Fuzzy Arbitrary Order System: Fuzzy Fractional Differential Equations and ApplicationsISBN: 9781119004110
272 pages
August 2016

Description
Presents 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 a few. The fundamentals of fractional differential equations and the basic preliminaries of fuzzy fractional differential equations are first introduced, followed by numerical solutions, comparisons of various methods, and simulated results. In addition, fuzzy ordinary, partial, linear, and nonlinear fractional differential equations are addressed to solve uncertainty in physical systems. In addition, this book features:
 Basic preliminaries of fuzzy set theory, an introduction of fuzzy arbitrary order differential equations, and various analytical and numerical procedures for solving associated problems
 Coverage on a variety of fuzzy fractional differential equations including structural, diffusion, and chemical problems as well as heat equations and biomathematical applications
 Discussions on how to model physical problems in terms of nonprobabilistic methods and provides systematic coverage of fuzzy fractional differential equations and its applications
 Uncertainties in systems and processes with a fuzzy concept
Fuzzy Arbitrary Order System: Fuzzy Fractional Differential Equations and Applications is an ideal resource for practitioners, researchers, and academicians in applied mathematics, physics, biology, engineering, computer science, and chemistry who need to model uncertain physical phenomena and problems. The book is appropriate for graduatelevel courses on fractional differential equations for students majoring in applied mathematics, engineering, physics, and computer science.
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 nTerm 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 ArbitraryOrder Heat Equation 41
5.2 Solution of Fuzzy ArbitraryOrder 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 ArbitraryOrder Predator–Prey Equations 49
6.1.1 Particular Case 51
6.2 Numerical Results of Fuzzy ArbitraryOrder Predator–Prey Equations 54
Bibliography 65
7 Fuzzy Fractional Chemical Problems 67
7.1 ArbitraryOrder Rossler’s Systems 67
7.2 HPM Solution of Uncertain ArbitraryOrder 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 FractionalOrder Diffusion Equation 121
9.1.1 DoubleParametricBased Solution of Uncertain
FractionalOrder 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 ParametricBased 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 DoubleParametricBased 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 DoubleParametricBased 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 DoubleParametricBased 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
Author Information
Snehashish Chakraverty, PhD, is Professor and Head of the Department of Mathematics at the National Institute of Technology, Rourkela in India. The author of five books and approximately 140 journal articles, his research interests include mathematical modeling, machine intelligence, uncertainty modeling, numerical analysis, and differential equations.
Smita Tapaswini, PhD, is Assistant Professor in the Department of Mathematics at the Kalinga Institute of Industrial Technology University in India and is also PostDoctoral Fellow at the College of Mathematics and Statistics at Chongqing University in China. Her research interests include fuzzy differential equations, fuzzy fractional differential equations, and numerical analysis.
Diptiranjan Behera, PhD, is PostDoctoral Fellow at the Institute of Reliability Engineering in the School of Mechatronics Engineering at the University of Electronic Science and Technology of China. His current research interests include interval and fuzzy mathematics, fuzzy finite element methods, and fuzzy structural analysis.