Fractional Calculus

Fractional Calculus

by Behzad Ghanbari
Released June 2024
Publisher(s): Morgan Kaufmann
ISBN: 9780443315015

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Book description

Fractional Calculus: Bridging Theory with Computational and Contemporary Advances is an authoritative and comprehensive guide that delves into the world of fractional calculus, offering a unique blend of theoretical foundations, numerical algorithms, practical applications, and innovative perspectives. This book explores the mathematical framework of fractional calculus and its relevance across various disciplines, providing readers with a deep understanding of this rapidly growing field. The author presents a rigorous yet accessible approach to fractional calculus, making it suitable for mathematicians, researchers, academics, graduate students, and professionals in engineering and applied sciences. The book covers a wide range of topics, including numerical methods for fractional calculus equations, fractional differential equations, fractal dynamics, and fractional control systems. It also explores applications in areas such as physics, engineering, signal processing, and data analysis. Fractional Calculus: Bridging Theory with Computational and Contemporary Advances equips readers with the necessary tools to tackle challenging problems involving fractional calculus, empowering them to apply these techniques in their research, professional work, or academic pursuits. The book provides a comprehensive introduction to the fundamentals of fractional calculus, explaining the theoretical concepts and key definitions in a clear and accessible manner. This helps readers build a strong foundation in the subject. The book then covers a range of numerical algorithms specifically designed for fractional calculus problems, explaining the underlying principles, step-by-step implementation, and computational aspects of these algorithms. This enables readers to apply numerical techniques to solve fractional calculus problems effectively. The book also provides examples that illustrate how fractional calculus is applied to solve real-world problems, providing readers with insights into the wide-ranging applications of the subject.
  • Provides a comprehensive introduction to the fundamentals of fractional calculus, explaining the theoretical concepts and key definitions in a clear and accessible manner
  • Covers a range of numerical algorithms specifically designed for fractional calculus problems
  • Includes practical examples and case studies from various fields such as physics, biology, finance, and signal processing

Table of contents

  1. Cover image
  2. Title page
  3. Table of Contents
  4. Copyright
  5. Dedication
  6. List of figures
  7. Biography
  8. Preface
  9. 1: Introduction to fractional calculus
    1. Abstract
    2. 1.1. What is fractional calculus?
    3. 1.2. Some prerequisite special functions
    4. 1.3. Riemann–Liouville fractional differentiation
    5. 1.4. Caputo fractional derivative
    6. 1.5. Grünwald–Letnikov derivative
    7. 1.6. Caputo–Fabrizio fractional differentiation
    8. 1.7. The Atangana–Baleanu fractional derivative and integral
    9. 1.8. Riesz fractional derivatives
    10. 1.9. The Atangana–Gómez fractional derivative
    11. 1.10. General fractional derivatives
    12. 1.11. Fractal-fractional operators
    13. References
  10. 2: Exploring numerical algorithms for fractional model solutions
    1. Abstract
    2. 2.1. Discretization of fractional integrals
    3. 2.2. Discretization of fractional derivatives
    4. 2.3. Product-integration rules for Caputo fractional problems
    5. 2.4. An explicit technique for a Caputo–Fabrizio Cauchy problem
    6. 2.5. A trapezoidal-based technique for the Caputo–Fabrizio problem
    7. 2.6. A trapezoidal-based technique the AB–Cauchy problem
    8. 2.7. A discretization of the time fractal-fractional models
    9. 2.8. A survey on numerical methods for fractal-fractional models
    10. 2.9. An efficient numerical approach for fractional diffusion PDEs
    11. References
  11. 3: Analytical methods for solving fractional differential equations
    1. Abstract
    2. 3.1. Solving the resonant Davey–Stewartson system using the M-fractional derivative
    3. 3.2. Solving the generalized Schrödinger's equation using the β-fractional derivative
    4. 3.3. Solving the third-order generalized Schrödinger equation using the conformable fractional derivative
    5. 3.4. On analytical solutions to a new integrable nonlinear Schrödinger equation via local fractal calculus
    6. References
  12. 4: Elucidating chaos in dynamical systems via fractional calculus
    1. Abstract
    2. 4.1. Dynamics of fractional-order chaotic systems
    3. 4.2. Applications of chaos indicators in fractional dynamics systems
    4. 4.3. On the chaotic systems including the fractal-fractional derivative
    5. References
  13. 5: Fractional frameworks for mathematical biology
    1. Abstract
    2. 5.1. Mathematical models of phytoplankton dynamics
    3. 5.2. A survey on fractional prey–predator models
    4. 5.3. Disease modeling using fractional differential equations
    5. 5.4. A review on epidemic models in sight of fractional calculus
    6. References
  14. 6: Fractional calculus perspective on noise removal in images
    1. Abstract
    2. 6.1. Fractional denoising masks for Gaussian noise
    3. 6.2. Some performance metrics in image processing
    4. 6.3. A generalized approach to fractional masks
    5. 6.4. Fractional masks using Prabhakar fractional calculus
    6. 6.5. Applications of adaptive strategies in fractional image processing
    7. 6.6. Fractional denoising masks for salt-and-pepper noise
    8. References
  15. 7: Novel fractional calculus based approaches for edge detection
    1. Abstract
    2. 7.1. Some fractional-based ideas in edge detection
    3. 7.2. The main results
    4. 7.3. Numerical experiments and discussions
    5. References
  16. 8: Some fractional calculus based approaches for image enhancement
    1. Abstract
    2. 8.1. Some mathematical background
    3. 8.2. The main materials
    4. References
  17. References
    1. References
  18. Index

Product information

  • Title: Fractional Calculus
  • Author(s): Behzad Ghanbari
  • Release date: June 2024
  • Publisher(s): Morgan Kaufmann
  • ISBN: 9780443315015