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