Mittag-Leffler Functions, Related Topics and Applications (Springer Monographs in Mathematics)

The 2nd edition of this book is essentially an extended version of the 1st and provides a very sound overview of the most important special functions of Fractional Calculus. It has been updated with material from many recent papers and includes several surveys of important results known before the publication of the 1st edition, but not covered there. As a result of researchers' and scientists' increasing interest in pure as well as applied mathematics in non-conventional models, particularly those using fractional calculus, Mittag-Leffler functions have caught the interest of the scientific community. Focusing on the theory of Mittag-Leffler functions, this volume offers a self-contained, comprehensive treatment, ranging from rather elementary matters to the latest research results. In addition to the theory the authors devote some sections of the work to applications, treating various situations and processes in viscoelasticity, physics, hydrodynamics, diffusion and wave phenomena, as well as stochastics. In particular, the Mittag-Leffler functions make it possible to describe phenomena in processes that progress or decay too slowly to be represented by classical functions like the exponential function and related special functions. The book is intended for a broad audience, comprising graduate students, university instructors and scientists in the field of pure and applied mathematics, as well as researchers in applied sciences like mathematical physics, theoretical chemistry, bio-mathematics, control theory and several other related areas.-- Provided by publisher Preface to the Second Edition Preface to the First Edition Contents 1 Introduction 2 Historical Overview of the Mittag-Leffler Functions 2.1 A Few Biographical Notes On Gösta Magnus Mittag-Leffler 2.2 The Contents of the Five Papers by Mittag-Leffler on New Functions 2.3 Further History of Mittag-Leffler Functions 3 The Classical Mittag-Leffler Function 3.1 Definition and Basic Properties 3.2 Relations to Elementary and Special Functions 3.3 Recurrence and Differential Relations 3.4 Integral Representations and Asymptotics 3.5 Distribution of Zeros 3.6 Further Analytic Properties 3.6.1 Additional Integral Properties 3.6.2 Mittag-Leffler Summation of Power Series 3.6.3 Mittag-Leffler Reproducing Kernel Hilbert Spaces 3.7 The Mittag-Leffler Function of a Real Variable 3.7.1 Integral Transforms 3.7.2 The Complete Monotonicity Property 3.7.3 Relation to Fractional Calculus 3.8 Historical and Bibliographical Notes 3.9 Exercises 4 The Two-Parametric Mittag-Leffler Function 4.1 Series Representation and Properties of Coefficients 4.2 Explicit Formulas. Relations to Elementary and Special Functions 4.3 Differential and Recurrence Relations 4.4 Integral Relations and Asymptotics 4.5 The Two-Parametric Mittag-Leffler Function as an Entire Function 4.6 Distribution of Zeros 4.6.1 Distributions of Zeros and Inverse Problems for Differential Equations in Banach Spaces 4.7 Computations With the Two-Parametric Mittag-Leffler Function 4.8 Further Analytic Properties 4.8.1 Additional Integral and Differential Formulas 4.8.2 Geometric Properties of the Mittag-Leffler Function 4.8.3 An Extension for Negative Values of the First Parameter 4.9 The Two-Parametric Mittag-Leffler Function of a Real Variable 4.9.1 Integral Transforms of the Two-Parametric Mittag-Leffler Function 4.9.2 The Complete Monotonicity Property 4.9.3 Relations to the Fractional Calculus 4.10 Historical and Bibliographical Notes 4.11 Exercises 5 Mittag-Leffler Functions with Three Parameters 5.1 The Prabhakar (Three-Parametric Mittag-Leffler) Function 5.1.1 Definition and Basic Properties 5.1.2 Integral Representations and Asymptotics 5.1.3 Expansion on the Negative Semi-axes 5.1.4 Integral Transforms of the Prabhakar Function 5.1.5 Complete Monotonicity of the Prabhakar Function 5.1.6 Fractional Integrals and Derivatives of the Prabhakar Function 5.1.7 Relations to the Fox–Wright Function, H-function and Other Special Functions 5.2 The Kilbas–Saigo (Three-Parametric Mittag-Leffler) Function 5.2.1 Definition and Basic Properties 5.2.2 The Order and Type of the Entire Function Eα,m,l(z) 5.2.3 Recurrence Relations for Eα,m,l(z) 5.2.4 Connection of En,m,l(z) with Functions of Hypergeometric Type 5.2.5 Differentiation Properties of En,m,l(z) 5.2.6 Complete Monotonicity of the Kilbas–Saigo Function 5.2.7 Fractional Integration of the Kilbas–Saigo Function 5.2.8 Fractional Differentiation of the Kilbas–Saigo Function 5.3 The Le Roy Type Function 5.3.1 Definition and Main Analytic Properties 5.3.2 Integral Representations of the Le Roy Type Function 5.3.3 Laplace Transforms of the Le Roy Type Function 5.3.4 The Asymptotic Expansion on the Negative Semi-axis 5.3.5 Extension to Negative Values of the Parameter α 5.4 Historical and Bibliographical Notes 5.5 Exercises 6 Multi-index and Multi-variable Mittag-Leffler Functions 6.1 The Four-Parametric Mittag-Leffler Function: The Luchko ... 