# FRACTIONAL INTEGRALS AND DERIVATIVES. Theory and Applications

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1 FRACTIONAL INTEGRALS AND DERIVATIVES Theory and Applications Stefan G. Samko Rostov State University, Russia Anatoly A. Kilbas Belorussian State University, Minsk, Belarus Oleg I. Marichev Belorussian State University, Minsk, Belarus Gordon and Breach Science Publishers Switzerland Australia Belgium France Germany Great Britain India Japan Malaysia Netherlands Russia Singapore USA

2 Foreword Preface to the English edition Preface Introduction Notation of the main forms of fractional integrals and derivatives Brief historical outline xv xvii xix xxiii xxv xxvii Chapter 1 Fractional Integrals and Derivatives on an Interval 1 1. Preliminaries The spaces H x and H x (p), The spaces L p and L p (p) Some special functions Integral transforms Riemann-Liouville Fractional Integrals and Derivatives The Abel integral equation On the solvability of the Abel equation in the space of integrable functions Definition of fractional integrals and derivatives and their simplest properties Fractional integrals and derivatives of complex order Fractional integrals of some elementary functions Fractional integration and differentiation as reciprocal operations Composition formulae. Connection with semigroups of operators 46

3 vi CONTENTS 3. The Fractional Integrals of Holder and Summable Functions Mapping properties in the space H x Mapping properties in the space HQ(P) Mapping properties in the space L p Mapping properties in the space L p (p) Bibliographical Remarks and Additional Information to Chapter Historical notes Survey of other results (relating to 1-3) 84 Chapter 2 Fractional Integrals and Derivatives on the Real Axis and Half-Axis The Main Properties of Fractional Integrals and Derivatives Definitions and elementary properties Fractional integrals of Holderian functions Fractional integrals of summable functions The Marchaud fractional derivative The finite part of integrals due to Hadamard Properties of finite differences and Marchaud fractional derivatives of order a > Connection with fractional power of operators Representation of Function by Fractional Integrals of Lp-Functions The space I a (L p ) Inversion of fractional integrals of L p -functions Characterization of the space I a (L p ) Sufficiency conditions for the representability of functions by fractional integrals On the integral modulus of continuity of J a (L p )-functions Integral Transforms of Fractional Integrals and Derivatives The Fourier transform The Laplace transform The Mellin transform Fractional Integrals and Derivatives of Generalized Functions Preliminary ideas The case of the axis R 1. Lizorkin's space of test functions

4 vii 8.3. Schwartz's approach The case of the half-axis. The approach via the adjoint operator McBride's spaces The case of an interval Bibliographical Remarks and Additional Information to Chapter Historical notes Survey of other results (relating to 5-8) Tables of fractional integrals and derivatives 172 Chapter 3 Further Properties of Fractional Integrals and Derivatives Compositions of Fractional Integrals and Derivatives with Weights Compositions of two one-sided integrals with power weights Compositions of two-sided integrals with power weights Compositions of several integrals with power weights Compositions with exponential and power-exponential weights Connection between Fractional Integrals and the Singular Operator The singular operator S The case of the whole line The case of an interval and a half-axis Some other composition relations Fractional Integrals of the Potential Type The real axis. The Riesz and Feller potentials On the "truncation" of the Riesz potential to the half-axis The case of the half-axis The case of a finite interval Functions Representable by Fractional Integrals on an Interval The Marchaud fractional derivative on an interval Characterization of fractional integrals of functions in L p Continuation, restriction and "sewing" of fractional integrals Characterization of fractional integrals of Holderian functions 238

