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| Preface | |
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| Acknowledgments | |
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| Special Functions of Preface | |
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| Acknowledgements | |
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| Special Functions of the Fractional Calculuslt | |
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| Gamma Function | |
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| Mittag-Leffler Function | |
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| Wright Function | |
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| Fractional Derivatives and Integralslt | |
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| The Name of the Game | |
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| Gruuml;nwald-Letnikov Fractional Derivatives | |
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| Riemann-Liouville Fractional Derivatives | |
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| Some Other Approaches | |
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| Sequential Fractional Derivatives | |
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| Left and Right Fractional Derivatives | |
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| Properties of Fractional Derivatives | |
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| Laplace Transforms of Fractional Derivatives | |
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| Fourier Transforms of Fractional Derivatives | |
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| Mellin Transforms of Fractional Derivatives | |
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| Existence and Uniqueness Theoremslt | |
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| Linear Fractional Differential Equations | |
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| Fractional Differential Equation of a General Form | |
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| Existence and Uniqueness Theorem as a Method of Solution | |
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| Dependence of a Solution on Initial Conditions | |
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| The Laplace Transform Methodlt | |
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| Standard Fractional Differential Equations | |
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| Sequential Fractional Differential Equations | |
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| Fractional Green's Functionlt | |
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| Definition and Some Properties | |
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| Term Equation | |
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| Term Equation | |
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| Term Equation. | |
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| Term Equation | |
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| Calculation of Heat Load Intensity Change in Blast Furnace Walls | |
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| Finite-Part Integrals and Fractional Derivatives | |
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| General Case: nlt;/i>-term Equation | |
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| Other Methods for the Solution of Fractional-order Equationslt | |
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| The Mellin Transform Method | |
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| Power Series Method | |
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| Babenko's Symbolic Calculus Method | |
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| Method of Orthogonal Polynomials | |
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| Numerical Evaluation of Fractional Derivativeslt | |
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| Approximation of Fractional Derivatives | |
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| The "Short-Memory" Principle | |
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| Order of Approximation | |
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| Computation of Coefficients | |
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| Higher-order Approximations | |
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| Numerical Solution of Fractional Differential Equationslt | |
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| Initial Conditions: Which Problem to Solve? Numerical Solution | |
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| Examples of Numerical Solutions | |
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| The "Short-Memory" Principle in Initial Value Problems for Fractional Differential Equations | |
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| Fractional-Order Systems and Controllers | |
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| Fractional-Order Systems and Fractional-Order Controllerslt | |
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| Example | |
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| on Viscoelasticity | |
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| Bode's Analysis of Feedback Amplifiers | |
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| Fractional Capacitor Theory | |
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| Electrical Circuits | |
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| Electroanalytical Chemistry | |
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| Electrode-Electrolyte Interface | |
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| Fractional Multipoles | |
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| Biology | |
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| Fractional Diffusion Equations | |
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| Control Theory | |
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| Fitting of Experimental Data | |
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| The "Fractional-Order" Physics? Bibliography | |
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| Tables of Fractional Derivativeslt | |
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| Index | |