mb
Release : 2022-05-31
Category : Mathematics
ISBN : 9783031024115

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Numerical Integration of Space Fractional Partial Differential Equations by Younes Salehi,William E. Schiesser Pdf

Partial differential equations (PDEs) are one of the most used widely forms of mathematics in science and engineering. PDEs can have partial derivatives with respect to (1) an initial value variable, typically time, and (2) boundary value variables, typically spatial variables. Therefore, two fractional PDEs can be considered, (1) fractional in time (TFPDEs), and (2) fractional in space (SFPDEs). The two volumes are directed to the development and use of SFPDEs, with the discussion divided as: Vol 1: Introduction to Algorithms and Computer Coding in R Vol 2: Applications from Classical Integer PDEs. Various definitions of space fractional derivatives have been proposed. We focus on the Caputo derivative, with occasional reference to the Riemann-Liouville derivative. The Caputo derivative is defined as a convolution integral. Thus, rather than being local (with a value at a particular point in space), the Caputo derivative is non-local (it is based on an integration in space), which is one of the reasons that it has properties not shared by integer derivatives. A principal objective of the two volumes is to provide the reader with a set of documented R routines that are discussed in detail, and can be downloaded and executed without having to first study the details of the relevant numerical analysis and then code a set of routines. In the first volume, the emphasis is on basic concepts of SFPDEs and the associated numerical algorithms. The presentation is not as formal mathematics, e.g., theorems and proofs. Rather, the presentation is by examples of SFPDEs, including a detailed discussion of the algorithms for computing numerical solutions to SFPDEs and a detailed explanation of the associated source code.

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 : 50,5 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.

Generalized Fractional Order Differential Equations Arising in Physical Models

Santanu Saha Ray,Subhadarshan Sahoo
Generalized Fractional Order Differential Equations Arising in Physical Models Book PDF

Author : Santanu Saha Ray,Subhadarshan Sahoo
Publisher : CRC Press
Page : 351 pages
File Size : 46,5 Mb
Release : 2018-11-13
Category : Mathematics
ISBN : 9780429771798

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Generalized Fractional Order Differential Equations Arising in Physical Models by Santanu Saha Ray,Subhadarshan Sahoo Pdf

This book analyzes the various semi-analytical and analytical methods for finding approximate and exact solutions of fractional order partial differential equations. It explores approximate and exact solutions obtained by various analytical methods for fractional order partial differential equations arising in physical models.

Handbook of Fractional Calculus for Engineering and Science

Harendra Singh,H. M. Srivastava,Juan J. Nieto
Handbook of Fractional Calculus for Engineering and Science Book PDF

Author : Harendra Singh,H. M. Srivastava,Juan J. Nieto
Publisher : CRC Press
Page : 236 pages
File Size : 50,7 Mb
Release : 2022-02-17
Category : Technology & Engineering
ISBN : 9781000540109

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Handbook of Fractional Calculus for Engineering and Science by Harendra Singh,H. M. Srivastava,Juan J. Nieto Pdf

Fractional calculus is used to model many real-life situations from science and engineering. The book includes different topics associated with such equations and their relevance and significance in various scientific areas of study and research. In this book readers will find several important and useful methods and techniques for solving various types of fractional-order models in science and engineering. The book should be useful for graduate students, PhD students, researchers and educators interested in mathematical modelling, physical sciences, engineering sciences, applied mathematical sciences, applied sciences, and so on. This Handbook: Provides reliable methods for solving fractional-order models in science and engineering. Contains efficient numerical methods and algorithms for engineering-related equations. Contains comparison of various methods for accuracy and validity. Demonstrates the applicability of fractional calculus in science and engineering. Examines qualitative as well as quantitative properties of solutions of various types of science- and engineering-related equations. Readers will find this book to be useful and valuable in increasing and updating their knowledge in this field and will be it will be helpful for engineers, mathematicians, scientist and researchers working on various real-life problems.

