Special Functions and Analysis of Differential Equations

Cover
Half Title
Title Page
Copyright Page
Table of Contents
Preface
Editors
Contributors
1 A Chebyshev Spatial Discretization Method for Solving Fractional Fokker–Planck Equation with Riesz Derivatives
	1.1 Introduction
	1.2 Preliminaries and Problem Statement
		1.2.1 Fractional Calculus
		1.2.2 Polynomial Interpolation
		1.2.3 Problem Statement
	1.3 Chebyshev Spatial Discretization Method for Solving Linear SFFPEs
		1.3.1 Chebyshev Operational Matrix of Fractional Operators
		1.3.2 Spatial Discretization of SFFPEs
	1.4 Convergence Analysis
	1.5 Numerical Examples
	1.6 Conclusion
	References
2 Special Functions and Their Link with Nonlinear Rod Theory
	2.1 Introduction
		2.1.1 A Short Sketch of the Elastica History
	2.2 A Supported Rod Inflected by Axial Thrust
		2.2.1 The Problem
		2.2.2 The Elastica Nonlinear ODE
		2.2.3 Phase Portrait Analysis
		2.2.4 Integration
	2.3 Cantilever Laded at Its Tip: Elastica Parametrized via Elliptic Functions
		2.3.1 The Problem
		2.3.2 A 3D Rod Model
		2.3.3 A Two-Points Boundary Value Problem
		2.3.4 Rod Local Rotations Parametrized through the Arc: ɸ = ɸ(s)
			2.3.4.1 The Elastica Arc Length
			2.3.4.2 The Free End Rotation ɸ0
		2.3.5 Elastica Coordinates x(s), y(s) Parametrized through Its Arc
		2.3.6 A Meaningful Generalization about Loads
	2.4 The Cantilever Deflections by Means of the Lauricella Hypergeometric Functions
		2.4.1 The Main Assumptions
		2.4.2 Cantilever Inflected by a Constant Bending Couple at Its Tip
		2.4.3 The Heavy Cantilever
		2.4.4 A Tip Sheared Horizontal Cantilever
			2.4.4.1 A Hypergeometric Treatment
			2.4.4.2 A Treatment by Elliptic Integrals
			2.4.4.3 Consistency with the Literature
			2.4.4.4 How to Compute the Tip Position after the Strain
		2.4.5 The Cantilever Loaded by Sinusoidal Bending Moment
		2.4.6 The Cantilever Inflected by Hydrostatic Pressure
	2.5 A Thin Heavy Flagpole Bent under a Transverse Wind: Its Elastica through the Bessel Functions
		2.5.1 The Problem
		2.5.2 The Heavy Flagpole under a Transverse Wind
		2.5.3 The Analytical Solution of the Third Order ODE
	2.6 Curvature Effects on the Statically Redundant Reactions: The Heavy Cantilever
		2.6.1 Statement of the Problem
		2.6.2 Hypergeometric Tools
		2.6.3 The Statically Indeterminate Heavy Cantilever Supported by a Roller
			2.6.3.1 First Subsystem: The Heavy Cantilever
			2.6.3.2 Second Subsystem: The Tip-Sheared Cantilever
			2.6.3.3 Consistence and Detection of the Statically Redundant Unknown
		2.6.4 Conclusions about the Statically Indeterminate Unknowns
	References
3 Second Kind Chebyshev Wavelets for Solving the Variable-Order Space-Time Fractional Telegraph Equation
	3.1 Introduction
	3.2 Definitions and Mathematical Preliminaries
	3.3 The SKCWs and Their Properties
		3.3.1 Wavelets and the SKCWs
		3.3.2 Function Approximation
		3.3.3 Convergence and Error Analysis
		3.3.4 The Operational Matrix of Variable-Order Fractional Derivative (OMV-FD)
	3.4 The Proposed Method
	3.5 Illustrative Examples
	3.6 Conclusion
	References
4 Hyers–Ulam–Rassias Stabilities of Some Classes of Fractional Differential Equations
	4.1 Introduction
	4.2 Preliminaries Results
	4.3 Hyers–Ulam–Rassias Stability in a Finite Interval
	4.4 Hyers–Ulam Stability in a Finite Interval
	4.5 Hyers–Ulam–Rassias Stability in an Infinite Interval
	4.6 Conclusions
	Acknowledgments
	References
