Fractional Order Systems and Applications in Engineering

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Fractional Order Systems and Applications in Engineering
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Contents
Contributors
Preface
1 Complete synchronization of the time-fractional Chua reaction–diffusion system
1.1 Introduction
1.1.1 General overview
1.1.2 The Chua chaotic system
1.1.3 Fractional Chua systems
1.2 Proposed system model
1.3 Free diffusion case: equilibria and stability
1.4 Spatio-temporal synchronization
1.5 Results and discussion
1.6 Concluding remarks
References
2 New nonsymmetric and parametric divergences with particular cases
2.1 Introduction
2.2 New information divergence measures
2.3 Divergence measures in the form of new series
2.4 Relations among divergence measures
2.5 Parametric measure of information
2.6 Concluding remarks
References
3 Analytical solutions of some fractional diffusion boundary value problems
3.1 Introduction
3.2 Definitions
3.3 Elementary results
3.4 Temperature distribution in finite solid circular cylinder
3.5 Particular case
3.6 Temperature distribution in finite hollow circular cylinder
3.7 Particular cases
References
4 An enhanced hybrid stochastic fractal search FOPID for speed control of DC motor
4.1 Introduction
4.2 Modeling of DC motor
4.3 Controller design and objective function
4.3.1 Controller design
4.3.2 Objective function
4.4 Proposed hybrid stochastic fractal search
4.5 Proposed HSFS-FOPID approach for speed control of DC motor
4.6 Modeling of DC motor with HSFS-FOPID
4.7 Robustness analysis
4.8 Conclusion
References
5 Fractional dynamics and metrics of deadly pandemic diseases
5.1 Introduction
5.2 Fractional model formulation
5.3 Basic reproduction number
5.4 Stability of equilibrium states
5.5 Discussion of results
5.6 Conclusion
References
6 A numerical technique for solving fractional Benjamin–Bona–Mahony–Burgers equations with bibliometric analysis
6.1 Introduction
6.2 Preliminaries
6.2.1 Fractional calculus
6.2.2 Müntz–Legendre polynomials and their properties
6.2.2.1 Function approximation
6.3 Riemann–Liouville pseudooperational matrix of Müntz–Legendre polynomials
6.4 Computational method
6.5 Convergence analysis
6.6 Numerical examples
6.7 Conclusion
References
7 Some roots and paths in the fractional calculus developing environment
7.1 Fractional calculus: historical touch and mathematical context. Definitions
7.2 New families of evolution equations. Dirac equations and fractional calculus
7.3 Numerical algorithms. Cloud computing
7.3.1 Numerical algorithms
7.3.2 Cloud computing
7.3.2.1 Cloud computing infrastructure
7.3.2.2 Experiments
7.3.2.3 Execution and cost model
Experiment set A
Experiment set B
7.4 Atmospheric dust modeling
7.5 Electromagnetic waves. Fractal structures and metamaterials
7.5.1 Metamaterials
Funding
Acknowledgments
References
8 Accruement of nonlinear dynamical system and its dynamics: electronics and cryptographic engineering
8.1 Introduction
8.2 Systems and their properties
8.2.1 Lyapunov spectrum and Kaplan–Yorke dimension
8.3 Analog circuit imitations
8.4 Cryptography and security analysis
NPCR and UACI
Correlation
Mean square error (MSE)
Peak signal-to-noise ratio (PSNR)
Entropy
8.5 Conclusion
References
9 Some new integral inequalities via generalized proportional fractional integral operators for the classes of m-logarithmically convex functions
9.1 Introduction
9.2 New integral inequalities via generalized proportional integral operators
References
10 Application and optimization of a robust fractional-order FOPI-FOPID automatic generation controller for a multiarea interconnected hybrid power system
10.1 Introduction
10.2 Investigated PS
10.3 Controller
10.3.1 Fractional-order calculus
10.3.2 Integer-order and fractional-order controllers
10.4 Optimization algorithm
10.4.1 Hooke–Jeeves method
10.5 Results and analysis
10.5.1 Comparison of AGC performances of integral, PI, PID, FOPID, and FOPI-FOPID cascade controllers
10.5.2 Study of the performance of the proposed FOPI-FOPID cascade controller against the random variation in load
10.5.3 Study of the performance of the proposed FOPI-FOPID cascade controller against the random variation in solar radiation
10.5.4 Study of the performance of the proposed FOPI-FOPID cascade controller against the variation in wind turbine speed reference
10.5.5 Study of the performance of the proposed FOPI-FOPID cascade controller for different performance indices used in optimization
10.5.6 Study of the performance of the proposed FOPI-FOPID cascade controller against variation in power system block parameters
