Introduction to Fractional Calculus

Name: Introduction to Fractional Calculus
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M. Mathai A.(Editor)

Nova Science Publishers Inc

Published on 7. January 2019

274 pages

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This is a modified version of Module 10 of the Centre for Mathematical and Statistical Sciences (CMSS). CMSS modules are notes prepared on various topics with many examples from real-life situations and exercises so that the subject matter becomes interesting to students. These modules are used for undergraduate level courses and graduate level training in various topics at CMSS. Aside from Module 8, these modules were developed by Dr. A. M. Mathai, Director of CMSS and Emeritus Professor of Mathematics and Statistics, McGill University, Canada. Module 8 is based on the lecture notes of Professor W. J. Anderson of McGill University, developed for his undergraduate course (Mathematics 447). Professor Dr. Hans J. Haubold has been a research collaborator of Dr. A.M. Mathai's since 1984, mainly in the areas of astrophysics, special functions and statistical distribution theory. He is also a lifetime member of CMSS and a Professor at CMSS. A large number of papers have been published jointly in these areas since 1984. The following monographs and books have been brought out in conjunction with this joint research: Modern Problems in Nuclear and Neutrino Astrophysics (A.M. Mathai and H.J. Haubold, 1988, Akademie-Verlag, Berlin); Special Functions for Applied Scientists (A.M.Mathai and H.J. Haubold, 2008, Springer, New York); and The H-Function: Theory and Applications (A.M.Mathai, R.K. Saxena and H.J. Haubold, 2010, Springer, New York). These CMSS modules are printed at CMSS Press and published by CMSS. Copies are made available to students free of charge, and to researchers and others at production cost. For the preparation of the initial drafts of all these modules, financial assistance was made available from the Department of Science and Technology, the Government of India (DST), New Delhi under project number SR/S4/MS:287/05. Hence, the authors would like to express their thanks and gratitude to DST, the Government of India, for its financial assistance.

