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Computational Fractional Dynamical Systems 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…mehr
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.
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Autorenporträt
Snehashish Chakraverty, Senior Professor, Department of Mathematics (Applied Mathematics Group), National Institute of Technology Rourkela, Odisha, India.
Rajarama Mohan Jena, Senior Research Fellow, Department of Mathematics, National Institute of Technology Rourkela, Odisha, India.
Subrat Kumar Jena, Senior Research Fellow, Department of Mathematics, National Institute of Technology Rourkela, Odisha, India.
Inhaltsangabe
Preface
Acknowledgments
About the Authors
Introduction to Fractional Calculus
1.1. Introduction
1.2. Birth of fractional calculus
1.3. Useful mathematical functions
1.3.1. The gamma function
1.3.2. The beta function
1.3.3. The Mittag-Leffler function
1.3.4. The Mellin-Ross function
1.3.5. The Wright function
1.3.6. The error function
1.3.7. The hypergeometric function
1.3.8. The H-function
1.4. Riemann-Liouville fractional integral and derivative
1.5. Caputo fractional derivative
1.6. Grünwald-Letnikov fractional derivative and integral
1.7. Riesz fractional derivative and integral
1.8. Modified Riemann-Liouville derivative
1.9. Local fractional derivative
1.9.1. Local fractional continuity of a function
1.9.2. Local fractional derivative
References
Recent Trends in Fractional Dynamical Models and Mathematical Methods
2.1. Introduction
2.2. Fractional calculus: A generalization of integer-order calculus
2.3. Fractional derivatives of some functions and their graphical illustrations
2.4. Applications of fractional calculus
2.4.1. N.H. Abel and Tautochronous problem
2.4.2. Ultrasonic wave propagation in human cancellous bone
2.4.3. Modeling of speech signals using fractional calculus
2.4.4. Modeling the cardiac tissue electrode interface using fractional calculus
2.4.5. Application of fractional calculus to the sound waves propagation in rigid porous Materials
2.4.6. Fractional calculus for lateral and longitudinal control of autonomous vehicles
2.4.7. Application of fractional calculus in the theory of viscoelasticity
2.4.8. Fractional differentiation for edge detection
2.4.9. Wave propagation in viscoelastic horns using a fractional calculus rheology model
2.4.10. Application of fractional calculus to fluid mechanics
2.4.11. Radioactivity, exponential decay and population growth
2.4.12. The Harmonic oscillator
2.5. Overview of some analytical/numerical methods
2.5.1. Fractional Adams-Bashforth/Moulton methods
2.5.2. Fractional Euler method
2.5.3. Finite difference method
2.5.4. Finite element method
2.5.5. Finite volume method
2.5.6. Meshless method
2.5.7. Reproducing kernel Hilbert space method
2.5.8. Wavelet method
2.5.9. The Sine-Gordon expansion method
2.5.10. The Jacobi elliptic equation method
2.5.11. The generalized Kudryashov method
References
Adomian Decomposition Method (ADM)
3.1. Introduction
3.2. Basic Idea of ADM
3.3. Numerical Examples
References
Adomian Decomposition Transform Method
4.1. Introduction
4.2. Transform methods for the Caputo sense derivatives