10.6 Conclusion
References
11 Fourth-order fractional diffusion equations: constructs and memory kernel effects
11.1 Introduction
11.1.1 Aim
11.1.2 Chapter organization
11.2 Preliminaries
11.2.1 Memory functions and related fractional operators
11.2.1.1 Singular memories
11.2.1.2 Nonsingular memories
11.2.2 Boltzmann superposition principle (fading memory concept)
11.2.3 The multiple integral-balance method
11.2.3.1 Basic concept
11.2.3.2 Application to time-fractional equations
11.3 Solutions (to formally fractionalized models)
11.3.1 General approach
11.3.2 Suggested general form of the solution and underlying physical concept
11.3.3 MIM to integer-order 4th-order diffusion equation: some notes
11.3.4 Caputo-type derivative with a singular power-law kernel
11.3.5 Caputo-type derivatives with nonsingular kernels
11.3.5.1 Exponential kernel (Caputo–Fabrizio operator)
11.3.5.2 Mittag-Leffler function as a kernel (AB derivative)
11.4 Solutions (to systematically fractionalized models)
11.4.1 Toward the high-order flux-gradient relationships with memory
11.4.2 Power-law memory kernel
11.4.3 Exponential memory kernel
11.4.4 Mittag-Leffler memory kernel
11.5 A necessary discussion: what do the models developed mean
11.5.1 Where the causality principle is satisfied?
11.5.2 What MIM solutions allow us to detect in these models?
11.6 Conclusions
References
12 Analysis of COVID-19 outbreak using GIS and SEIR model
12.1 Introduction
12.1.1 Heat maps in Geographical Information System (GIS)
12.2 Material and methods
12.2.1 Generation of heat maps
12.2.2 SEIR model
12.3 Results and discussion
12.3.1 Heat maps
12.3.1.1 January 2020
12.3.1.2 February 2020
12.3.1.3 March 2020
12.3.1.4 April 2020
12.3.1.5 May 2020
12.3.1.6 June 2020
12.3.1.7 July 2020 and mid-August (2020)
12.3.2 SEIR model
Conclusions
References
13 Hidden chaotic attractors in fractional-order discrete-time systems
13.1 Introduction
13.2 Basic tools
13.3 Fractional-order Hénon-like map with no equilibrium points
13.3.1 Dynamical analysis
13.3.2 Entropy analysis
13.4 Conclusion
References
14 Dynamical investigation and simulation of an incommensurate fractional-order model of COVID-19 outbreak with nonlinear saturated incidence rate
14.1 Introduction
14.2 Background of Atangana–Baleanu operators
14.3 The SQIR model
14.3.1 Fractional-order model
14.4 Existence and uniqueness results
14.5 Basic reproduction number, existence, and stability of equilibria
14.5.1 Local stability analysis
14.6 Numerical approximation
14.7 Simulation and calibration of the fractional-order model
14.7.1 Estimation of model parameters and best fit of fractional orders
14.7.2 Sensitivity analysis of model parameters with respect to R0
14.7.3 Impact of fractional-order derivation
14.8 Conclusion
References
15 Weak Pontryagin's maximum principle for optimal control problems involving a general analytic kernel
15.1 Introduction
15.2 Preliminaries
15.3 Fundamental properties
15.4 Main results
15.4.1 Continuity of solutions of control differential equations
15.4.2 Differentiability of solutions
15.4.3 Pontryagin's maximum principle
15.4.4 Calculus of variations
Isoperimetric problems
Acknowledgments
References
16 Computational half-sweep preconditioned Gauss–Seidel method for time-fractional diffusion equations
16.1 Introduction
16.2 Preliminaries
16.3 Caputo's finite difference approximation
16.4 Analysis of stability of time fractional diffusion equations
16.5 Formulation of half-sweep preconditioned Gauss–Seidel
16.6 Numerical experiment
16.7 Conclusions
References
17 Operational matrix approach for solving variable-order fractional integro-differential equations
17.1 Introduction
17.2 Preliminaries and notations
17.3 Fourth kind Chebyshev polynomials and their shifted ones
17.3.1 Fourth kind Chebyshev polynomials
17.3.2 The fourth kind shifted Chebyshev polynomials
17.4 Approach function
17.5 Convergence estimate
17.6 Operational matrices of differentiation and integration for solving VO-FIDEs
17.6.1 Operational matrix of differentiation
17.6.2 Operational matrix of integration
17.6.3 Solving VO-FIDEs via operational matrices
17.7 Computational examples and results analysis
17.8 Concluding remarks
Acknowledgment
References
18 On basic Humbert confluent hypergeometric functions
18.1 Introduction, definitions, basic concepts, and notations
18.2 q-Contiguous function relations and q-recursion formulas for q-Humbert confluent hypergeometric functions Φ1
18.3 q-Contiguous function relations and q-recursion formulas for Φ2
18.4 q-Contiguous function relations and q-recursion formulas for Φ3
18.5 q-Contiguous function relations and q-recursion formulas for Ψ1
18.6 q-Contiguous function relations and q-recursion formulas of q-Humbert confluent hypergeometric functions Ψ2
18.7 q-Contiguous function relations and q-recursion formulas for Ξ1
18.8 q-Contiguous function relations and q-recursion formulas for Ξ2
References
19 Derivatives of Horn's hypergeometric functions G1, G2, Γ1, and Γ2 with respect to their parameters
19.1 Introduction, basic concepts, notations, and preliminaries
19.2 Derivatives of Φ1 with respect to their parameters
19.3 Derivatives of G2 with respect to their parameters
19.4 Derivatives of Γ1 with respect to their parameters
19.5 Derivatives of Γ2 with respect to the parameters
19.6 Applications
19.7 Concluding remarks and observations
Acknowledgments
References
Index
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