6.1.1 Definition and Special Cases 6.1.2 Basic Properties 6.1.3 Integral Representations and Asymptotics 6.1.4 Extended Four-Parametric Mittag-Leffler Functions 6.1.5 Relations to the Wright Function and the H-Function 6.1.6 Integral Transforms of the Four-Parametric Mittag-Leffler Function 6.1.7 Integral Transforms with the Four-Parametric Mittag-Leffler Function in the Kernel 6.1.8 Relations to the Fractional Calculus 6.2 The Four-Parametric Mittag-Leffler Function: A Generalization ... 6.2.1 Definition and General Properties 6.2.2 The Four-Parametric Mittag-Leffler Function of a Real Variable 6.3 Mittag-Leffler Functions with 2n Parameters 6.3.1 Definition and Basic Properties 6.3.2 Representations in Terms of Hypergeometric Functions 6.3.3 Integral Representations and Asymptotics 6.3.4 Extension of the 2n-Parametric Mittag-Leffler Function 6.3.5 Relations to the Wright Function and to the H-Function 6.3.6 Integral Transforms with the Multi-parametric Mittag-Leffler Functions 6.3.7 Relations to the Fractional Calculus 6.4 Mittag-Leffler Functions of Several Variables 6.4.1 Integral Representations 6.4.2 Asymptotic Behavior for Large Values of Arguments 6.5 Mittag-Leffler Functions with Matrix Arguments 6.6 Historical and Bibliographical Notes 6.7 Exercises 7 The Classical Wright Function 7.1 Definition and Basic Properties 7.2 Relations to Elementary and Special Functions 7.3 Integral Representations and Asymptotics 7.4 Distribution of Zeros 7.5 Further Analytic Properties 7.5.1 Additional Properties of the Wright Function in the Complex Plane 7.5.2 Geometric Properties of the Wright Function 7.5.3 Auxiliary Functions of the Wright Type 7.6 The Wright Function of a Real Variable 7.6.1 Relation to Fractional Calculus 7.6.2 Laplace Transforms of the Mittag-Leffler and the Wright Functions 7.6.3 Mainardi's Approach to the Wright Functions of the Second Kind 7.7 Historical and Bibliographical Notes 7.8 Exercises 8 Applications to Fractional Order Equations 8.1 Fractional Order Integral Equations 8.1.1 The Abel Integral Equation 8.1.2 Other Integral Equations Whose Solutions Are Represented Via Generalized Mittag-Leffler Functions 8.2 Fractional Ordinary Differential Equations 8.2.1 Fractional Ordinary Differential Equations with Constant Coefficients 8.2.2 Ordinary FDEs with Variable Coefficients 8.2.3 Other Types of Ordinary Fractional Differential Equations 8.3 Optimal Control for Equations with Fractional Derivatives and Integrals 8.3.1 Linear Fractional-Order Controllers 8.3.2 Nonlinear Fractional-Order Controllers 8.3.3 Modification of the Control Actions in Fractional-Order PID Controllers 8.3.4 Further Possible Modifications of the Fractional-Order PID Controllers 8.4 Differential Equations with Fractional Partial Derivatives 8.4.1 Cauchy-Type Problems for Differential Equations with Riemann–Liouville Fractional Partial Derivatives 8.4.2 The Cauchy Problem for Differential Equations with Caputo Fractional Partial Derivatives 8.5 Numerical Methods for the Solution of Fractional Differential Equations 8.5.1 Direct Numerical Methods 8.5.2 Indirect Numerical Methods 8.5.3 Other Numerical Methods 8.6 Historical and Bibliographical Notes 8.7 Exercises 9 Applications to Deterministic Models 9.1 Fractional Relaxation and Oscillations 9.1.1 Simple Fractional Relaxation and Oscillation 9.1.2 The Composite Fractional Relaxation and Oscillations 9.2 Examples of Applications of the Fractional Calculus in Physical Models 9.2.1 Linear Visco-Elasticity 9.2.2 The Use of Fractional Calculus in Linear Viscoelasticity 9.2.3 The General Fractional Operator Equation 9.3 The Fractional Dielectric Models 9.3.1 The Main Models for Anomalous Dielectric Relaxation 9.3.2 The Cole–Cole Model 9.3.3 The Davidson–Cole Model 9.3.4 The Havriliak–Negami Model 9.4 The Fractional Calculus in the Basset Problem 9.4.1 The Equation of Motion for the Basset Problem 9.4.2 The Generalized Basset Problem 9.5 Other Deterministic Fractional Models 9.6 Historical and Bibliographical Notes 9.7 Exercises 10 Applications to Stochastic Models 10.1 Introduction 10.2 The Mittag-Leffler Process According to Pillai 10.3 Elements of Renewal Theory and Continuous Time Random Walks (CTRWs) 10.3.1 Renewal Processes 10.3.2 Continuous Time Random Walks (CTRWs) 10.3.3 The Renewal Process as a Special CTRW 10.4 The Poisson Process and Its Fractional Generalization (The Renewal Process of Mittag-Leffler Type) 10.4.1 The Mittag-Leffler Waiting Time Density 10.4.2 The