Table of Contents
Cover
Title Page
Copyright
Preface
Acknowledgments
Chapter 1: Preliminaries of Fuzzy Set Theory
Bibliography
Chapter 2: Basics of Fractional and Fuzzy Fractional Differential Equations
Bibliography
Chapter 3: Analytical Methods for Fuzzy Fractional Differential Equations (FFDES)
3.1 n -Term Linear Fuzzy Fractional Linear Differential Equations
3.2 Proposed Methods
Bibliography
Chapter 4: Numerical Methods for Fuzzy Fractional Differential Equations
4.1 Homotopy Perturbation Method (HPM) (He, 1999a, 2000a)
4.2 Adomian Decomposition Method (ADM) (Adomian, 1984, 1994)
4.3 Variational Iteration Method (VIM) (He, 1999b, 2000b)
Bibliography
Chapter 5: Fuzzy Fractional Heat Equations
5.1 Arbitrary-Order Heat Equation
5.2 Solution of Fuzzy Arbitrary-Order Heat Equations by HPM
5.3 Numerical Examples
5.4 Numerical Results
Bibliography
Chapter 6: Fuzzy Fractional Biomathematical Applications
6.1 Fuzzy Arbitrary-Order Predator–Prey Equations
6.2 Numerical Results of Fuzzy Arbitrary-Order Predator–Prey Equations
Bibliography
Chapter 7: Fuzzy Fractional Chemical Problems
7.1 Arbitrary-Order Rossler's Systems
7.2 HPM Solution of Uncertain Arbitrary-Order Rossler's System
7.3 Particular Case
7.4 Numerical Results
Bibliography
Chapter 8: Fuzzy Fractional Structural Problems
8.1 Fuzzy Fractionally Damped Discrete System
8.2 Uncertain Response Analysis
8.3 Numerical Results
8.4 Fuzzy Fractionally Damped Continuous System
8.5 Uncertain Response Analysis
8.6 Numerical Results
Bibliography
Chapter 9: Fuzzy Fractional Diffusion Problems
9.1 Fuzzy Fractional-Order Diffusion Equation
9.2 Numerical Results of Fuzzy Fractional Diffusion Equation
Bibliography
Chapter 10: Uncertain Fractional Fornberg–Whitham Equations
10.1 Parametric-Based Interval Fractional Fornberg–Whitham Equation
10.2 Solution by VIM (He, 1999, 2000)
10.3 Solution Bounds for Different Interval Initial Conditions
10.4 Numerical Results
Bibliography
Chapter 11: Fuzzy Fractional Vibration Equation of Large Membrane
11.1 Double-Parametric-Based Solution of Uncertain Vibration Equation of Large Membrane
11.2 Solutions of Fuzzy Vibration Equation of Large Membrane
11.3 Case Studies (Solution Bounds for Particular Cases)
11.4 Numerical Results for Fuzzy Fractional Vibration Equation for Large Membrane
Bibliography
Chapter 12: Fuzzy Fractional Telegraph Equations
12.1 Double-Parametric-Based Fuzzy Fractional Telegraph Equations
12.2 Solutions of Fuzzy Telegraph Equations Using Homotopy Perturbation Method
12.3 Solution Bounds for Particular Cases
12.4 Numerical Results for Fuzzy Fractional Telegraph Equations
Bibliography
Chapter 13: Fuzzy Fokker–Planck Equation with Space and Time Fractional Derivatives
13.1 Fuzzy Fractional Fokker–Planck Equation with Space and Time Fractional Derivatives
13.2 Double-Parametric-Based Solution of Uncertain Fractional Fokker–Planck Equation
13.3 Case Studies Using HPM and ADM
13.4 Numerical Results of Fuzzy Fractional Fokker–Planck Equation
Bibliography
Chapter 14: Fuzzy Fractional Bagley–Torvik Equations
14.1 Various Types of Fuzzy Fractional Bagley–Torvik Equations
14.2 Results and Discussions
Bibliography
Appendix A
A.1 Fractionally Damped Spring–Mass System (Problem 1)
A.2 Fractionally Damped Beam (Problem 2)
Bibliography
Index
End User License Agreement
Pages
ix
x
xi
xiii
xiv
1
2
3
4
5
6
7
9
10
11
12
13
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
257
258
Guide
Cover
Table of Contents
Preface
Begin Reading
List of Illustrations
Chapter 1: Preliminaries of Fuzzy Set Theory
Figure 1.1 Triangular fuzzy number
Figure 1.2 Trapezoidal fuzzy number
Figure 1.3 Gaussian fuzzy number
Chapter 5: Fuzzy Fractional Heat Equations
Figure 5.1 Solution of fuzzy arbitrary (fractional)-order heat equations of Example 4.1 for at , and for (a) , (2) , and (c)
Figure 5.2 Solution of fuzzy arbitrary (fractional)-order heat equations of Example 5.2 for at and for (a) , (b) , and (c)