5 Applications of Fractional Derivatives to Heat Transfer in Channel Flow of Nanofluids
	5.1 Introduction
	5.2 Mathematical Modeling
	5.3 Solution of the Problem
		5.3.1 Solution of Energy Equation
		5.3.2 Solution of Momentum Equation
	5.4 Parametric Studies
	5.5 Concluding Remarks
	Acknowledgment
	Appendix 5.A
	References
6 The Hyperbolic Maximum Principle Approach to the Construction of Generalized Convolutions
	6.1 Introduction
	6.2 Preliminaries
		6.2.1 Solutions of the Sturm–Liouville Equation
		6.2.2 Sturm–Liouville Type Transforms
		6.2.3 Diffusion Processes
	6.3 The Hyperbolic Equation ℓ[sub(x)]f = ℓ[sub(y)]f
		6.3.1 Existence and Uniqueness of Solution
		6.3.2 Maximum Principle and Positivity of Solution
	6.4 Sturm–Liouville Translation and Convolution
		6.4.1 Definition and First Properties
		6.4.2 Sturm–Liouville Transform of Measures
	6.5 The Product Formula
	6.6 Harmonic Analysis on L[sub(p)] Spaces
	6.7 Applications to Probability Theory
		6.7.1 Infinite Divisibility of Measures and the Lévy–Khintchine Representation
		6.7.2 Convolution Semigroups and Their Contraction Properties
		6.7.3 Additive and Lévy Processes
	6.8 Examples
	Acknowledgments
	References
7 Elements of Aomoto’s Generalized Hypergeometric Functions and a Novel Perspective on Gauss’ Hypergeometric Differential Equation
	7.1 Introduction
	7.2 Elements of Aomoto’s Generalized Hypergeometric Functions
		7.2.1 Definition
		7.2.2 Integral Representation of F(Z) and Twisted Cohomology
		7.2.3 Twisted Homology and Twisted Cycles
		7.2.4 Differential Equations of F(Z)
		7.2.5 Nonprojected Formulation
	7.3 Generalized Hypergeometric Functions on Gr(2, n + 1)
	7.4 Reduction to Gauss’ Hypergeometric Function
		7.4.1 Basics of Gauss’ Hypergeometric Function
		7.4.2 Reduction to Gauss’ Hypergeometric Function 1: From Defining Equations
		7.4.3 Reduction to Gauss’ Hypergeometric Function 2: Use of Twisted Cohomology
		7.4.4 Reduction to Gauss’ Hypergeometric Function 3: Permutation Invariance
		7.4.5 Summary
	References
8 Around Boundary Functions of the Right Half-Plane and the Unit Disc
	8.1 Prerequisites
	8.2 RHP vs. D
		8.2.1 Laurent Expansion to q-Expansion
		8.2.2 RHP → D
		8.2.3 Fourier Series as an Intrinsic Property of the Monolog
		8.2.4 Boundary Functions of Certain Lambert Series
		8.2.5 Control Theory in the Unit Disc
	8.3 Robust Controller
		8.3.1 GNP
		8.3.2 Robust Stabilizer
	References
9 The Stankovich Integral Transform and Its Applications
	9.1 Introduction
	9.2 Transforms Definition
	9.3 Properties
		9.3.1 Adjointness Property
		9.3.2 Transforms of Power Functions
		9.3.3 Convolution Property
		9.3.4 Composition Rules
		9.3.5 Laplace Transform
		9.3.6 Mellin Transform
		9.3.7 Fractional Differentiation and Integration
		9.3.8 Limit Behavior
		9.3.9 Transforms of Some Special Functions
			9.3.9.1 Mittag-Leffler Function
			9.3.9.2 Exponential, Trigonometric, and Hyperbolic Functions
			9.3.9.3 Wright Functions
			9.3.9.4 Nu-Function
	9.4 Application
		9.4.1 Integral Representation of the Wright Function
		9.4.2 Evaluation of Improper Integrals
		9.4.3 Fractional Differential Equations
			9.4.3.1 Equations with Riemann–Liouville Derivatives
			9.4.3.2 Equations with Caputo and Weyl Derivatives
			9.4.3.3 Fundamental Solution for a Higher-Order Parabolic Equation
	Acknowledgments
	References
10 Electric Current as a Continuous Flow
	10.1 Introduction
	10.2 Maxwell Equations as Wave Equations
	10.3 A Titbit about Differential Forms
