Description
Book SynopsisPresents a systematic treatment of fuzzy fractional differential equations as well as newly developed computational methods to model uncertain physical problems
Complete with comprehensive results and solutions, Fuzzy Arbitrary Order System: Fuzzy Fractional Differential Equations and Applications details newly developed methods of fuzzy computational techniquesneeded to model solve uncertainty. Fuzzy differential equations are solved via various analytical andnumerical methodologies, and this book presents their importance for problem solving, prototypeengineering design, and systems testing in uncertain environments.
In recent years, modeling of differential equations for arbitrary and fractional order systems has been increasing in its applicability, and as such, the authors feature examples from a variety of disciplines to illustrate the practicality and importance of the methods within physics, applied mathematics, engineering, and chemistry, to name
Table of Contents
PREFACE ix
ACKNOWLEDGMENTS xiii
1 Preliminaries of Fuzzy Set Theory 1
Bibliography 7
2 Basics of Fractional and Fuzzy Fractional Differential Equations 9
Bibliography 12
3 Analytical Methods for Fuzzy Fractional Differential Equations (FFDES) 15
3.1 n-Term Linear Fuzzy Fractional Linear Differential Equations 16
3.2 Proposed Methods 18
Bibliography 28
4 Numerical Methods for Fuzzy Fractional Differential Equations 31
4.1 Homotopy Perturbation Method (HPM) 31
4.2 Adomian Decomposition Method (ADM) 35
4.3 Variational Iteration Method (VIM) 37
Bibliography 39
5 Fuzzy Fractional Heat Equations 41
5.1 Arbitrary-Order Heat Equation 41
5.2 Solution of Fuzzy Arbitrary-Order Heat Equations by HPM 41
5.3 Numerical Examples 43
5.4 Numerical Results 45
Bibliography 47
6 Fuzzy Fractional Biomathematical Applications 49
6.1 Fuzzy Arbitrary-Order Predator–Prey Equations 49
6.1.1 Particular Case 51
6.2 Numerical Results of Fuzzy Arbitrary-Order Predator–Prey Equations 54
Bibliography 65
7 Fuzzy Fractional Chemical Problems 67
7.1 Arbitrary-Order Rossler’s Systems 67
7.2 HPM Solution of Uncertain Arbitrary-Order Rossler’s System 68
7.3 Particular Case 71
7.3.1 Special Case 73
7.4 Numerical Results 78
Bibliography 83
8 Fuzzy Fractional Structural Problems 87
8.1 Fuzzy Fractionally Damped Discrete System 88
8.2 Uncertain Response Analysis 90
8.2.1 Uncertain Step Function Response 90
8.2.2 Uncertain Impulse Function Response 93
8.3 Numerical Results 96
8.3.1 Case Studies for Uncertain Step Function Response 97
8.3.2 Case Studies for Uncertain Impulse Function Response 100
8.4 Fuzzy Fractionally Damped Continuous System 101
8.5 Uncertain Response Analysis 110
8.5.1 Unit step Function Response 110
8.5.2 Unit Impulse Function Response 111
8.6 Numerical Results 112
8.6.1 Case Studies for Fuzzy Unit Step Response 114
8.6.2 Case Studies for Fuzzy Unit Impulse Response 115
Bibliography 118
9 Fuzzy Fractional Diffusion Problems 121
9.1 Fuzzy Fractional-Order Diffusion Equation 121
9.1.1 Double-Parametric-Based Solution of Uncertain
Fractional-Order Diffusion Equation 123
9.1.2 Solution Bounds for Different External Forces 125
9.2 Numerical Results of Fuzzy Fractional Diffusion Equation 130
Bibliography 139
10 Uncertain Fractional Fornberg–Whitham Equations 141
10.1 Parametric-Based Interval Fractional Fornberg–Whitham
Equation 141
10.2 Solution by VIM 143
10.3 Solution Bounds for Different Interval Initial Conditions 145
10.4 Numerical Results 148
Bibliography 152
11 Fuzzy Fractional Vibration Equation of Large Membrane 155
11.1 Double-Parametric-Based Solution of Uncertain Vibration Equation of Large Membrane 156
11.2 Solutions of Fuzzy Vibration Equation of Large Membrane 158
11.3 Case Studies (Solution Bounds for Particular Cases) 160
11.4 Numerical Results for Fuzzy Fractional Vibration Equation for Large Membrane 172
Bibliography 188
12 Fuzzy Fractional Telegraph Equations 191
12.1 Double-Parametric-Based Fuzzy Fractional Telegraph Equations 191
12.2 Solutions of Fuzzy Telegraph Equations Using Homotopy Perturbation Method 194
12.3 Solution Bounds for Particular Cases 195
12.4 Numerical Results for Fuzzy Fractional Telegraph Equations 199
Bibliography 205
13 Fuzzy Fokker–Planck Equation with Space and Time Fractional Derivatives 207
13.1 Fuzzy Fractional Fokker–Planck Equation with Space and Time Fractional Derivatives 207
13.2 Double-Parametric-Based Solution of Uncertain Fractional Fokker–Planck Equation 209
13.2.1 Solution by HPM 209
13.2.2 Solution By ADM 210
13.3 Case Studies Using HPM and ADM 211
13.3.1 Using HPM 211
13.3.2 Using ADM 215
13.4 Numerical Results of Fuzzy Fractional Fokker–Planck Equation 218
Bibliography 220
14 Fuzzy Fractional Bagley–Torvik Equations 223
14.1 Various Types of Fuzzy Fractional Bagley–Torvik Equations 223
14.2 Results and Discussions 231
Bibliography 241
APPENDIX A 243
A.1 Fractionally Damped Spring–Mass System (Problem 1) 243
A.1.1 Response Analysis 246
A.1.2 Analytical Solution Using Fractional Green’s Function 247
A.2 Fractionally Damped Beam (Problem 2) 248
A.2.1 Response Analysis 250
A.2.2 Numerical Results 251
Bibliography 255
INDEX 257
