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