7.2.1 The Age of the Earth Problem......Page 2
7.3.3 Fractional Impedance Model......Page 5
7.5…Future Travels......Page 7
Acknowledgments......Page 8
Cover......Page 1
7.3.2 Fractional Dynamics Model......Page 4
Preface......Page 6
6.4.1 Introduction......Page 10
6.4.3 The Eigenfunctions and Frequency Response......Page 12
1 A Travel Through the World of Fractional Calculus......Page 13
2.1.1 A Brief Historical Overview......Page 16
6.5.4 A Simple Example......Page 18
2.1.2 Current Formulations......Page 19
2.1.3 A Signal Processing Point of View......Page 20
2.1.4 Overview......Page 21
2.2.1 On the Grünwald--Letnikov Derivative......Page 22
4.10.4.1 Riemann--Liouville......Page 24
2.5…Properties......Page 28
2.9.3 The Causal Logarithm......Page 29
2.9.4 Consequences in the Laplace Transform Domain......Page 30
2.6.1 Additivity and Commutativity of the Orders......Page 31
2.7.1 The Exponential......Page 33
2.7.4 The Complex Sinusoid......Page 36
3.5.2 The Power Function......Page 14
5.8.2 Type 2 Derivative......Page 15
5.8.3 The Integer Order Cases......Page 17
4.10.4.3 The Rational Order Case......Page 26
Fractional Calculus forScientists and Engineers......Page 3
Contents......Page 9
2.2.3 Integer Order Derivatives......Page 25
2.3…Definition of Fractional Derivative......Page 27
6.4.2 The General Formulation......Page 11
2.2.2 Difference Definitions......Page 23
2.7.2 The Constant Function......Page 34
2.7.3 The LT of the Fractional Derivative......Page 35
2.8…Starting from the Transfer Function......Page 37
2.9.1 The Causal Power Function......Page 39
2.9.2 The Causal Exponential......Page 43
2.9.3 The Causal Logarithm......Page 44
2.9.4 Consequences in the Laplace Transform Domain......Page 45
2.10…Riemann--Liouville and Caputo Derivatives......Page 46
2.11.1 Periodic Functions......Page 48
2.13…Conclusions......Page 51
3.1…Introduction......Page 53
3.2.1 Positive Integer Order......Page 54
3.2.2 Fractional Order......Page 56
3.2.3.1 Repeated differencing......Page 57
3.2.3.2 Inversion......Page 58
3.3…Obtaining the Generalized Cauchy Formula......Page 59
3.4.1 General Formulation......Page 60
3.5.1 The Exponential Function......Page 64
3.5.2 The Power Function......Page 66
3.5.3.1 theta = 0: Forward Derivative......Page 67
3.5.4 Derivatives of Some Causal Functions......Page 68
3.6…Derivatives of Functions with Laplace Transform......Page 69
3.7.1 RL and C Derivatives in the Complex Plane......Page 70
3.7.2 Half Plane Derivatives......Page 73
3.8…Conclusions......Page 78
4.1…Introduction......Page 80
4.2…Description......Page 81
4.3…From the Transfer Function to the Impulse Response......Page 82
4.4.1 By the Inversion Integral......Page 84
4.4.2 By Series Expansion......Page 86
4.4.3 Rational Case......Page 87
4.5…Stability of Fractional Linear Time Invariant Continuous-Time Systems......Page 88
4.6…Examples of Simple FLTI Systems......Page 89
4.7.1 Introduction......Page 91
4.7.2 The Initialization Problem......Page 95
4.9…An Example......Page 97
4.9.1 The Initial-Value Theorem......Page 98
4.10.2 Step by Step Differentiation......Page 100
4.10.3 Examples......Page 101
4.10.4.1 Riemann--Liouville......Page 103
4.10.4.2 Caputo......Page 104
4.10.4.3 The Rational Order Case......Page 105
4.12…Conclusions......Page 106
5.1…Motivation......Page 109
5.2…Integer Order Two-Sided Differences and Derivatives......Page 112
5.3…Integral Representations for the Integer Order Two-Sided Differences......Page 113
5.4…Fractional Central Differences......Page 115
5.5…Integral Representations for the Fractional Two-Sided Differences......Page 116
5.6…The Fractional Two-Sided Derivatives......Page 117
5.7…Integral Formulae......Page 118
5.8.1 Type 1 Derivative......Page 121
5.8.2 Type 2 Derivative......Page 123
5.8.3 The Integer Order Cases......Page 125
5.8.4 Other Properties of the Central Derivatives......Page 126
5.9.1 Some Computational Issues......Page 127
5.10…Conclusions......Page 128
6.1…Introduction......Page 130
6.2.1 The ‘‘Below t’’ Case......Page 131
6.2.2 The ‘‘Above t’’ Case......Page 135
6.3…Integral Formulations......Page 137
6.4.1 Introduction......Page 139
6.4.2 The General Formulation......Page 140
6.4.3 The Eigenfunctions and Frequency Response......Page 141
6.5.1 The Uniform Orders Case......Page 142
6.5.3 The Fractional Order System......Page 145
6.5.4 A Simple Example......Page 147
6.5.5 Additional Comments......Page 149
6.6…Conclusions......Page 150
References......Page 151
7.1…Some Considerations......Page 152
7.2.1 The Age of the Earth Problem......Page 153
7.3.1 General Considerations......Page 154
7.3.2 Fractional Dynamics Model......Page 155
7.3.3 Fractional Impedance Model......Page 156
7.3.4 Additional Comments......Page 157
7.5…Future Travels......Page 158
References......Page 159