Fractional Calculus : Theory

The first volume of this two-volume book, presents history, the mathematical modeling and the applications of fractional order systems, and contains mathematical and theoretical studies and research related to this domain. This volume is made up of 11 chapters. The first chapter presents an analysis of the Caputo derivative and the pseudo state representation with the infinite state approach. The second chapter studies the stability of a class of fractional Cauchy problems. The third chapter shows how to solve fractional order differential equations and fractional order partial differential equations using modern matrix algebraic approaches. Following this chapter, chapter four proposes another analytical method to solve differential equations with local fractional derivative operators. Concerning chapter five, it presents the extended Borel transform and its related fractional analysis. After presenting the analytical resolution methods for fractional calculus, chapter six shows the essentials of fractional calculus on discrete settings. The initialization of such systems is shown in chapter seven. In fact, this chapter presents a generalized application of the Hankel operator for initialization of fractional order systems. The last four chapters show some new studies and applications of non-integer calculus. In fact, chapter eight presents the fractional reaction-transport equations and evanescent continuous time random walks. Chapter nine shows a novel approach in the exponential integrators for fractional differential equations. Chapter ten presents the non-fragile tuning of fractional order PD controllers for integrating time delay systems. At the end, chapter eleven proposes a discrete finite-dimensional approximation of linear infinite dimensional systems. To sum up, this volume presents a mathematical and theoretical study of fractional calculus along with a stability study and some applications. This volume ends up with some new techniques and methods applied in fractional calculus. This volume will be followed up by a second volume that focuses on the applications of fractional calculus in several engineering domains. FRACTIONAL CALCULUS: THEORY FRACTIONAL CALCULUS: THEORY LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA CONTENTS FOREWORD PREFACE ABOUT THE EDITORS Chapter 1: ANALYSIS OF THE CAPUTO DERIVATIVE AND PSEUDO STATE REPRESENTATION WITH THE INFINITE STATE APPROACH Abstract Introduction Riemann-Liouville Integral and Infinite State Approach Laplace Transform of the Caputo Derivative Initial Conditions for FDE/FDS Simulation The Transition Matrix of a Commensurate Order System Conclusion References Chapter 2: STABILITY OF A CLASS OF FRACTIONAL CAUCHY PROBLEM Abstract Methods Existence of Solutions Ulam-Hyers Stability Conclusion References Chapter 3: NUMERICAL SOLUTION OF FRACTIONAL ORDER DIFFERENTIAL EQUATIONS VIA MATRIX-BASED METHODS Abstract 1. Introduction 2. Fractional Integration and DifferentiationMatrices 3. Discretization and Solution of FDE 4. Numerical Examples 5. Conclusion Acknowledgments References Chapter 4: ON ANALYTICAL METHODS FOR DIFFERENTIAL EQUATIONS WITH LOCAL FRACTIONAL DERIVATIVE OPERATORS Abstract Introduction Local Fractional Derivatives and Integrals Local Fractional Integral Transforms Applications Conclusion References Chapter 5: EXTENDED BOREL TRANSFORM AND FRACTIONAL CALCULUS Abstract 1. Introduction 2. Review of Fractional Calculus 3. Integral Transform R 4. Inverse and Adjoint of R 5. Extended Borel Transform 6. R and the Extended Borel Transform 7. Applications to Fractional Differential Equations 8. Fractional Euler Type Equations 9. Fractional Analog of Regular Singularity Appendix. R[ecs] as a Generalized Function References Chapter 6: INTRODUCTION TO STABILITY THEORY OF LINEAR FRACTIONAL DIFFERENCE EQUATIONS Abstract 1. Introduction 2. Basic Properties of Fractional Difference Operators 3. Linear Fractional Difference Equations 4. Stability of Single-TermFractional Difference Equations 5. Stability of Multi-Term Fractional Difference Equations 6. Conclusion Acknowledgment References Chapter 7: USING THE HANKEL OPERATOR TO INITIALIZE FRACTIONAL-ORDER SYSTEMS Abstract 1. Introduction 2. Impulse Response Approximation 3. Output-Based Initialization 4. Output-Based Initialization Examples 5. Conclusion References Chapter 8: FRACTIONAL REACTION-TRANSPORT EQUATIONS ARISING FROM EVANESCENT CONTINUOUS TIME RANDOM WALKS Abstract 1. Introduction 2. Integral Equation for a CTRW with Evanescence 3. Fractional Diffusion Equations with Evanescence 4. Applications 5. Summary and Outlook References Chapter 9: EXPONENTIAL INTEGRATORS FOR FRACTIONAL DIFFERENTIAL EQUATIONS Abstract Differential equations 2. Linear Systems of FDEs: Theoretical Aspects 3. Exponential Integrators: A Short ReviewThe history 4. Exponential Product Integration for FDEs 5. Evaluation of the Matrix Mittag–Leffler Function 6. Application to Time–Fractional Partial Differential Equations References Chapter 10: NON-FRAGILE TUNING OF FRACTIONAL-ORDER PD CONTROLLERS FOR INTEGRATING AND DOUBLE INTEGRATING TIME-DELAY SYSTEMS Abstract 1. Introduction 2. Positive Stability Region 3. Non-Fragile Tuning of a PDm Controller with a Fixed m 4. Non-Fragile Tuning of a PDm Controller with a Tunable Order Conclusion References Chapter 11: ON DISCRETE, FINITE-DIMENSIONAL APPROXIMATION OF LINEAR, INFINITE DIMENSIONAL SYSTEMS Abstract Introduction Motivating Examples Rational Approximations of Transfer Functions of LFOS Conclusion Acknowledgment References INDEX