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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 | |