Fractional Calculus : Models and Numerical Methods.
Title:
Fractional Calculus : Models and Numerical Methods.
Author:
Diethelm, Kai.
ISBN:
9789814355216
Personal Author:
Physical Description:
1 online resource (426 pages)
Series:
Series on Complexity, Nonlinearity and Chaos
Contents:
Contents -- Preface -- 1. Preliminaries -- 1.1 Fourier and Laplace Transforms -- 1.2 Special Functions and Their Properties -- 1.2.1 The Gamma function and related special functions -- 1.2.2 Hypergeometric functions -- 1.2.3 Mittag-Leffler functions -- 1.3 Fractional Operators -- 1.3.1 Riemann-Liouville fractional integrals and fractional derivatives -- 1.3.2 Caputo fractional derivatives -- 1.3.3 Liouville fractional integrals and fractional derivatives. Marchaud derivatives -- 1.3.4 Generalized exponential functions -- 1.3.5 Hadamard type fractional integrals and fractional derivatives -- 1.3.6 Fractional integrals and fractional derivatives of a function with respect to another function -- 1.3.7 Grunwald-Letnikov fractional derivatives -- 2. A Survey of Numerical Methods for the Solution of Ordinary and Partial Fractional Differential Equations -- 2.1 Approximation of Fractional Operators -- 2.1.1 Methods based on quadrature theory -- 2.1.2 Grunwald-Letnikov methods -- 2.1.3 Lubich's fractional linear multistep methods -- 2.2 Direct Methods for Fractional ODEs -- 2.2.1 The basic idea -- 2.2.2 Quadrature-based direct methods -- 2.3 Indirect Methods for Fractional ODEs -- 2.3.1 The basic idea -- 2.3.2 An Adams-type predictor-corrector method -- 2.3.3 The Cao-Burrage-Abdullah approach -- 2.4 Linear Multistep Methods -- 2.5 Other Methods -- 2.6 Methods for Terminal Value Problems -- 2.7 Methods for Multi-Term FDE and Multi-Order FDS -- 2.8 Extension to Fractional PDEs -- 2.8.1 General formulation of the problem -- 2.8.2 Examples -- 3. Efficient Numerical Methods -- 3.1 Methods for Ordinary Differential Equations -- 3.1.1 Dealing with non-locality -- 3.1.2 Parallelization of algorithms -- 3.1.3 When and when not to use fractional linear multistep formulas -- 3.1.4 The use of series expansions -- 3.1.5 Adams methods for multi-order equations.
3.1.6 Two classes of singular equations as application examples -- 3.2 Methods for Partial Differential Equations -- 3.2.1 The method of lines -- 3.2.2 BDFs for time-fractional equations -- 3.2.3 Other methods -- 3.2.4 Methods for equations with space-fractional operators -- 4. Generalized Stirling Numbers and Applications -- 4.1 Introduction -- 4.2 Stirling Functions s(a, k), a C -- 4.2.1 Equivalent definitions -- 4.2.2 Multiple sum representations. The Riemann Zeta function -- 4.3 General Stirling Functions s(α, β) with Complex Arguments -- 4.3.1 Definition and main result -- 4.3.2 Differentiability of the s(α, β) -- The zeta function encore -- 4.3.3 Recurrence relations for s(α, β) -- 4.4 Stirling Functions of the Second Kind S(α, k) -- 4.4.1 Stirling functions S(a, k), a 0, and their representations by Liouville and Marchaud fractional derivatives -- 4.4.2 Stirling functions S(α, k), α < 0, and their representations by Liouville fractional integrals -- 4.4.3 Stirling functions S(a, k), a C, and their representations -- 4.4.4 Stirling functions S(a, k), a C, and recurrence relations -- 4.4.5 Further properties and first applications of Stirling functions S(a, k), a C -- 4.4.6 Applications of Stirling functions S(a, k) (a C) to Hadamard-type fractional operators -- 4.5 Generalized Stirling Functions S(n, ß), ß C -- 4.5.1 Definition and some basic properties -- 4.5.2 Main properties -- 4.6 Generalized Stirling Functions S(a, ß), a, ß C -- 4.6.1 Basic properties -- 4.6.2 Representations by Liouville fractional operators -- 4.6.3 First application -- 4.6.4 Special examples -- 4.7 Connections Between s(α, β) and S(α, k) -- 4.7.1 Coincidence relations -- 4.7.2 Results from sampling analysis -- 4.7.3 Generalized orthogonality properties -- 4.7.4 The s(α, k) connecting two types of fractional derivatives.