	10.4 Algebraic Introduction to Cohomology
	10.5 Vectorial Stokes Theorem
	Acknowledgments
	Dedication
	References
11 On New Integral Inequalities Involving Generalized Fractional Integral Operators
	11.1 Introduction
	11.2 Concepts of Fractional Integral Operators
	11.3 Integral Inequalities via Fractional Integral Operators
	11.4 New Results via Generalized Fractional Integral Operators
	References
12 A Note on Fox’s H Function in the Light of Braaksma’s Results
	12.1 Introduction
	12.2 Braaksma Revisited
	12.3 Expansion in the Neighborhood of the Singular Point
	References
13 Categories and Zeta & Möbius Functions: Applications to Universal Fractional Operators
	13.1 Riemann Zeta Function and Heuristic Approach of Riemann Hypothesis
		13.1.1 Introduction to Zeta Riemann Function Based on Universal Transfer Function: N-Measure in Prime Ξ(s)-Space
	13.2 Dual Structures in Category Theory
		13.2.1 An Outlook about Category Theory
		13.2.2 Lattices and Exponentiation: First Step to Applications in Physics
		13.2.3 Monads: Categorical Foundations of Coarse Graining
		13.2.4 Kan Extension and Functorial Division
		13.2.5 Adjunction, Order Structures: Emergence of a Pair of Scaling Parameters
	13.3 Physical Application: Fractional Differentiation, α-Exponential, and Arrow of Time
		13.3.1 From Non-Additive Choquet Integrals to Non-Integer Derivatives
			13.3.1.1 Set Derivative and Additive Systems
			13.3.1.2 Toward Non-Integer Derivatives
		13.3.2 Countable versus Real Representation: Hamel Basis, Cauchy Additive Functional, and Extended Autosimilarity
		13.3.3 Kan Extension and Completion of the TEISI and CRONE Models
			13.3.3.1 Kan Extension and Completion of the Universal Dynamic Models through the Construction of a Topos
	13.4 Conclusions and Outlook
	Acknowledgments
	References
14 New Contour Surfaces to the (2+1)-Dimensional Boussinesq Dynamical Equation
	14.1 Introduction
	14.2 General Properties of MEFM
	14.3 Implementation of the Method
	14.4 Conclusions
	Conflict of Interest
	References
15 Statistical Approach of Mixed Convective Flow of Third-Grade Fluid towards an Exponentially Stretching Surface with Convective Boundary Condition
	15.1 Introduction
	15.2 Mathematical Formulation
	15.3 Homotopy Analytic Solutions
		15.3.1 Zero[sup(th)]-Order Deformation Problem
		15.3.2 m[sup(th)]-Order Deformation Problems
	15.4 Convergence Analysis
	15.5 Results and Discussion
		15.5.1 Analysis of Velocity Profile
		15.5.2 Analysis of Temperature Profile
		15.5.3 Skin Friction and Nusselt Number
	15.6 Statistical Paradigm
	15.7 Probable Error
		15.7.1 Statistical Proclamation
	15.8 Concluding Remarks
	References
16 Solvability of the Boundary-Value Problem for a Third-Order Linear Loaded Differential Equation with the Caputo Fractional Derivative
	16.1 Introduction and Formulation of the Problem
	16.2 Representation of Solution of the Equation
	16.3 The Main Results
	16.4 Conclusion
	Acknowledgments
	References
17 Chaotic Systems and Synchronization Involving Fractional Conformable Operators of the Riemann–Liouville Type
	17.1 Introduction
	17.2 Mathematical Preliminaries
	17.3 Design of the Slave System
	17.4 Numerical Method for Fractional Conformable Derivative in the RL Sense
	17.5 Examples
		17.5.1 Moore–Spiegel system
		17.5.2 Arneodo’s System
		17.5.3 Van der Pol Oscillator (VPO)
		17.5.4 Chua’s Circuit Sine Function Approach
	17.6 Conclusions
	Acknowledgments
	Conflicts of Interest
	References
Index