Poisson Process 10.4.3 The Renewal Process of Mittag-Leffler Type 10.4.4 Thinning of a Renewal Process 10.5 Fractional Diffusion and Subordination Processes 10.5.1 Renewal Process with Reward 10.5.2 Limit of the Mittag-Leffler Renewal Process 10.5.3 Subordination in the Space-Time Fractional Diffusion Equation 10.5.4 The Rescaling and Respeeding Concept Revisited. Universality of the Mittag-Leffler Density 10.6 The Wright M-Functions in Probability 10.6.1 The Absolute Moments of Order δ 10.6.2 The Characteristic Function 10.6.3 Relations with Lévy Stable Distributions 10.6.4 The Wright mathbbM-Function in Two Variables 10.7 Historical and Bibliographical Notes 10.8 Exercises Appendix A The Eulerian Functions A.1 The Gamma Function A.1.1 Analytic Continuation A.1.2 The Graph of the Gamma Function on the Real Axis A.1.3 The Reflection or Complementary Formula A.1.4 The Multiplication Formulas A.1.5 Pochhammer's Symbols A.1.6 Hankel Integral Representations A.1.7 Notable Integrals via the Gamma Function A.1.8 Asymptotic Formulas A.1.9 Infinite Products A.2 The Beta Function A.2.1 Euler's Integral Representation A.2.2 Symmetry A.2.3 Trigonometric Integral Representation A.2.4 Relation to the Gamma Function A.2.5 Other Integral Representations A.2.6 Notable Integrals via the Beta Function A.3 Historical and Bibliographical Notes A.4 Exercises Appendix B The Basics of Entire Functions B.1 Definition and Series Representations B.2 Growth of Entire Functions. Order, Type and Indicator Function B.3 Weierstrass Canonical Representation. Distribution of Zeros B.4 Entire Functions of Completely Regular Growth B.5 Historical and Bibliographical Notes B.6 Exercises Appendix C Integral Transforms C.1 Fourier Type Transforms C.2 The Laplace Transform C.3 The Mellin Transform C.4 Simple Examples and Tables of Transforms of Basic Elementary and Special Functions C.5 Historical and Bibliographical Notes C.6 Exercises Appendix D The Mellin–Barnes Integral D.1 Definition. Contour of Integration D.2 Asymptotic Methods for the Mellin–Barnes Integral D.3 Historical and Bibliographical Notes D.4 Exercises Appendix E Elements of Fractional Calculus E.1 The Riemann–Liouville Fractional Calculus E.2 The Caputo Fractional Calculus E.3 The Marchaud Fractional Calculus E.4 The Erdélyi–Kober Fractional Calculus E.5 The Hadamard Fractional Calculus E.6 The Grünwald–Letnikov Fractional Calculus E.7 The Riesz Fractional Calculus E.8 Historical and Bibliographical Notes Appendix F Higher Transcendental Functions F.1 Hypergeometric Functions F.1.1 Classical Gauss Hypergeometric Functions F.1.2 Euler Integral Representation. Mellin–Barnes Integral Representation F.1.3 Basic Properties of Hypergeometric Functions F.1.4 The Hypergeometric Differential Equation F.1.5 Kummer's and Tricomi's Confluent Hypergeometric Functions F.1.6 Generalized Hypergeometric Functions and their Properties F.2 Wright Functions F.2.1 The Classical Wright Function F.2.2 The Bessel–Wright Function. Generalized Wright Functions and Fox–Wright Functions F.3 Meijer G-Functions F.3.1 Definition via Integrals. Existence F.3.2 Basic Properties of the Meijer G-Functions F.3.3 Special Cases F.3.4 Relations to Fractional Calculus F.3.5 Integral Transforms of G-Functions F.4 Fox H-Functions F.4.1 Definition via Integrals. Existence F.4.2 Series Representations and Asymptotics. Recurrence Relations F.4.3 Special Cases F.4.4 Relations to Fractional Calculus F.4.5 Integral Transforms of H-Functions F.5 Historical and Bibliographical Notes F.6 Exercises Appendix References Index As a result of researchers’ and scientists’ increasing interest in pure as well as applied mathematics in non-conventional models, particularly those using fractional calculus, Mittag-Leffler functions have recently caught the interest of the scientific community. Focusing on the theory of the Mittag-Leffler functions, the present volume offers a self-contained, comprehensive treatment, ranging from rather elementary matters to the latest research results. In addition to the theory the authors devote some sections of the work to the applications, treating various situations and processes in viscoelasticity, physics, hydrodynamics, diffusion and wave phenomena, as well as stochastics. In particular the Mittag-Leffler functions allow us to describe phenomena in processes that progress or decay too slowly to be represented by classical functions like the exponential function and its successors. The book is intended for a broad audience, comprising graduate students, university instructors and scientists in the field of pure and applied mathematics, as well as researchers in applied sciences like mathematical physics, theoretical chemistry, bio-mathematics, theory of control, and several other related areas.