Special Functions and Analysis of Differential Equations

Praveen Agarwal, Ravi P Agarwal, Michael Ruzhansky

• Publisher: Chapman and Hall/CRC
• Year: 2020
• Print ISBN: 9780367334727
• eBook ISBN: 9781000078589
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

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.

1
A Chebyshev Spatial Discretization Method for Solving Fractional Fokker–Planck Equation with Riesz Derivatives

Arman Dabiri

Southern Illinois University

Behrouz Parsa Moghadam

Islamic Azad University

J. A. Tenreiro Machado

Institute of Engineering of Porto

CONTENTS

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

1.1 Introduction

Einstein [1] proposed that the Brownian motion, representing continuous small random fluctuations in the position of particles, can be described by the linear evolution of the mean square displacement using a diffusion constant, i.e., 〈(Δx)²〉 ∝ t. The theory yields a partial differential equation (PDE) initially introduced by Fokker and Planck [2], and presently called the Fokker–Planck equation (FPE). This PDE is constructed by examining the time evolution of the velocity distri...

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
• 2 Special Functions and Their Link with Nonlinear Rod Theory
• 3 Second Kind Chebyshev Wavelets for Solving the Variable-Order Space-Time Fractional Telegraph Equation
• 4 Hyers–Ulam–Rassias Stabilities of Some Classes of Fractional Differential Equations
• 5 Applications of Fractional Derivatives to Heat Transfer in Channel Flow of Nanofluids
• 6 The Hyperbolic Maximum Principle Approach to the Construction of Generalized Convolutions
• 7 Elements of Aomoto’s Generalized Hypergeometric Functions and a Novel Perspective on Gauss’ Hypergeometric Differential Equation
• 8 Around Boundary Functions of the Right Half-Plane and the Unit Disc
• 9 The Stankovich Integral Transform and Its Applications
• 10 Electric Current as a Continuous Flow
• 11 On New Integral Inequalities Involving Generalized Fractional Integral Operators
• 12 A Note on Fox’s H Function in the Light of Braaksma’s Results
• 13 Categories and Zeta & M¨obius Functions: Applications to Universal Fractional Operators
• 14 New Contour Surfaces to the (2+1)-Dimensional Boussinesq Dynamical Equation
• 15 Statistical Approach of Mixed Convective Flow of Third-Grade Fluid towards an Exponentially Stretching Surface with Convective Boundary Condition
• 16 Solvability of the Boundary-Value Problem for a Third-Order Linear Loaded Differential Equation with the Caputo Fractional Derivative
• 17 Chaotic Systems and Synchronization Involving Fractional Conformable Operators of the Riemann Liouville Type
• Index