دانلود کتاب Numerical Methods for Fractal-Fractional Differential Equations and Engineering

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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