Chapter 6: Fuzzy Fractional Biomathematical Applications
Figure 6.1 Fuzzy solution of fractional-order predator–prey equations for (a) and (b) where and when
Figure 6.2 Fuzzy solution of fractional-order predator–prey equations for (a) and (b) where and when
Figure 6.3 Fuzzy solution of fractional-order predator–prey equations for (a) and (b) where and when
Figure 6.4 Fuzzy solution of fractional-order predator–prey equations for (a) and (b) where and when
Figure 6.5 Interval solution of fractional predator–prey equations for and with at (a) and (b)
Figure 6.6 Interval solution of fractional predator–prey equations for and with at (a) and (b)
Figure 6.7 Interval solution of fractional predator–prey equations for and with at (a) and (b)
Figure 6.8 Interval solution of fractional predator–prey equations for and with at (a) and (b)
Chapter 7: Fuzzy Fractional Chemical Problems
Figure 7.1 Uncertain (Fuzzy) solution of arbitrary order Rossler's system for and at (a) (b) (c)
Figure 7.2 Uncertain (Fuzzy) solution of arbitrary-order Rossler's system for (a) , (b) , and (c) where when
Figure 7.3 Uncertain (Fuzzy) solution of arbitrary-order Rossler's system for (a) , (b) , and (c) where when
Figure 7.4 Uncertain (fuzzy) solution of arbitrary-order Rossler's system for (a) (b) , and (c) where when
Figure 7.5 Crisp solution of arbitrary order Rossler's system for , and at (a) (b) (c) (d)
Chapter 8: Fuzzy Fractional Structural Problems
Figure 8.1 Triangular fuzzy response subject to unit step load for Case 1 with natural frequency (a) , (b) , and damping ratio
Figure 8.2 Trapezoidal fuzzy response subject to unit step load for Case 2 with natural frequency (a) , (b) , and damping ratio
Figure 8.3 Gaussian fuzzy response subject to unit step load for Case 3 with natural frequency (a) , (b) , and damping ratio
Figure 8.4 Uncertain but bounded (interval) response subject to unit step load for Case 1 when (a) , (b) with crisp analytical solution (- - -) by Podlubny (1999) where natural frequency and damping ratio
Figure 8.5 Uncertain but bounded (interval) response subject to unit step load for Case 2 when (a) , (b) with crisp analytical solution (- - -) by Podlubny (1999) where natural frequency and damping ratio
Figure 8.6 Uncertain but bounded (interval) response subject to unit step load for Case 3 when (a) , (b) with crisp analytical solution (- - -) by Podlubny (1999) where natural frequency and damping ratio
Figure 8.7 Uncertain but bounded (interval) response subject to unit step load for Case 1 when (a) , (b) with crisp analytical solution (- - -) by Podlubny (1999) where natural frequency and damping ratio
Figure 8.9 Uncertain but bounded (interval) response subject to unit step load for Case 3 when (a) , (b) with crisp analytical solution (- - -) by Podlubny (1999) where natural frequency and damping ratio
Figure 8.10 Triangular fuzzy response subject to unit impulse load for Case 1 with natural frequency (a) , (b) and damping ratio
Figure 8.11 Trapezoidal fuzzy response subject to unit impulse load for Case 2 with natural frequency (a) , (b) and damping ratio
Figure 8.12 Gaussian fuzzy response subject to unit impulse load for Case 3 with natural frequency (a) , (b) and damping ratio
Figure 8.13 Fuzzy unit step response for , , and
Figure 8.14 Fuzzy unit step response for , , and
Figure 8.15 Interval unit step response for (a) (b) with , , and
Figure 8.16 Interval unit step response for (a) , (b) with , , , and
Figure 8.17 Interval unit step response for (a) , (b) with , , , and
Figure 8.18 Interval unit step response for (a) , (b) with , , , and
Figure 8.19 Fuzzy unit impulse response for , , and (Case 1)
Figure 8.20 Fuzzy unit impulse response for , , and (Case 2)
Figure 8.21 Fuzzy unit impulse response for , , and (Case 3)
Figure 8.22 Fuzzy unit impulse response for , , and (Case 4)
Chapter 9: Fuzzy Fractional Diffusion Problems
Figure 9.1 Fuzzy solution for Case 1 using HPM
