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| Preliminaries | |
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| Spaces of Integrable, Absolutely Continuous, and Continuous Functions | |
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| Generalized Functions | |
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| Fourier Transforms | |
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| Laplace and Mellin Transforms | |
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| The Gamma Function and Related Special Functions | |
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| Hypergeometric Functions | |
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| Bessel Functions | |
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| Classical Mittag-Leffler Functions | |
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| Generalized Mittag-Leffler Functions | |
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| Functions of the Mittag-Leffler Type | |
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| Wright Functions | |
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| The H-Function | |
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| Fixed Point Theorems | |
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| Fractional Integrals and Fractional Derivatives | |
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| Riemann-Liouville Fractional Integrals and Fractional Derivatives | |
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| Liouville Fractional Integrals and Fractional Derivatives on the Half-Axis | |
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| Liouville Fractional Integrals and Fractional Derivatives on the Real Axis | |
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| Caputo Fractional Derivatives | |
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| Fractional Integrals and Fractional Derivatives of a Function with Respect to Another Function | |
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| Erdelyi-Kober Type Fractional Integrals and Fractional Derivatives | |
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| Hadamard Type Fractional Integrals and Fractional Derivatives | |
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| Grunwald-Letnikov Fractional Derivatives | |
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| Partial and Mixed Fractional Integrals and Fractional Derivatives | |
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| Riesz Fractional Integro-Differentiation | |
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| Comments and Observations | |
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| Ordinary Fractional Differential Equations. Existence and Uniqueness Theorems | |
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| Introduction and a Brief Overview of Results | |
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| Equations with the Riemann-Liouville Fractional Derivative in the Space of Summable Functions | |
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| Equivalence of the Cauchy Type Problem and the Volterra Integral Equation | |
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| Existence and Uniqueness of the Solution to the Cauchy Type Problem | |
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| The Weighted Cauchy Type Problem | |
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| Generalized Cauchy Type Problems | |
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| Cauchy Type Problems for Linear Equations | |
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| Miscellaneous Examples | |
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| Equations with the Riemann-Liouville Fractional Derivative in the Space of Continuous Functions. Global Solution | |
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| Equivalence of the Cauchy Type Problem and the Volterra Integral Equation | |
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| Existence and Uniqueness of the Global Solution to the Cauchy Type Problem | |
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| The Weighted Cauchy Type Problem | |
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| Generalized Cauchy Type Problems | |
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| Cauchy Type Problems for Linear Equations | |
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| More Exact Spaces | |
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| Further Examples | |
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| Equations with the Riemann-Liouville Fractional Derivative in the Space of Continuous Functions. Semi-Global and Local Solutions | |
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| The Cauchy Type Problem with Initial Conditions at the Endpoint of the Interval. Semi-Global Solution | |
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| The Cauchy Problem with Initial Conditions at the Inner Point of the Interval. Preliminaries | |
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| Equivalence of the Cauchy Problem and the Volterra Integral Equation | |
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| The Cauchy Problem with Initial Conditions at the Inner Point of the Interval. The Uniqueness of Semi-Global and Local Solutions | |
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| A Set of Examples | |
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| Equations with the Caputo Derivative in the Space of Continuously Differentiable Functions | |
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| The Cauchy Problem with Initial Conditions at the Endpoint of the Interval. Global Solution | |
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| The Cauchy Problems with Initial Conditions at the End and Inner Points of the Interval. Semi-Global and Local Solutions | |
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| Illustrative Examples | |
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| Equations with the Hadamard Fractional Derivative in the Space of Continuous Functions | |
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| Methods for Explicitly Solving Fractional Differential Equations | |
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| Method of Reduction to Volterra Integral Equations | |
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| The Cauchy Type Problems for Differential Equations with the Riemann-Liouville Fractional Derivatives | |
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| The Cauchy Problems for Ordinary Differential Equations | |
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| The Cauchy Problems for Differential Equations with the Caputo Fractional Derivatives | |
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| The Cauchy Type Problems for Differential Equations with Hadamard Fractional Derivatives | |
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| Compositional Method | |
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| Preliminaries | |
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| Compositional Relations | |
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| Homogeneous Differential Equations of Fractional Order with Riemann-Liouville Fractional Derivatives | |
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| Nonhomogeneous Differential Equations of Fractional Order with Riemann-Liouville and Liouville Fractional Derivatives with a Quasi-Polynomial Free Term | |
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| Differential Equations of Order 1/2 | |
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| Cauchy Type Problem for Nonhomogeneous Differential Equations with Riemann-Liouville Fractional Derivatives and with a Quasi-Polynomial Free Term | |
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| Solutions to Homogeneous Fractional Differential Equations with Liouville Fractional Derivatives in Terms of Bessel-Type Functions | |
