This book is about the simulation and modeling of novel chaotic systems within the frame of fractal-fractional operators. The methods used, their convergence, stability, and error analysis are given, and this is the first book to offer mathematical modeling and simulations of chaotic problems with a wide range of fractal-fractional operators, to find solutions.
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 chaotic problems and simulations using multiple fractal-fractional operators and concentrates on models that display chaos. The book details how these systems can be predictable for a while and then can appear to become random.
Practitioners, engineers, researchers, and senior undergraduate and graduate students from mathematics and engineering disciplines will find this book of interest._
Author(s): Muhammad Altaf Khan, Abdon Atangana
Series: Mathematics and its Applications
Publisher: CRC Press
Year: 2023
Language: English
Pages: 431
City: Boca Raton
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
21.4.1. Euler-based numerical scheme for FF-Caputo operator
21.4.2. Euler-based numerical scheme for FF-CF operator
21.4.3. Euler-based numerical scheme for FF Atangana-Baleanu operator
21.5. Conclusion
Chapter 22: The Hidden Attractors Model with Fractal-Fractional Operators
22.1. Introduction
22.2. Model and its analysis
22.3. Existence and uniqueness
22.3.1. Equilibrium points and their analysis
22.4. Numerical procedure for the chaotic model
22.4.1. Numerical scheme with Euler for FF-Caputo operator
22.4.2. Numerical scheme with Euler FF Caputo-Fabrizio operator
22.4.3. Numerical scheme with Euler FF Atangana-Baleanu
22.5. Conclusion
Chapter 23: An SIR Epidemic Model with Fractal-Fractional Derivative
23.1. Introduction
23.2. Model formulation
23.3. Positivity of the model
23.4. Existence and uniqueness
23.4.1. Equilibrium points and their analysis
23.4.2. Global stability
23.5. Numerical results and the schemes
23.5.1. Euler scheme with power law case
23.5.2. Euler scheme with exponential kernel
23.5.3. Euler scheme with Mittag-Leffler Kernel
23.6. Conclusion
Chapter 24: Application of Fractal-Fractional Operators to COVID-19 Infection
24.1. Introduction
24.2. Mathematical model
24.2.1. Fractal-fractional order COVID-19 model
24.3. Existence and uniqueness
24.4. Equilibrium points and their analysis
24.5. Data fitting, numerical schemes, and their graphical results
24.5.1. Numerical scheme for COVID-19 infection model with power law
24.5.2. Numerical scheme for COVID-19 infection model with the exponential kernel
24.5.3. Numerical scheme for COVID model with Mittag-Leffler Kernel
24.6. Conclusion
References
Index
