Numerical methods for fractal-fractional differential equations and engineering : simulations and modeling

Numerical methods for fractal-fractional differential equations and engineering simulations and modeling

"Numerical Methods for Fractal-Fractional Differential Equations and Engineering: Simulations and Modeling provides details for stability, convergence, and analysis along with numerical methods and their solution procedures for fractal-fractional operators. The book offers applications to chaot...

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Detalles Bibliográficos
Otros Autores: Khan, Muhammad Altaf (Researcher in applied mathematics), author (author), Atangana, Abdon, author
Formato: Libro electrónico
Idioma:Inglés
Publicado: Boca Raton : CRC Press 2023.
Edición:First edition
Colección:Mathematics and Its Applications Series
Materias:
Fractional differential equations.
Fractional calculus.
Fractional differential equations > Numerical solutions.
Ver en Biblioteca Universitat Ramon Llull:https://discovery.url.edu/permalink/34CSUC_URL/1im36ta/alma991009784621806719
Tabla de Contenidos:
  • Cover
  • Half Title
  • Series Page
  • Title Page
  • Copyright Page
  • Dedication
  • Contents
  • Preface
  • Acknowledgement
  • Contributors
  • Chapter 1: Basic Principle of Nonlocalities
  • 1.1. Introduction
  • 1.2. Chaotic dynamics
  • 1.3. Strange attractors
  • 1.4. Some important concepts
  • 1.5. Some important concepts of numerical approximation
  • 1.5.1. Interpolation
  • 1.5.2. Linear interpolation
  • 1.5.3. Lagrange interpolation
  • 1.5.4. Middle point method
  • 1.6. Basic Reproduction number
  • 1.7. Stable
  • 1.7.1. Unstable
  • 1.7.2. Asymptotically stable
  • Chapter 2: Basic of Fractional Operators
  • 2.1. Introduction
  • 2.2. Some properties of the fractional operators
  • 2.3. Fundamental theorem of fractional calculus
  • 2.4. Fractal-Fractional operators
  • Chapter 3: Definitions of Fractal-Fractional Operators with Numerical Approximations
  • 3.1. Introduction
  • 3.2. Numerical schemes for fractal-fractional derivative
  • 3.2.1. Numerical scheme for Caputo fractal-fractional model
  • 3.2.2. Numerical scheme for Caputo-Fabrizio fractal-fractional operator
  • 3.2.3. Numerical scheme for Atangana-Baleanu fractal-fractional operator
  • 3.3. Numerical solution of fractional differential equations (FDEs)
  • 3.3.1. Numerical schemes for Atangana-Baleanu FDEs
  • Chapter 4: Error Analysis
  • 4.1. Introduction
  • 4.2. Error analysis for fractal-fractional RL Cauchy problems
  • 4.3. Error analysis for fractal-fractional CF cauchy problem
  • 4.4. Error analysis for fractal-fractional cauchy problem with Mittag-Leffler Kernel
  • Chapter 5: Existence and Uniqueness of Fractal Fractional Differential Equations
  • 5.1. Introduction
  • 5.2. Existence and uniqueness for power law case
  • 5.3. Existence and uniqueness for Mittag-Leffler case
  • 5.4. Existence and uniqueness for exponential case.
