Fractional Calculus: Theory

Roy Abi Zeid Daou (Editor)
Lebanese German University, Sahel Alma Campus, Keserwane, Lebanon

Xavier Moreau (Editor)
Université Bordeaux, 351, cours de la Libération, Talence – France

Series: Mathematics Research Developments




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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. (Imprint: Nova)



About the Editors

Chapter 1 - Analysis of the Caputo Derivative and Pseudo State Representation with the Infinite State Approach (pp. 1-26)
Jean-Claude Trigeassou, Nezha Maamri and Alain Oustaloup (Département de Physique de l’Université Fédérale de Kazan Kremlevskaia, Russia)

Chapter 2 - Stability of a Class of Fractional Cauchy Problem (pp. 27-42)
Rabha W. Ibrahim (Department of Mathematics, Shivaji University, Kolhapur, India)

Chapter 3 - Numerical Solution of Fractional Order Differential Equations via Matrix-Based Methods (pp. 43-64)
Matthew Harker and Paul O’Leary (Dipartimento di Ingegneria Elettricaedell Informazione–Politecnico di Bari, Italy)

Chapter 4 - On Analytical Methods for Differential Equations with Local Fractional Derivative Operators (pp. 65-88)
Xiao-Jun Yang, Dumitru Baleanu and J. A. Tenreiro Machado (University Mouloud Mammeri of Tizi-Ouzou, Algeria and others)

Chapter 5 - Extended Borel Transform and Fractional Calculus (pp. 1-26)
Akira Asada (University of Bordeaux, IMS Laboratory, UMR 5218 CNRS, 33405 Talence, Bordeaux, France and others)

Chapter 6 - Introduction to Stability Theory of Linear Fractional Difference Equations (pp. 117-162)
Jan Čermák and Tomáš Kisela (University of Bordeaux, Laboratory IMS, UMR 5218 CNRS, 33405 Talence, Bordeaux, France and others)

Chapter 7 - Using the Hankel Operator to Initialize Fractional-Order Systems (pp. 163-182)
Jay L. Adams, Robert J. Veillette and Tom T. Hartley (Faculty of Electrical Engineering, Sharif University of Technology, Tehran, and Faculty of Electrical Engineering, Hamedan University of Technology, Hamedan, Iran)

Chapter 8 - Fractional Reaction-transport Equations Arising from Evanescent Continuous Time Random Walks (pp. 183-202)
E. Abad, S. B. Yuste and K. Lindenberg (Research Center on Dynamics of Solids, Voronezh State University of Architecture and Civil Engineering, Russia)

Chapter 9 - Exponential Integrators for Fractional Differential Equations (pp. 203-236)
Roberto Garrappa and Marina Popolizio (Institute of Mathematics and Computer Science, Jan Długosz University, Poland)

Chapter 10 - Non-Fragile Tuning of Fractional-Order PD Controllers for Integrating and Double Integrating Time-Delay Systems (pp. 237-256)
MirSaleh Bahavarnia and Mohammad Saleh Tavazoei (Ghent University, Department of Electrical Energy, Systems and Automation, Belgium)

Chapter 11 - On Discrete, Finite-Dimensional Approximation of Linear, Infinite Dimensional Systems (pp. 257-274)
Milan R. Rapaić, Tomislav B. Šekara and Mihailo P. Lazarević (GECAD, Portugal)


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