Cover; Half-Title Page; Dedication; Title Page; Copyright Page; Contents; Foreword; Preface; PART 1: Simulation and Identification of Fractional Differential Equations (FDEs) and Systems (FDSs); 1. The Fractional Integrator; 1.1. Introduction; 1.2. Simulation and modeling of integer order ordinary differential equations; 1.2.1. Simulation with analog computers; 1.2.2. Simulation with digital computers; 1.2.3. Initial conditions; 1.2.4. State space representation and simulation diagram; 1.2.5. Concluding remarks; 1.3. Origin of fractional integration: repeated integration
1.4. Riemann-Liouville integration1.4.1. Definition; 1.4.2. Laplace transform of the Riemann-Liouville integral; 1.4.3. Fractional integration operator; 1.4.4. Fractional differentiation; 1.5. Simulation of FDEs with a fractional integrator; 1.5.1. Simulation of a one-derivative FDE; 1.5.2. FDE; 1.5.3. Simulation of the general linear FDE; A.1. Appendix; A.1.1. Lord Kelvin's principle; A.1.2. A brief history of analog computing; A.1.3. Interpretation of the RK2 algorithm; A.1.4. The gamma function; 2. Frequency Approach to the Synthesis of the Fractional Integrator; 2.1. Introduction
3.2. Simulation with the Grünwald-Letnikov approach3.2.1. Euler's technique; 3.2.2. The Grünwald-Letnikov fractional derivative; 3.2.3. Numerical simulation with the Grünwald-Letnikov integrator; 3.2.4. Some specificities of the Grünwald-Letnikov integrator; 3.2.5. Short memory principle; 3.3. Simulation with infinite state approach; 3.4. Caputo's initialization; 3.5. Numerical simulations; 3.5.1. Introduction; 3.5.2. Comparison of discrete impulse responses (DIRs); 3.5.3. Simulation accuracy; 3.5.4. Static error caused by the short memory principle; 3.5.5. Caputo's initialization
3.5.6. ConclusionA. 3. Appendix: Mittag-Leffler function; A.3.1. Definition; A.3.2. Laplace transform; A.3.3. Unit step response of 1/sn + a; A.3.4. Caputo's initialization; 4. Fractional Modeling of the Diffusive Interface; 4.1. Introduction; 4.2. Heat transfer and diffusive model of the plane wall; 4.2.1. Heat transfer; 4.2.2. Physical model of the diffusive interface; 4.2.3. Frequency analysis of the diffusive phenomenon; 4.2.4. Time analysis of the diffusive phenomenon; 4.2.5. Conclusion; 4.3. Fractional commensurate order models; 4.3.1. Physical origin
Notes
4.3.2. Analysis of physical commensurate order models