  • 5.5. Existence and uniqueness for the case with Delta-Dirac Kernel
  • Chapter 6: A Numerical Solution of Fractal-Fractional ODE with Linear Interpolation
  • 6.1. Introduction
  • 6.2. Case with the Delta-Dirac Kernel
  • 6.2.1. Examples of fractal differential equations
  • 6.3. The case of power law kernel
  • 6.4. Case with exponential decay kernel
  • 6.4.1. Examples of fractal-fractional with exponential decay function
  • 6.5. Case with generalised Mittag-Leffler Kernel
  • Chapter 7: Numerical Scheme of Fractal-Fractional ODE with Middle Point Interpolation
  • 7.1. Introduction
  • 7.2. Numerical scheme for Delta-Dirac case
  • 7.3. Numerical scheme for exponential case
  • 7.4. Numerical scheme for power law case
  • 7.5. Numerical scheme for the Mittag-Leffler case
  • Chapter 8: Fractal-Fractional Euler Method
  • 8.1. Introduction
  • 8.2. Euler method with Dirac-Delta
  • 8.3. Fractal-fractional Euler method with the exponential kernel
  • 8.4. Fractal-fractional Euler method for power law kernel
  • 8.5. Fractal-fractional Euler method with the generalised Mittag-Leffler
  • Chapter 9: Application of Fractal-Fractional Operators to a Chaotic Model
  • 9.1. Introduction
  • 9.2. Model
  • 9.2.1. Fixed points
  • 9.3. Existence and uniqueness
  • 9.4. Stability of the used numerical scheme
  • 9.5. Case for power law
  • 9.6. Numerical schemes and its simulations
  • 9.6.1. Numerical procedure in the sense of fractal-fractional-Caputo operator
  • 9.6.2. Numerical procedure for fractal-fractional Caputo-Fabrizio operator
  • 9.6.3. Numerical procedure for fractal-fractional Atangana-Baleanu operator
  • 9.7. Numerical results
  • 9.8. Conclusion
  • Chapter 10: Fractal-Fractional Modified Chua Chaotic Attractor
  • 10.1. Introduction
  • 10.2. Model framework
  • 10.3. Existence and uniqueness conditions
  • 10.4. Consistency of the scheme.
  • 10.4.1. For the case of power law
  • 10.5. Numerical procedure for the chaotic model
  • 10.5.1. Numerical procedure in the sense of fractal-fractional-Caputo operator
  • 10.5.2. Numerical procedure for fractal-fractional Caputo-Fabrizio operator
  • 10.5.3. Numerical procedure for fractal-fractional Atangana-Baleanu operator
  • 10.6. Numerical results
  • 10.7. Conclusion
  • Chapter 11: Application of Fractal-Fractional Operators to Study a New Chaotic Model
  • 11.1. Introduction
  • 11.2. Model framework
  • 11.3. Existence and Uniqueness
  • 11.3.1. Equilibrium points and its analysis
  • 11.4. Numerical procedure for the chaotic model
  • 11.4.1. Numerical procedure in the sense of fractal-fractional-Caputo operator
  • 11.4.2. Numerical procedure for fractal-fractional Caputo-Fabrizio operator
  • 11.4.3. Numerical procedure for fractal-fractional Atangana-Baleanu operator
  • 11.5. Numerical results
  • 11.6. Conclusion
  • Chapter 12: Fractal-Fractional Operators and Their Application to a Chaotic System with Sinusoidal Component
  • 12.1. Introduction
  • 12.2. Model descriptions
  • 12.3. Existence and Uniqueness
  • 12.4. Equilibrium points
  • 12.5. Numerical procedure for the chaotic model
  • 12.5.1. Numerical procedure in the sense of fractal-fractional-Caputo operator
  • 12.5.2. Numerical procedure for fractal-fractional Caputo-Fabrizio operator
  • 12.5.3. Numerical procedure for fractal-fractional Atangana-Baleanu operator
  • 12.6. Numerical results
  • 12.7. Conclusion
  • Chapter 13: Application of Fractal-Fractional Operators to Four-Scroll Chaotic System
  • 13.1. Introduction
  • 13.2. Model descriptions
  • 13.3. Existence and uniqueness
  • 13.4. Equilibrium points
  • 13.5. Numerical procedure for the chaotic model
  • 13.5.1. Numerical scheme for power law kernel using linear interpolation.