Figure 9.2 Fuzzy solution for Case 2 using HPM
Figure 9.3 Fuzzy solution for Case 3 using HPM
Figure 9.4 Interval solutions for Case 1 using HPM
Figure 9.5 Interval solutions for Case 2 using HPM
Figure 9.6 Interval solutions for Case 3 using HPM
Figure 9.7 Fuzzy solution for Case 4 using ADM
Figure 9.8 Fuzzy solution for Case 5 using ADM
Figure 9.9 Fuzzy solution for Case 6 using ADM
Figure 9.10 Interval solutions for of Case 4 using ADM
Figure 9.11 Interval solutions for of Case 4 using ADM
Figure 9.12 Interval solutions for of Case 5 using ADM
Figure 9.13 Interval solutions for of Case 5 using ADM
Figure 9.14 Interval solutions for of Case 6 using ADM
Figure 9.15 Interval solutions for of Case 6 using ADM
Chapter 10: Uncertain Fractional Fornberg–Whitham Equations
Figure 10.1 Interval solutions for Case 1 at
Figure 10.2 Interval solutions for Case 2 at
Figure 10.3 Interval solutions for Case 1 at
Figure 10.4 Interval solutions for Case 2 at (Abidi and Omrani, 2011)
Figure 10.5 Comparison solutions for Case 1 at and
Figure 10.6 Comparison solutions for Case 2 at and
Figure 10.7 The surface shows the approximate solution of Case 1 for
Figure 10.8 The surface shows the approximate solution of Case 2 for
Chapter 11: Fuzzy Fractional Vibration Equation of Large Membrane
Figure 11.1 Fuzzy displacement at and of Case 1
Figure 11.2 Fuzzy solution at and of Case 2
Figure 11.3 Fuzzy solution at and of Case 3
Figure 11.4 Fuzzy solution at and of Case 4
Figure 11.5 Fuzzy solution at and of Case 5
Figure 11.6 Interval solution of Case 1
Figure 11.7 Interval solution of Case 2
Figure 11.8 Interval solution of Case 3
Figure 11.9 Interval solution of Case 4
Figure 11.10 Interval solution of Case 5
Figure 11.11 Interval solution of Case 1 at
Figure 11.12 Interval solution of Case 2 at
Figure 11.13 Interval solution of Case 3 at
Figure 11.14 Interval solution of Case 4 at
Figure 11.15 Interval solution of Case 5 at
Figure 11.16 Fuzzy displacement at and of Case 6
Figure 11.17 Fuzzy solution at and of Case 7
Figure 11.18 Fuzzy solution at and of Case 8
Figure 11.19 Fuzzy solution at and of Case 9
Figure 11.20 Fuzzy solution at and of Case 10
Figure 11.21 Interval solution of Case 6 for
Figure 11.22 Interval solution of Case 7 for
Figure 11.23 Interval solution of Case 8 for
Figure 11.24 Interval solution of Case 9 for
Figure 11.25 Interval solution of Case 10 for
Figure 11.26 Interval solution of Case 6 at for
Figure 11.27 Interval solution of Case 7 at for
Figure 11.28 Interval solution of Case 8 at for
Figure 11.29 Interval solution of Case 9 at for
Figure 11.30 Interval solution of Case 10 at for
Chapter 12: Fuzzy Fractional Telegraph Equations
Figure 12.1 Fuzzy solution for Case 1 at and
Figure 12.2 Fuzzy solution for Case 2 at and
Figure 12.3 Fuzzy solution for Case 1 at and
Figure 12.4 Fuzzy solution for Case 2 at and
Figure 12.5 Interval solution for Case 1 at and
Figure 12.6 Interval solution for Case 2 at and
Figure 12.7 Interval solution for Case 1 at and
Figure 12.8 Interval solution for Case 2 at and
Figure 12.9 Interval solution for Case 1 at and
Figure 12.10 Interval solution for Case 1 at and
Figure 12.11 Interval solution for Case 2 at and
Figure 12.12 Interval solution for Case 2 at and
Chapter 13: Fuzzy Fokker–Planck Equation with Space and Time Fractional Derivatives
Figure 13.1 Fuzzy solution for Case 1 using HPM
Figure 13.2 Fuzzy solution for Case 2 using HPM
Chapter 14: Fuzzy Fractional Bagley–Torvik Equations
Figure 14.1 (a) Fuzzy solution of Case 1 HPM for . (b) Interval solution of Case 1 HPM for , 0.5, and 1
Figure 14.2 (a) Fuzzy solution of Case 2 HPM for , 0.5, and 1. Interval solution of Case 2 HPM for , 0.5, and 1
Figure 14.3 (a) Fuzzy solution of Case 3 HPM for , 0.5, and 1. (b) Interval solution of Case 3 HPM for , 0.5, and 1
Appendix A
Figure A.1 Unit step responses along with natural frequency (a) , (b) , and damping ratio