4.7.5 The representation of a general fractional difference operator via s(α, k) -- 5. Fractional Variational Principles -- 5.1 Fractional Euler-Lagrange Equations -- 5.1.1 Introduction and survey of results -- 5.1.2 Fractional Euler-Lagrange equations for discrete and continuous systems -- 5.1.2.1 Fractional Euler-Lagrange equations for discrete systems -- 5.1.3 Fractional Lagrangian formulation of field systems -- 5.1.4 Fractional Euler-Lagrange equations with delay -- 5.1.4.1 Riemann-Liouville fractional Euler-Lagrange equations with delay -- 5.1.4.2 Caputo fractional Euler-Lagrange equations with delay -- 5.1.5 Fractional discrete Euler-Lagrange equations -- 5.1.6 Fractional Lagrange-Finsler geometry -- 5.1.7 Applications -- 5.1.7.2 Multi-order and multi-term fractional variational formulations with Hilfer derivatives -- 5.1.7.3 A fractional Lagrangian approach of Schrodinger equations -- 5.1.7.4 Fractional Lagrangians which differ by a fractional Riesz derivative -- 5.1.7.5 Euler-Lagrange equations in fractional space -- 5.1.7.7 Fractional Faddeev-Jackiw formalism -- 5.1.7.8 Fractional variational calculus with generalized boundary condition -- 5.2 Fractional Hamiltonian Dynamics -- 5.2.1 Introduction and overview of results -- 5.2.2 Fractional Hamiltonian analysis for discrete and continuous systems -- 5.2.2.1 A direct method within Riemann-Liouville fractional derivatives -- 5.2.2.2 A direct method with Caputo fractional derivatives -- 5.2.2.3 A direct method within Riesz-Caputo fractional derivatives -- 5.2.3 Fractional Hamiltonian formulation for constrained systems -- 5.2.3.1 Fractional Hessian matrix -- 5.2.3.2 The reduced phase space -- 5.2.3.3 Fractional Ostrogradski's approach -- 5.2.4 Applications -- 5.2.4.1 Discrete fractional constrained systems -- 5.2.4.2 Fractional Hamiltonian formulation in fractional time.
5.2.4.3 Fractional Nambu mechanics -- 5.2.4.4 A fractional supersymmetric model -- 5.2.4.5 Fractional optimal control formulation -- 5.2.4.6 The fractional optimal control approach with delay -- 5.2.4.7 Fractional multi time Hamiltonian equations -- 5.2.4.8 Hamilton-Jacobi formulation with Caputo fractional derivative -- 5.2.4.9 Fractional dynamics on extended phase space -- 6. CTRW and Fractional Diffusion Models -- 6.1 Introduction -- 6.2 The Definition of Continuous-Time Random Walks -- 6.3 Fractional Diffusion and Limit Theorems -- 7. Applications of CTRW to Finance and Economics -- 7.1 Introduction -- 7.2 Models of Price Fluctuations in Financial Markets -- 7.3 Simulation -- 7.4 Option Pricing -- 7.5 Other Applications -- Appendix A Source Codes -- A.1 The Adams-Bashforth-Moulton Method -- A.2 Lubich's Fractional Backward Differentiation Formulas -- A.3 Time-fractional Diffusion Equations -- A.4 Computation of the Mittag-Leffler Function -- A.5 Monte Carlo simulation of CTRW -- Bibliography -- Index -- Advisory Board.
Abstract:
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.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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