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Intro
AN INTRODUCTION TOFRACTIONAL CALCULUS
AN INTRODUCTION TOFRACTIONAL CALCULUS
CONTENTS
Preface
Introduction
List of Symbols
Chapter 1Mathematical Preliminaries
1.1. Introduction
1.2. Fractional Integrals
1.3. A Note on Hypergeometric Series
1.4. Some Physical Interpretations
1.4.1. Fractional Integral as Fraction of a Total Probability
1.4.2. Left-Sided Riemann-Liouville Fractional Integral as the Density of
1.4.3. The Total Probability Coming from a Gamma Density
1.5. Semigroup Properties of Fractional Integral Operators
Exercises
1.6. Solution of a Simple Differential Equation
Exercises
Bibliography
Chapter 2The Mittag-Leffler Function
2.1. Introduction
2.2. The Mittag-Leffler Function and Its Properties
2.3. The Mellin-Barnes Representations of the Mittag-Leffler Function
2.3.1. Analytic Continuation
2.3.2. Some Properties of the Mittag-Leffler Function
Exercises
Exercises
2.5. The Mittag-Leffler Statistical DensitySeveral
2.5.1. The Mellin-Barnes Representation of the Mittag-Leffler Density
a density.2.5.2. Properties of the Mittag-Leffl
2.5.3. Series Representation of a L´evy Density
2.5.4 From the Mittag-Leffler Function to PathwayModel,Tsallis' Statistics and SuperstatisticsConsider the
Bibliography
the Mittag-Leffler FunctionR.P. Agarwal (1953):
Chapter 3Fractional Integrals andFractional Derivatives
3.1. Introduction
3.2. The Riemann-Liouville Left-Sided
3.3. The Laplace Transforms of Fractional Derivatives
Exercises
3.4. The Right-Sided Fractional
3.4.1. The Laplace Transforms of Right-Sided Fractional
3.4.2. The Mellin Transforms of Fractional Integrals and FractionalDerivatives
Exercises
3.5. The Kober Operators
Exercises
3.6. The Saigo Fractional Operators
Exercises
3.7. Some Convenient Notations
Bibliography
Additional Reading Materials
Chapter 4The Kober Fractional IntegralOperators and StatisticalDistributions
4.1. Introduction
4.2. Distributions of Products and Ratios
4.2.1. A Pathway Kober Operator of the Second Kind
4.2.2. Special Cases
4.2.3. Another Generalization of the Kober Operators of the Second Kind
4.2.4. A Generalization in Terms of Hypergeometric Series
4.2.5. The Mellin Transform of the Generalized Kober Operatorof the Second Kind
Exercises
4.3. The Kober Operator of the First Kind
4.3.1. A Pathway Generalization of the Kober Operator of the First Kind
4.3.2. Some Special Cases
4.3.3. Generalization of the Kober Operator of the First Kind in Terms ofHypergeometric Series
4.3.4. The Mellin Transform of the Generalized Kober Operatorof the First Kind
Exercises
4.4. The Riemann-Liouville Operators asMellin Convolutions
Exercises
4.5. The Kober Operators in the RealMatrix-Variate Case
Exercises
4.6. The Kober Fractional Integral Operator of the Second Kind for theReal Matrix-Variate Case
4.6.1. A Pathway Generalization of the Kober Operator of the SecondKind in the Matrix-Variate Case
Exercises
4.7. The M-transform of the Kober Operator of the Second Kind
Exercises
4.8. Generalization in Terms of Hypergeometric Series for the KoberOperator of the Second Kind in the RealMatrix-Variate Case
Exercises
4.9. The Kober Fractional Integral Operator of the First Kindin the Matrix-Variate Case
4.9.1. Pathway Extension of the Kober Operator of the First Kind in theMatrix-Variate Case
Exercises
Bibliography
Chapter 5The Kober Fractional IntegralOperators with Many Variablesand Statistical Distributions
5.1. Introduction
5.2. The Kober Fractional Integral Operator of the Second Kind
5.2.1. A Pathway Generalization of the Kober Operator of the SecondKind in the Multivariate Case
Exercises
5.3. The Mellin Transform in the Multivariate Case for the KoberOperator of the Second Kind
Exercises
5.4. The Kober Fractional Integral Operator of the First Kind for theMultivariate Case
Exercises
5.5. Fractional Integrals InvolvingMany Matrix Variables
Exercises
5.6. Fractional Integral Operators of the First Kind in the Case of ManyMatrix Variables
Exercises
5.7. M-transforms for the Fractional Integral Operators in theMany Matrix-Variate Case
Bibliography
Chapter 6Fractional DifferentialEquations
6.1. Introduction
6.2. Fractional Relaxation
6.2.1. Generalization of Fractional Relaxation
6.2.2. The Cauchy Problem
Exercises
6.3. Fractional Diffusion
Exercises
6.4. Fractional Differencing
6.4.1. Methods for Solving Fractional Differential Equations
6.4.2. Adomian DecompositionMethod
6.4.3. Daftardar-Gejji Method
Additional Problems
Bibliography
Additional Reading Materials
Chapter 7Fractional Calculus in theComplex Case
7.1. Introduction
7.2. Some Jacobians in the Complex Domain
Exercises
7.3. Fractional Integrals in the ComplexMatrix-Variate Case
7.3.1. The Kober Fractional Integral Operator of the Second Kindof Order and Parameter
7.3.2. The Second Kind Riemann-Liouville and Weyl Fractional IntegralOperators in the Complex Matrix-Variate Case
7.3.3. The Saigo and Related Fractional Integral Operatorsof the Second Kind
7.3.4. M-transforms for Fractional Integrals of the Second Kind
Exercises
7.4. The Left-Sided or First Kind Fractional Integral Operatorof Order in the ComplexMatrix-Variate Case
7.4.1. The Kober Fractional Integral Operator of Order andParameter of the First Kind for the Complex Matrix-Variate Case
7.4.2. The Riemann-Liouville and theWeyl Fractional Integral Operatorsof the First Kind of Order for the Complex Matrix-Variate Case
Exercises
Bibliography
Chapter 8Fractional Derivatives in theMatrix-Variate Case
8.1. Introduction
8.2. Some Differential Operators in the Real Matrix-Variate Case
8.3. Some Basic Results for Real Matrix-Variate Case
8.3.1. Some Operators for Hypergeometric Series with Matrix Argument
8.3.2. Special Cases
Exercises
8.4. Fractional Derivatives Involving Zonal Polynomials
Exercises
Bibliography
Appendix
A1. The H-function
A2. The G-function or Meijer's G-function
A3. Wright's Function
About the Authors
Author Index
Subject Index
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Introduction to Fractional Calculus

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