Fractional Differential Equations

Praveen Agarwal,Carlo Cattani,Shaher Momani
Fractional Differential Equations Book PDF

Author : Praveen Agarwal,Carlo Cattani,Shaher Momani
Publisher : Elsevier
Page : 272 pages
File Size : 53,7 Mb
Release : 2024-05-10
Category : Mathematics
ISBN : 9780443154249

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Fractional Differential Equations by Praveen Agarwal,Carlo Cattani,Shaher Momani Pdf

Fractional Differential Equations: Theoretical Aspects and Applications presents the latest mathematical and conceptual developments in the field of Fractional Calculus and explores the scope of applications in research science and computational modelling. Fractional derivatives arise as a generalization of integer order derivatives and have a long history: their origin can be found in the work of G. W. Leibniz and L. Euler. Shortly after being introduced, the new theory turned out to be very attractive for many famous mathematicians and scientists, including P. S. Laplace, B. Riemann, J. Liouville, N. H. Abel, and J. B. J. Fourier, due to the numerous possibilities it offered for applications.Fractional Calculus, the field of mathematics dealing with operators of differentiation and integration of arbitrary real or even complex order, extends many of the modelling capabilities of conventional calculus and integer-order differential equations and finds its application in various scientific areas, such as physics, mechanics, engineering, economics, finance, biology, and chemistry, among others. However, many aspects from the theoretical and practical point of view have still to be developed in relation with models based on fractional operators. Efficient analytical and numerical methods have been developed but still need particular attention. Fractional Differential Equations: Theoretical Aspects and Applications delves into these methods and applied computational modelling techniques, including analysis of equations involving fractional derivatives, fractional derivatives and the wave equation, analysis of FDE on groups, direct and inverse problems, functional inequalities, and computational methods for FDEs in physics and engineering. Other modelling techniques and applications explored by the authors include general fractional derivatives involving the special functions in analysis, fractional derivatives with respect to another function in analysis, new fractional operators in real-world applications, fractional order dynamical systems, hidden attractors in complex systems, nonlinear dynamics and chaos in engineering applications, quantum chaos, and self-excited attractors. Provides the most recent and up-to-date developments in the theory and scientific applications Fractional Differential Equations Includes transportable computer source codes for readers in MATLAB, with code descriptions as it relates to the mathematical modelling and applications Provides readers with a comprehensive foundational reference for this key topic in computational modeling, which is a mathematical underpinning for most areas of scientific and engineering research

Numerical Methods for Fractional Calculus

Changpin Li,Fanhai Zeng
Numerical Methods for Fractional Calculus Book PDF

Author : Changpin Li,Fanhai Zeng
Publisher : Chapman and Hall/CRC
Page : 0 pages
File Size : 42,5 Mb
Release : 2015-05-19
Category : Mathematics
ISBN : 1482253801

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Numerical Methods for Fractional Calculus by Changpin Li,Fanhai Zeng Pdf

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 properties of fractional integrals and derivatives before covering numerical methods for fractional integrals and derivatives. It then discusses finite difference methods for both FODEs and FPDEs, including the Euler and linear multistep methods. The final chapter shows how to solve FPDEs by using the finite element method. This book provides efficient and reliable numerical methods for solving fractional calculus problems. It offers a primer for readers to further develop cutting-edge research in numerical fractional calculus. MATLAB® functions are available on the book’s CRC Press web page.

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 : 41,7 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 Solutions for Partial Differential Equations

Victor Grigor'e Ganzha,Evgenii Vasilev Vorozhtsov
Numerical Solutions for Partial Differential Equations Book PDF

Author : Victor Grigor'e Ganzha,Evgenii Vasilev Vorozhtsov
Publisher : CRC Press
Page : 215 pages
File Size : 46,6 Mb
Release : 2017-11-22
Category : Mathematics
ISBN : 9781351427500

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Numerical Solutions for Partial Differential Equations by Victor Grigor'e Ganzha,Evgenii Vasilev Vorozhtsov Pdf

Partial differential equations (PDEs) play an important role in the natural sciences and technology, because they describe the way systems (natural and other) behave. The inherent suitability of PDEs to characterizing the nature, motion, and evolution of systems, has led to their wide-ranging use in numerical models that are developed in order to analyze systems that are not otherwise easily studied. Numerical Solutions for Partial Differential Equations contains all the details necessary for the reader to understand the principles and applications of advanced numerical methods for solving PDEs. In addition, it shows how the modern computer system algebra Mathematica® can be used for the analytic investigation of such numerical properties as stability, approximation, and dispersion.