Fractional Behaviours Modelling: Analysis and Application of Several Unusual Tools (Intelligent Systems, Control and Automation: Science and Engineering, 101)

Preface
Contents
1 Introduction
	References
2 Power-Law Type Dynamic Behaviours
	2.1 Introduction
	2.2 Definitions of Power-Law Type Long Memory Behaviours
		2.2.1 Spectral Density and Autocorrelation Functions of the Input Output Signals of an LTI System
		2.2.2 Spectral Density and Autocorrelation Functions of the Input Output Signals of an LTI System Exhibiting a Power-Law Type (or Fractional) Impulse Response
	2.3 Some Examples
	References
3 Fractional Order Models
	3.1 Introduction
	3.2 Fractional Integration
	3.3 Fractional Differentiation
	3.4 Frequency Response of Fractional Integration and Differentiation Operators
	3.5 Fractional Models Definition
	3.6 How to Take into Account Initial Conditions?
		3.6.1 Modal Decomposition of a Fractional Order System
	3.7 Some Drawbacks Associated to Fractional Models
	References
4 Introduction of New Kernels
	4.1 Introduction
	4.2 Kernels 1ν(t)
	4.3 Kernel 2ν ( t )
	4.4 Kernel 3ν(t)
	4.5 Kernel 4ν(t)
	4.6 Conclusion
	References
5 Volterra Equation
	5.1 Introduction
	5.2 Pseudo State Space Description: A Particular Case of the Volterra Equations
	5.3 A Volterra-Equation-Based Model for Power-Law Type Long Memory Behaviour
		5.3.1 A First Kernel
		5.3.2 A Second Kernel
		5.3.3 A Third Kernel
		5.3.4 A Fourth Kernel
		5.3.5 A Fifth Kernel
	5.4 Conclusion
	References
6 Non-linear Models
	6.1 Introduction
		6.1.1 Fractional Behaviours and Non-linear Models
		6.1.2 Sand Heap Growth: An Example of Fractional Behaviour Produced by a Nonlinear System
	6.2 Application to Adsorption
		6.2.1 Introduction
		6.2.2 Random Sequential Adsorption (RSA)
		6.2.3 Dynamical Modelling of RSA
		6.2.4 Addition of a Phenomenon of Desorption
		6.2.5 Application on Experimental Data
	6.3 Conclusion
	References
7 Partial Differential Equations with Spatially Variable Coefficients
	7.1 Introduction
	7.2 Prior Art on the Approximation of Fractional Order Integrators and the Resulting Electrical Networks
	7.3 Beyond Geometric Distribution
	7.4 Extension to Cauer Type Networks
	7.5 Heat Equation with Spatially Variable Coefficients for Power-Law Type Long Memory Behaviour Modelling
	7.6 Discussions Around Some Other Distributions for Further
	References
8 Conclusion
	References
Appendix A Impulse Response of Some Transfer Functions that Exhibit Power-Law Type Long Memory Behaviours
References
Appendix B Demonstration of Theorem 1
Index