  • 13.5.2. Numerical scheme for exponential decay kernel using linear interpolations
  • 13.5.3. Numerical scheme for generalised Mittag-Leffler Kernel using linear interpolations
  • 13.6. Numerical results
  • 13.7. Conclusion
  • Chapter 14: Application of Fractal-Fractional Operators to a Novel Chaotic Model
  • 14.1. Introduction
  • 14.2. Model descriptions
  • 14.3. Existence and uniqueness
  • 14.3.1. Equilibrium points and their analysis
  • 14.4. Numerical schemes based on linear interpolations
  • 14.5. Numerical scheme for power law kernel
  • 14.5.1. Numerical scheme for exponential decay kernel using linear interpolations
  • 14.5.2. Numerical scheme for generalised Mittag-Leffler Kernel using linear interpolations
  • 14.6. Conclusion
  • Chapter 15: A 4D Chaotic System under Fractal-Fractional Operators
  • 15.1. Introduction
  • 15.2. Model details
  • 15.3. Existence and uniqueness
  • 15.4. Schemes based on linear interpolations
  • 15.4.1. Numerical scheme for power law kernel using linear interpolations
  • 15.4.2. Numerical scheme for exponential decay kernel using linear interpolations
  • 15.4.3. Numerical scheme for generalised Mittag-Leffler Kernel using linear interpolations
  • 15.5. Conclusion
  • Chapter 16: Self-Excited and Hidden Attractors through Fractal-Fractional Operators
  • 16.1. Introduction
  • 16.2. Chaotic model and its dynamical behaviour
  • 16.3. Existence and uniqueness
  • 16.4. Equilibrium points analysis
  • 16.5. Numerical procedure for the chaotic model
  • 16.6. Numerical scheme for power law kernel
  • 16.6.1. Numerical scheme for exponential decay kernel using linear interpolations
  • 16.6.2. Numerical scheme for generalised Mittag-Leffler Kernel using linear interpolations
  • 16.7. Conclusion
  • Chapter 17: Dynamical Analysis of a Chaotic Model in Fractal-Fractional Operators
  • 17.1. Introduction
  • 17.2. Model descriptions.
  • 17.3. Existence and uniqueness
  • 17.3.1. Model analysis
  • 17.4. Numerical schemes based on middle-point interpolations
  • 17.4.1. Numerical scheme for power law case
  • 17.4.2. Numerical scheme based on middle-point interpolation for exponential case
  • 17.4.3. Numerical scheme for the Mittag-Leffler case
  • 17.5. Conclusion
  • Chapter 18: A Chaotic Cancer Model in Fractal-Fractional Operators
  • 18.1. Introduction
  • 18.2. Model framework
  • 18.3. Existence and uniqueness
  • 18.3.1. Equilibrium points
  • 18.4. Numerical procedure for the chaotic model
  • 18.4.1. Numerical scheme for power law case
  • 18.4.2. Numerical scheme for exponential case
  • 18.4.3. Numerical scheme for the Mittag-Leffler case
  • 18.5. Conclusion
  • Chapter 19: A Multiple Chaotic Attractor Model under Fractal-Fractional Operators
  • 19.1. Introduction
  • 19.2. Model descriptions
  • 19.3. Existence and uniqueness
  • 19.3.1. Equilibria and their stability
  • 19.4. Numerical procedure for the chaotic model
  • 19.4.1. Numerical scheme for power law case
  • 19.4.2. Numerical scheme for exponential case
  • 19.4.3. Numerical scheme for the Mittag-Leffler case
  • 19.5. Conclusion
  • Chapter 20: The Dynamics of Multiple Chaotic Attractor with Fractal-Fractional Operators
  • 20.1. Introduction
  • 20.2. Model descriptions
  • 20.3. Existence and uniqueness of the model
  • 20.4. Numerical procedure for the chaotic model
  • 20.4.1. Numerical scheme for power law case
  • 20.4.2. Numerical scheme for exponential case
  • 20.4.3. Numerical scheme for the Mittag-Leffler case
  • 20.5. Conclusion
  • Chapter 21: Dynamics of 3D Chaotic Systems with Fractal-Fractional Operators
  • 21.1. Introduction
  • 21.2. Model descriptions and their analysis
  • 21.3. Existence and uniqueness
  • 21.3.1. Equilibrium points and their analysis.
  • 21.4. Numerical procedure for the chaotic model using Euler-based method.

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