| Description |
1 online resource : illustrations |
| Contents |
Front Cover -- Computation and Modeling for Fractional Order Systems -- Copyright -- Contents -- List of contributors -- 1 Response time and accuracy modeling through the lens of fractional dynamics -- 1.1 Introduction -- 1.1.1 Historical foundation and applications of sequential sampling theory -- 1.1.2 Lévy flight models as an extension of diffusion models -- 1.2 Lévy-Brownian model as a model with both Lévy and diffusion properties -- 1.3 A tutorial on how to fit the Lévy-Brownian model -- 1.3.1 First-passage time approximation -- 1.3.2 Likelihood construction |
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1.4 Fitting to experimental data -- 1.5 Discussion -- 1.6 Conclusion -- References -- 2 An efficient analytical method for the fractional order Sharma-Tasso-Olever equation by means of the Caputo-Fabrizio deriv... -- 2.1 Introduction -- 2.2 Progress of fractional derivatives in the absence of singular kernel -- 2.3 Fundamental scheme of the modified form of HATM with new derivative -- 2.4 Analysis of MHATM with Caputo-Fabrizio derivative -- 2.5 Numerical solution of the time-fractional STO equation -- 2.5.1 Numerical discussion |
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3.2.3.1 Diffusion-flux relationship: the fading memory concept -- 3.2.3.2 Boltzmann's superposition -- 3.2.3.3 Simple heat conduction example: Cattaneo's approach -- 3.2.3.4 Extended fading memory concept -- 3.3 Kernel effects on the constitutive equations -- 3.3.1 Caputo type fractional operators: the general concept -- 3.3.1.1 Example 1: exponential memory -- 3.3.1.2 Example 2: Mittag-Leffler (one-parameter) memory -- 3.3.1.3 Example 3: Prabhakar memory kernel -- 3.3.1.4 Example 4: Rabotnov kernel as a memory -- 3.3.2 Volterra equation approach |
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3.3.2.1 The concept and Riemann-Liouville operators -- 3.3.2.2 Example 5: exponential memory -- 3.3.2.3 Example 6: Mittag-Leffler (one-parameter) function as a kernel -- 3.3.2.4 Example 7: Prabhakar kernel as a memory -- 3.3.2.5 Example 8: Rabotnov kernel as a memory -- 3.4 Final comments and outcomes -- Appendix 3.A Mittag-Leffler functions and fractional operators -- 3.A.1 Mittag-Leffler functions and related kernels -- 3.A.1.1 One-parameter Mittag-Leffler function -- 3.A.1.2 Two-parameter Mittag-Leffler function -- 3.A.1.3 Three-parameter Mittag-Leffler function -- 3.A.1.4 Prabhakar kernel |
| Summary |
Computation and Modeling for Fractional Order Systems provides readers with problem-solving techniques for obtaining exact and/or approximate solutions of governing equations arising in fractional dynamical systems presented using various analytical, semi-analytical, and numerical methods. In this regard, this book brings together contemporary and computationally efficient methods for investigating real-world fractional order systems in one volume. Fractional calculus has gained increasing popularity and relevance over the last few decades, due to its well-established applications in various fields of science and engineering. It deals with the differential and integral operators with non-integral powers. Fractional differential equations are the pillar of various systems occurring in a wide range of science and engineering disciplines, namely physics, chemical engineering, mathematical biology, financial mathematics, structural mechanics, control theory, circuit analysis, and biomechanics, among others. The fractional derivative has also been used in various other physical problems, such as frequency-dependent damping behavior of structures, motion of a plate in a Newtonian fluid, PID controller for the control of dynamical systems, and many others. The mathematical models in electromagnetics, rheology, viscoelasticity, electrochemistry, control theory, Brownian motion, signal and image processing, fluid dynamics, financial mathematics, and material science are well defined by fractional-order differential equations. Generally, these physical models are demonstrated either by ordinary or partial differential equations. However, modeling these problems by fractional differential equations, on the other hand, can make the physics of the systems more feasible and practical in some cases. In order to know the behavior of these systems, we need to study the solutions of the governing fractional models. The exact solution of fractional differential equations may not always be possible using known classical methods. Generally, the physical models occurring in nature comprise complex phenomena, and it is sometimes challenging to obtain the solution (both analytical and numerical) of nonlinear differential equations of fractional order. Various aspects of mathematical modeling that may include deterministic or uncertain (viz. fuzzy or interval or stochastic) scenarios along with fractional order (singular/non-singular kernels) are important to understand the dynamical systems. Computation and Modeling for Fractional Order Systems covers various types of fractional order models in deterministic and non-deterministic scenarios. Various analytical/semi-analytical/numerical methods are applied for solving real-life fractional order problems. The comprehensive descriptions of different recently developed fractional singular, non-singular, fractal-fractional, and discrete fractional operators, along with computationally efficient methods, are included for the reader to understand how these may be applied to real-world systems, and a wide variety of dynamical systems such as deterministic, stochastic, continuous, and discrete are addressed by the authors of the book |
| Bibliography |
Includes bibliographical references and index |
| Subject |
Fractional differential equations.
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Fractional calculus.
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| Genre/Form |
Electronic books
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| Form |
Electronic book
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| Author |
Chakraverty, Snehashish, editor
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Jena, Rajarama Mohan, editor
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| ISBN |
9780443154058 |
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0443154058 |
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