Special Functions in Fractional Calculus and Engineering

Cover
Half Title
Series Page
Title Page
Copyright Page
Table of Contents
Preface
Editors
Contributors
Chapter 1 An Introductory Overview of Special Functions and Their Associated Operators of Fractional Calculus
	1.1 Introduction, Definitions and Preliminaries
	1.2 Hypergeometric Functions: Extensions and Multivariate Generalizations
	1.3 The Zeta and Related Functions of Analytic Number Theory
	1.4 Extensions and Generalizations of the Mittag-Leffler-Type Functions
	1.5 Fractional Calculus and Its Applications
	1.6 Concluding Remarks and Observations
	Conflicts of Interest
	Bibliography
Chapter 2 Analytical Solutions for the Fluid Model Described by Fractional Derivative Operators Using Special Functions in Fractional Calculus
	2.1 Introduction
	2.2 Fractional Calculus Operators and Special Functions
	2.3 Fractional Model Under Consideration
	2.4 Solutions Procedures
		2.4.1 Solutions Procedures with Temperature Distrbution
		2.4.2 Solutions Procedures with Velocity Distribution
	2.5 Results and Discussion
	2.6 Conclusion
	Conflict of Interest
	References
Chapter 3 Special Functions and Exact Solutions for Fractional Diffusion Equations with Reaction Terms
	3.1 Introduction
	3.2 Diffusion-Reaction
		3.2.1 Case -K[sub(β)] [sup(t)]=δ(t),−1<η, μ ̸= 2
		3.2.2 Case -K[sub(β)] [sup(t)] = T[sup(−β)] /Γ(1−β), −1 < η, μ ̸= 2
		3.2.3 Case -K[sub(β)] [sup(t)]=N[sup(′)] [sub(β)] [sup(e)][sup(−β′T)] , −1 < η, μ ̸= 2
		3.2.4 Reaction Process – Arbitrary Reaction Rates
	3.3 Discussion and Conclusion
	Acknowledgment
	References
Chapter 4 Computable Solution of Fractional Kinetic Equations Associated with Incomplete ℵ-Functions and M-Series
	4.1 Introduction
	4.2 Generalized FKE Involving Incomplete ℵ-Functions and M-Series
	4.3 Special Cases
	4.4 Conclusions
	Declarations
	References
Chapter 5 Legendre Collocation Method for Generalized Fractional Advection Diffusion Equation
	5.1 Introduction
	5.2 Mathematical Background of Fractional Calculus
	5.3 Function Approximation Using Legendre Polynomials
		5.3.1 Approximation of a Two-Variable Function Using Legendre Polynomials
		5.3.2 Collocation Method for GFADE
	5.4 Convergence Analysis
	5.5 Error Analysis
	5.6 Numerical Results
	5.7 Conclusion
	References
Chapter 6 The Incomplete Generalized Mittag-Leffler Function and Fractional Calculus Operators
	6.1 Introduction, Definitions, and Preliminaries
	6.2 The Incomplete Generalized Mittag-Leffler Function
		6.2.1 Basic Properties of Ξ[sup(ρ,κ)] [sub(α, β)] (z)
	6.3 Incomplete Fox-H, Fox-Wright Representations and Mellin-Barnes Integrals of Ξ[sup(ρ,κ)] [sub(α, β)] (z)
	6.4 Integral Transforms Representations
		6.4.1 Laplace Transform
		6.4.2 Whittaker Transforms
		6.4.3 Euler-Beta Transform
	6.5 Fractional Calculus Operators
	6.6 Application to the Solution of Fractional Kinetic Equation
	6.7 Further Remarks and Observations
	Bibliography
Chapter 7 Numerical Solution of Fractional Order Diffusion Equation Using Fibonacci Neural Network
	7.1 Introduction
	7.2 Definitions
		7.2.1 Caputo Fractional Order Derivative
		7.2.2 Properties of Fibonacci Polynomial
	7.3 FNN and Method to Apply to Solve Considered Model
		7.3.1 Method to Use FNN to Solve Two-Dimensional FDE
		7.3.2 Learning Algorithm for FNN
	7.4 Numerical Example
	7.5 Solution of Two-Dimensional FDE
	7.6 Application of the Method in Engineering
	7.7 Conclusion
	Bibliography
Chapter 8 Analysis of a Class of Reaction-Diffusion Equation Using Spectral Scheme
	8.1 Introduction
	8.2 Preliminaries
	8.3 Basic Properties of Laguerre Polynomials
	8.4 Formulation of Opertional Matrix
	8.5 Approximation of Function
	8.6 Stability Analysis
	8.7 Numerical Examination of FNKGE
	8.8 Discussion of Outcomes
	8.9 Application of Model
	8.10 Conclusion
	References
Chapter 9 New Fractional Calculus Results for the Families of Extended Hurwitz-Lerch Zeta Function
	9.1 Introduction and Preliminaries
	9.2 Fractional Integral Operators of the Ф[sup(ρ,τ;κ)] [sub(λ,μ; ν)] (t,s,a)
	9.3 Fractional Differential Operators of the Ф[sup(ρ,τ;κ)] [sub(λ,μ; ν)] (t,s,a)
		9.3.1 Fractional Calculus Operators of the Ф[sub(λ,μ; ν)] (t,s,a)
		9.3.2 Fractional Calculus Operators of the Ф[sup(τ;κ)] [sub(μ; ν)] (t,s,a)
		9.3.3 Fractional Calculus Operators of the Ф[sup(*)] [sub(μ)] (t,s,a)
	9.4 Further Observations and Applications
	9.5 Concluding Remarks
	References
Chapter 10 Compact Difference Schemes for Solving the Equation of Fractional Oscillator Motion with Viscoelastic Damping
	10.1 Introduction
	10.2 Some Applications of Fractional Oscillator Motion with Viscoelastic Damping in Engineering
	10.3 Construction and Analysis of Scheme 1 for Riemann–Liouville with 0 < α < 1
		10.3.1 Construction of Scheme 1
		10.3.2 Analysis of Scheme 1
	10.4 Construction and Analysis of Scheme 2 for 1 < α < 2
		10.4.1 Construction of Scheme 2
		10.4.2 Analysis of Scheme 2
	10.5 Numerical Example
	10.6 Conclusions
	Author Contributions
	References
Chapter 11 Dynamics of the Dadras-Momeni System in the Frame of the Caputo-Fabrizio Fractional Derivative
	11.1 Introduction
	11.2 Preliminaries
	11.3 Model Formulation
	11.4 Existence and Uniqueness of Solutions for the Projected System
	11.5 Dynamics of the System (11.7)
	11.6 Design of Sliding Mode Controller
	11.7 Numerical Simulations by Single Step Adams-Bashforth-Moulton Method
	11.8 Conclusion
	References
Chapter 12 A Fractional Order Model with Non-Singular Mittag-Leffler Kernel
	12.1 Introduction
	12.2 Existence and Uniqueness of the Solutions
		12.2.1 Linear Growth
		12.2.2 Lipschitz Condition
	12.3 Numerical Method
	12.4 Results of the Simulation
	12.5 Conclusion
	References
Index