Special Functions and Analysis of Differential Equations

✍ Scribed by Praveen Agarwal (editor), Michael Ruzhansky (editor), Ravi P Agarwal (editor)

• Publisher: Routledge
• Year: 2020
• Tongue: English
• Leaves: 371
• Edition: 1
• Category: Library

✦ Synopsis

Differential Equations are very important tools in Mathematical Analysis. They are widely found in mathematics itself and in its applications to statistics, computing, electrical circuit analysis, dynamical systems, economics, biology, and so on. Recently there has been an increasing interest in and widely-extended use of differential equations and systems of fractional order (that is, of arbitrary order) as better models of phenomena in various physics, engineering, automatization, biology and biomedicine, chemistry, earth science, economics, nature, and so on. Now, new unified presentation and extensive development of special functions associated with fractional calculus are necessary tools, being related to the theory of differentiation and integration of arbitrary order (i.e., fractional calculus) and to the fractional order (or multi-order) differential and integral equations.

This book provides learners with the opportunity to develop an understanding of advancements of special functions and the skills needed to apply advanced mathematical techniques to solve complex differential equations and Partial Differential Equations (PDEs). Subject matters should be strongly related to special functions involving mathematical analysis and its numerous applications. The main objective of this book is to highlight the importance of fundamental results and techniques of the theory of complex analysis for differential equations and PDEs and emphasizes articles devoted to the mathematical treatment of questions arising in physics, chemistry, biology, and engineering, particularly those that stress analytical aspects and novel problems and their solutions.

Specific topics include but are not limited to

• Partial differential equations

• Least squares on first-order system

• Sequence and series in functional analysis

• Special functions related to fractional (non-integer) order control systems and equations

• Various special functions related to generalized fractional calculus

• Operational method in fractional calculus

• Functional analysis and operator theory

• Mathematical physics

• Applications of numerical analysis and applied mathematics

• Computational mathematics

• Mathematical modeling

This book provides the recent developments in special functions and differential equations and publishes high-quality, peer-reviewed book chapters in the area of nonlinear analysis, ordinary differential equations, partial differential equations, and related applications.

✦ Table of Contents

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