Figure A.2 Unit step responses along with natural frequency (a) , (b) , and damping ratio
Figure A.3 Impulse responses along with natural frequency (a) (b) , and damping ratio
Figure A.4 Impulse responses along with natural frequency (a) (b) , and damping ratio
List of Tables
Chapter 6: Fuzzy Fractional Biomathematical Applications
Table 6.1 Fuzzy Solution for
Table 6.2 Fuzzy Solution for
Table 6.3 Fuzzy Solution
Table 6.4 Fuzzy Solution for
Chapter 7: Fuzzy Fractional Chemical Problems
Table 7.1 Uncertain (Fuzzy) and Crisp Solution Using HPM for α = β = γ = 1/3
Table 7.2 Uncertain (Fuzzy) and Crisp Solution Using HPM for α = β = γ = 1/2
Table 7.3 Uncertain (Fuzzy) and Crisp Solution Using HPM for α = β = γ = 2/3
Chapter 8: Fuzzy Fractional Structural Problems
Table 8.1 Data for Fuzzy Initial Conditions
Table 8.2 r -Cut Representations of Fuzzy Initial Conditions
Chapter 9: Fuzzy Fractional Diffusion Problems
Table 9.1 Comparison of Present with HPM, Present with ADM and Godal et al. (2013) Solutions of Cases 1 and 4 for x = 0.5 and α = 0.4
Table 9.2 Comparison of Present with HPM, Present with ADM and Godal et al. (2013) Solutions of Cases 2 and 5 for x = 0.5 and α = 0.4
Chapter 10: Uncertain Fractional Fornberg–Whitham Equations
Table 10.1 Comparison Table for Case 1 for t = 1 and α = 1
Table 10.2 Comparison Table for Case 1 for x = 0 and α = 1
Table 10.3 Comparison Table for Case 2 for t = 1 and α = 1
Table 10.4 Comparison Table for Case 2 for x = 0 and α = 1
Chapter 13: Fuzzy Fokker–Planck Equation with Space and Time Fractional Derivatives
Table 13.1 Comparison of HPM, ADM, and Yildirim (2010) Solutions of Cases 1 and 3 for x = 0.5, γ = 0.5, and α = 1
Table 13.2 Comparison of HPM, ADM, and Yildirim (2010) Solutions of Cases 2 and 4 for x = 0.5, γ = 0.5, and α = 1
Chapter 14: Fuzzy Fractional Bagley–Torvik Equations
Table 14.1(a) HPM Solution of Case 1 for t = 0, t = 0.5, and t = 1
Table 14.1(b) HPM Solution of Case 1 for r = 0, r = 0.5, and r = 1
Table 14.2 HPM Solution of Case 2 for t = 0, t = 0.5, and t = 1
Table 14.2 HPM Solution of Case 2 for r = 0, r = 0.5, and r = 1
Table 14.3 HPM Solution of Case 3 for t = 0, t = 0.5, and t = 1
Table 14.3 HPM Solution of Case 3 for r = 0, r = 0.5, and r = 1
Table 14.4 Comparison Between HPM Solution and Podlubny (1999)
Fuzzy Arbitrary Order System
Fuzzy Fractional Differential Equations and Applications
Snehashish Chakraverty
Department of Mathematics, National Institute of Technology Rourkela, Odisha, India
Smita Tapaswini
College of Mathematics and Statistics, Chongqing University, Chongqing, P.R. China Department of Mathematics, KIIT University, Bhubaneswar, Odisha, India
Diptiranjan Behera
Institute of Reliability Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan, P.R. China
Copyright © 2016 by John Wiley & Sons, Inc. All rights reserved
Published by John Wiley & Sons, Inc., Hoboken, New Jersey
Published simultaneously in Canada
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permissions.
Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.
For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002.
Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com.
Library of Congress Cataloging-in-Publication Data:
Names: Chakraverty, Snehashish. | Tapaswini, Smita, 1987- | Behera, D. (Diptiranjan), 1988-
Title: Fuzzy arbitrary order system : fuzzy fractional differential equations and applications / by Snehashish Chakraverty, Smita Tapaswini, D. Behera.
Description: Hoboken, New Jersey : John Wiley and Sons, Inc., [2016] | Includes bibliographical references and index.
Identifiers: LCCN 2016013567 (print) | LCCN 2016015086 (ebook) | ISBN 9781119004110 (cloth) | ISBN 9781119004134 (pdf) | ISBN 9781119004172 (epub)
Subjects: LCSH: Fractional differential equations. | Fuzzy mathematics. | Differential equations.
Classification: LCC QA314 .C43 2016 (print) | LCC QA314 (ebook) | DDC 515/.352--dc23
LC record available at http://lccn.loc.gov/2016013567