Numerical Methods for Fractal-Fractional Differential Equations and Engineering

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