5 viii CONTENTS Fractional integration in the union of weighted Holder spaces Fractional integrals and derivatives of functions with a prescribed continuity modulus Miscellaneous Results for Fractional Integro-differentiation of Functions of a Real Variable Lipschitz spaces H p and H x Mapping properties of fractional integration in H p Fractional integrals and derivatives of functions which are given on the whole line and belong to H x on every finite interval Fractional derivatives of absolutely continuous functions The Riesz mean value theorem and inequalities for fractional integrals and derivatives Fractional integration and the summation of series and integrals The Generalized Leibniz Rule Fractional integro-differentiation of analytic functions on the real axis The generalized Leibniz rule Asymptotic Expansions of Fractional Integrals Definitions and properties of asymptotic expansions The case of a power asymptotic expansion The case of a power-logarithmic asymptotic expansion The case of a power-exponential asymptotic expansion The asymptotic solution of Abel's equation Bibliographical Remarks and Additional Information to Chapter Historical notes Survey of other results (relating to 10-16) 305 Chapter 4 - Other Forms of Fractional Integrals and Derivatives Direct Modifications and Generalizations of Riemann- Liouville Fractional Integrals Erdelyi-Kober-type operators Fractional integrals of a function by another function Hadamard fractional integro-differentiation 329

6 ix One-dimensional modification of Bessel fractional integrodifferentiation and the spaces H*' p = L' p The Chen fractional integral Dzherbashyan's generalized fractional integral Weyl Fractional Integrals and Derivatives of Periodic Functions Definitions. Connections with Fourier series Elementary properties of Weyl fractional integrals Other forms of fractional integration of periodic functions The coincidence of Weyl and Marchaud fractional derivatives The representability of periodic functions by the Weyl fractional integral Weyl fractional integration and differentiation in the space of Holderian functions Weyl fractional integrals and derivatives of periodic functions in H x The Bernstein inequality for fractional integrals of trigonometric polynomials An Approach to Fractional Integro-differentiation via Fractional Differences (The Grunwald-Letnikov Approach) Differences of a fractional order and their properties Coincidence of the Grunwald-Letnikov derivative with the Marchaud derivative. The periodic case Coincidence of the Grunwald-Letnikov derivative with the Marchaud derivative. The non-periodic case Grunwald-Letnikov fractional differentiation on a finite interval Operators with Power-Logarithmic Kernels Mapping properties in the space H x Mapping properties in the space HQ(P) Mapping properties in the space L p Mapping properties in the space L p (p) Asymptotic expansions Fractional Integrals and Derivatives in the Complex Plane Definitions and the main properties of fractional integrodifferentiation in the complex plane Fractional integro-differentiation of analytic functions Generalization of fractional integro-differentiation of analytic functions 426

7 23. Bibliographical Remarks and Additional Information to Chapter Historical notes Survey of other results (relating to 18-22) Answers to some questions put at the Conference on Fractional Calculus (New Haven, 1974) 455 Chapter 5 Fractional Integro-differentiation of Functions of Many Variables Partial and Mixed Integrals and Derivatives of Fractional Order The multidimensional Abel integral equation Partial and mixed fractional integrals and derivatives The case of two variables. Tensor product of operators Mapping properties of fractional integration operators in the spaces L p (R n ) (with mixed norm) Connection with a singular integral Partial and mixed fractional derivatives in the Marchaud form Characterization of fractional integrals of functions in Lp(R 2 ) Integral transform of fractional integrals and derivatives Lizorkin function space invariant relative to fractional integrodifferentiation Fractional derivatives and integrals of periodic functions of many variables Grunwald-Letnikov fractional differentiation Operators of the polypotential type Riesz Fractional Integro-differentiation Preliminaries The Riesz potential and its Fourier transform. Invariant Lizorkin space Mapping properties of the operator I a in the spaces L p (R n ) and L p (R n ;p) Riesz differentiation (hypersingular integrals) Unilateral Riesz potentials Hypersingular Integrals and the Space of Riesz Potentials Investigation of the normalizing constants d ni i(a) as functions of the parameter a 505

