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Record Nr. |
UNINA9910816788503321 |
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Autore |
Baleanu D (Dumitru) |
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Titolo |
Fractional calculus : models and numerical methods / / Dumitru Baleanu ... [et al.] |
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Pubbl/distr/stampa |
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Singapore, : World Scientific Pub. Co., 2012 |
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ISBN |
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1-280-66952-7 |
9786613646453 |
981-4355-21-6 |
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Edizione |
[1st ed.] |
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Descrizione fisica |
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1 online resource (426 p.) |
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Collana |
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Series on complexity, nonlinearity and chaos, , 2010-0019 ; ; v. 3 |
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Altri autori (Persone) |
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Disciplina |
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Soggetti |
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Fractional calculus |
Mathematical models |
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Lingua di pubblicazione |
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Formato |
Materiale a stampa |
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Livello bibliografico |
Monografia |
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Note generali |
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Description based upon print version of record. |
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Nota di bibliografia |
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Includes bibliographical references and index. |
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Nota di contenuto |
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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 function1.3.7 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 method2.3.3 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 |
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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 equations3.1.6 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 result4.3.2 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 |
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Sommario/riassunto |
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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 o |
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