An introduction to fractional calculus
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...
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| Main Authors | , |
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| Format | eBook Book |
| Language | English |
| Published |
New York
Nova Science
2017
Nova Science Publishers, Incorporated Nova Science Publishers, Inc |
| Edition | 1 |
| Series | Mathematics research developments |
| Subjects | |
| Online Access | Get full text |
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| Abstract | 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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| AbstractList | 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. |
| Author | Haubold, H. J. Mathai, A. M. |
| Author_xml | – sequence: 1 fullname: Mathai, A. M. – sequence: 2 fullname: Haubold, H. J. |
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| Notes | Includes bibliographical references and indexes |
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| PublicationPlace | New York |
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| Publisher | Nova Science Nova Science Publishers, Incorporated Nova Science Publishers, Inc |
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| Snippet | 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... |
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| SubjectTerms | Fractional calculus |
| TableOfContents | 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 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 -- Blank Page 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 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 |
| Title | An introduction to fractional calculus |
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