8 xi Convergence of the hypersingular integral for smooth functions and diminution of order / to / > 2[a/2] in the case of a non-centered difference The hypersingular integral as an inverse of a Riesz potential Hypersingular integrals with homogeneous characteristics Hypersingular integral with a homogeneous characteristic as a convolution with the distribution Representation of differential operators in partial derivatives by hypersingular integrals The space I a (L p ) of Riesz potentials and its characterization in terms of hypersingular integrals. The space L pr (R n ) Bessel Fractional Integro-differentiation The Bessel kernel and its properties Connections with Poisson, Gauss-Weierstrass and metaharmonic continuation semigroups The space of Bessel potentials The realization of (E A)"/ 2, a > 0, in terms of hypersingular integrals Other Forms of Multidimensional Fractional Integrodifferentiation Riesz potential with Lorentz distance (hyperbolic Riesz potentials) Parabolic potentials The realization of the fractional powers ( A x + 7)" and (E A x + - i) a i a > 0, in terms of a hypersingular integral Pyramidal analogues of mixed fractional integrals and derivatives Bibliographical Remarks and Additional Information to Chapter Historical notes Survey of other results (relating to 24-28) 584 Chapter 6 Applications to Integral Equations of the First Kind with Power and Power-Logarithmic Kernels The Generalized Abel Integral Equation The dominant singular integral equation The generalized Abel equation on the whole axis The generalized Abel equation on an interval The case of constant coefficients 622

9 xii CONTENTS 31. The Noether Nature of the Equation of the First Kind with Power-Type Kernels Preliminaries on Noether operators The equation on the axis Equations on a finite interval On the stability of solutions Equations with Power-Logarithmic Kernels Special Volterra functions and some of their properties The solution of equations with integer non-negative powers of logarithms The solution of equations with real powers of logarithms The Noether Nature of Equations of the First Kind with Power-Logarithmic Kernels Imbedding theorems for the ranges of the operators I"f and /*/ Connection between the operators with power-logarithmic kernels and singular operator The Noether nature of equation (33.1) Bibliographical Remarks and Additional Information to Chapter Historical notes Survey of other results (relating to 30-33) 687 Chapter 7 - Integral Equations of the First Kind with Special Functions as Kernels Some Equations with Homogeneous Kernels Involving Gauss and Legendre Functions Equations with the Gauss function Equations with the Legendre function Fractional Integrals and Derivatives as Integral Transforms Definition of the G-transform. The spaces SOT"}(L) and L ( 2 c ' y) and their characterization Existence, mapping properties and representations of the G-transform: Factorization of the G-transform Inversion of the G-transform The mapping properties, factorization and inversion of fractional integrals in the spaces 3Jt~^(L) and L\ e 720

10 xiii Other examples of factorization Mapping properties of the G-transform on fractional integrals and derivatives Index laws for fractional integrals and derivatives Equations with Non-Homogeneous Kernels Equations with difference kernels Generalized operators of Hankel and Erdelyi-Kober transforms Non-convolution operators with Bessel functions in kernels Equation of compositional type The W-transform and its inversion Application of fractional integrals to the inversion of the W-transform Applications of Fractional Integro-differentiation to the Investigation of Dual Integral Equations Dual Equations Triple equations Bibliographical Remarks and Additional Information to Chapter Historical notes Survey of other results (relating to 35-38) 775 Chapter 8 Applications to Differential Equations Integral Representations for Solution of Partial Differential Equations of the Second Order via Analytic Functions and Their Applications to Boundary Value Problems Preliminaries The representation of solutions of generalized Helmholtz two-axially symmetric equation Boundary value problems for the generalized Helmholtz twoaxially symmetric equation Euler-Poisson-Darboux Equation Representations for solutions of the Euler-Poisson-Darboux equation Classical and generalized solutions of the Cauchy problem The half-homogeneous Cauchy problem in multidimensional half-space The weighted Dirichlet and Neumann problems in a half-plane 826

11 xiv CONTENTS 42. Ordinary Differential Equations of Fractional Order The Cauchy-type problem for differential equations and systems of differential equations of fractional order of general form The Cauchy^type problem for linear differential equation of fractional order The Dirichlet-type problem for linear differential equation of fractional order Solution of the linear differential equation of fractional order with constant coefficients in the space of generalized functions The application of fractional differentiation to differential equations of integer order Bibliographical Remarks and Additional Information to Chapter Historical notes Survey of other results (relating to 40-42) 858 Bibliography 873 Author Index 953 Subject Index 965 Index of Symbols 973

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