Fuzzy Arbitrary Order System
Fuzzy Fractional Differential Equations and Applications

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 graduate-level courses on fractional differential equations for students majoring in applied mathematics, engineering, physics, and computer science.

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

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 Post-Doctoral 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 Post-Doctoral 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.