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| Operational Method | |
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| Liouville Fractional Integration and Differentiation Operators in Special Function Spaces on the Half-Axis | |
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| Operational Calculus for the Liouville Fractional Calculus Operators | |
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| Solutions to Cauchy Type Problems for Fractional Differential Equations with Liouville Fractional Derivatives | |
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| Other Results | |
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| Numerical Treatment | |
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| Integral Transform Method for Explicit Solutions to Fractional Differential Equations | |
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| Introduction and a Brief Survey of Results | |
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| Laplace Transform Method for Solving Ordinary Differential Equations with Liouville Fractional Derivatives | |
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| Homogeneous Equations with Constant Coefficients | |
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| Nonhomogeneous Equations with Constant Coefficients | |
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| Equations with Nonconstant Coefficients | |
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| Cauchy Type for Fractional Differential Equations | |
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| Laplace Transform Method for Solving Ordinary Differential Equations with Caputo Fractional Derivatives | |
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| Homogeneous Equations with Constant Coefficients | |
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| Nonhomogeneous Equations with Constant Coefficients | |
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| Cauchy Problems for Fractional Differential Equations | |
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| Mellin Transform Method for Solving Nonhomogeneous Fractional Differential Equations with Liouville Derivatives | |
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| General Approach to the Problems | |
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| Equations with Left-Sided Fractional Derivatives | |
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| Equations with Right-Sided Fractional Derivatives | |
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| Fourier Transform Method for Solving Nonhomogeneous Differential Equations with Riesz Fractional Derivatives | |
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| Multi-Dimensional Equations | |
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| One-Dimensional Equations | |
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| Partial Fractional Differential Equations | |
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| Overview of Results | |
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| Partial Differential Equations of Fractional Order | |
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| Fractional Partial Differential Diffusion Equations | |
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| Abstract Differential Equations of Fractional Order | |
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| Solution of Cauchy Type Problems for Fractional Diffusion-Wave Equations | |
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| Cauchy Type Problems for Two-Dimensional Equations | |
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| Cauchy Type Problems for Multi-Dimensional Equations | |
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| Solution of Cauchy Problems for Fractional Diffusion-Wave Equations | |
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| Cauchy Problems for Two-Dimensional Equations | |
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| Cauchy Problems for Multi-Dimensional Equations | |
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| Solution of Cauchy Problems for Fractional Evolution Equations | |
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| Solution of the Simplest Problem | |
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| Solution to the General Problem | |
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| Solutions of Cauchy Problems via the H-Functions | |
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| Sequential Linear Differential Equations of Fractional Order | |
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| Sequential Linear Differential Equations of Fractional Order | |
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| Solution of Linear Differential Equations with Constant Coefficients | |
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| General Solution in the Homogeneous Case | |
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| General Solution in the Non-Homogeneous Case. Fractional Green Function | |
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| Non-Sequential Linear Differential Equations with Constant Coefficients | |
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| Systems of Equations Associated with Riemann-Liouville and Caputo Derivatives | |
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| General Theory | |
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| General Solution for the Case of Constant Coefficients. Fractional Green Vectorial Function | |
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| Solution of Fractional Differential Equations with Variable Coefficients. Generalized Method of Frobenius | |
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| Introduction | |
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| Solutions Around an Ordinary Point of a Fractional Differential Equation of Order [alpha] | |
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| Solutions Around an Ordinary Point of a Fractional Differential Equation of Order 2[alpha] | |
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| Solution Around an [alpha]-Singular Point of a Fractional Differential Equation of Order [alpha] | |
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| Solution Around an [alpha]-Singular Point of a Fractional Differential Equation of Order 2[alpha] | |
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| Some Applications of Linear Ordinary Fractional Differential Equations | |
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| Dynamics of a Sphere Immersed in an Incompressible Viscous Fluid. Basset's Problem | |
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| Oscillatory Processes with Fractional Damping | |
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| Study of the Tension-Deformation Relationship of Viscoelastic Materials | |
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| Further Applications of Fractional Models | |
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| Preliminary Review | |
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| Historical Overview | |
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| Complex Systems | |
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| Fractional Integral and Fractional Derivative Operators | |
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| Fractional Model for the Super-Diffusion Processes | |
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| Dirac Equations for the Ordinary Diffusion Equation | |
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| Applications Describing Carrier Transport in Amorphous Semiconductors with Multiple Trapping | |
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| Bibliography | |
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| Subject Index | |