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FUZZY ARBITRARYORDER SYSTEM

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FUZZY ARBITRARYORDER SYSTEM

Fuzzy Fractional DifferentialEquations and Applications

SNEHASHISH CHAKRAVERTYDepartment of Mathematics, National Institute of Technology Rourkela, Odisha,India

SMITA TAPASWINICollege of Mathematics and Statistics, Chongqing University, Chongqing, P.R.ChinaDepartment of Mathematics, KIIT University, Bhubaneswar, Odisha, India

DIPTIRANJAN BEHERAInstitute of Reliability Engineering, University of Electronic Science andTechnology of China, Chengdu, Sichuan, P.R. China

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Copyright © 2016 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New JerseyPublished simultaneously in Canada

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form orby any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except aspermitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the priorwritten permission of the Publisher, or authorization through payment of the appropriate per-copy fee tothe 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 shouldbe addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permissions.

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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 equationsand 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) | ISBN9781119004110 (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) | DDC515/.352–dc23

LC record available at http://lccn.loc.gov/2016013567

Typeset in 10/12pt TimesLTStd by SPi Global, Chennai, India

Printed in the United States of America10 9 8 7 6 5 4 3 2 1

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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, 163.2 Proposed Methods, 18

Bibliography, 28

4 Numerical Methods for Fuzzy Fractional Differential Equations 31

4.1 Homotopy Perturbation Method (HPM), 314.2 Adomian Decomposition Method (ADM), 354.3 Variational Iteration Method (VIM), 37

Bibliography, 39

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vi CONTENTS

5 Fuzzy Fractional Heat Equations 41

5.1 Arbitrary-Order Heat Equation, 415.2 Solution of Fuzzy Arbitrary-Order Heat Equations by HPM, 415.3 Numerical Examples, 435.4 Numerical Results, 45

Bibliography, 47

6 Fuzzy Fractional Biomathematical Applications 49

6.1 Fuzzy Arbitrary-Order Predator–Prey Equations, 496.1.1 Particular Case, 51

6.2 Numerical Results of Fuzzy Arbitrary-Order Predator–PreyEquations, 54Bibliography, 65

7 Fuzzy Fractional Chemical Problems 67

7.1 Arbitrary-Order Rossler’s Systems, 677.2 HPM Solution of Uncertain Arbitrary-Order Rossler’s System, 687.3 Particular Case, 71

7.3.1 Special Case, 737.4 Numerical Results, 78

Bibliography, 83

8 Fuzzy Fractional Structural Problems 87

8.1 Fuzzy Fractionally Damped Discrete System, 888.2 Uncertain Response Analysis, 90

8.2.1 Uncertain Step Function Response, 908.2.2 Uncertain Impulse Function Response, 93

8.3 Numerical Results, 968.3.1 Case Studies for Uncertain Step Function Response, 978.3.2 Case Studies for Uncertain Impulse Function Response, 100

8.4 Fuzzy Fractionally Damped Continuous System, 1018.5 Uncertain Response Analysis, 110

8.5.1 Unit step Function Response, 1108.5.2 Unit Impulse Function Response, 111

8.6 Numerical Results, 1128.6.1 Case Studies for Fuzzy Unit Step Response, 1148.6.2 Case Studies for Fuzzy Unit Impulse Response, 115Bibliography, 118

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CONTENTS vii

9 Fuzzy Fractional Diffusion Problems 121

9.1 Fuzzy Fractional-Order Diffusion Equation, 1219.1.1 Double-Parametric-Based Solution of Uncertain

Fractional-Order Diffusion Equation, 1239.1.2 Solution Bounds for Different External Forces, 125

9.2 Numerical Results of Fuzzy Fractional Diffusion Equation, 130Bibliography, 139

10 Uncertain Fractional Fornberg–Whitham Equations 141

10.1 Parametric-Based Interval Fractional Fornberg–WhithamEquation, 141

10.2 Solution by VIM, 14310.3 Solution Bounds for Different Interval Initial Conditions, 14510.4 Numerical Results, 148

Bibliography, 152

11 Fuzzy Fractional Vibration Equation of Large Membrane 155

11.1 Double-Parametric-Based Solution of Uncertain Vibration Equationof Large Membrane, 156

11.2 Solutions of Fuzzy Vibration Equation of Large Membrane, 15811.3 Case Studies (Solution Bounds for Particular Cases), 16011.4 Numerical Results for Fuzzy Fractional Vibration Equation for Large

Membrane, 172Bibliography, 188

12 Fuzzy Fractional Telegraph Equations 191

12.1 Double-Parametric-Based Fuzzy Fractional Telegraph Equations, 19112.2 Solutions of Fuzzy Telegraph Equations Using Homotopy Perturbation

Method, 19412.3 Solution Bounds for Particular Cases, 19512.4 Numerical Results for Fuzzy Fractional Telegraph Equations, 199

Bibliography, 205

13 Fuzzy Fokker–Planck Equation with Space and Time FractionalDerivatives 207

13.1 Fuzzy Fractional Fokker–Planck Equation with Space and TimeFractional Derivatives, 207

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viii CONTENTS

13.2 Double-Parametric-Based Solution of Uncertain FractionalFokker–Planck Equation, 20913.2.1 Solution by HPM, 20913.2.2 Solution By ADM, 210

13.3 Case Studies Using HPM and ADM, 21113.3.1 Using HPM, 21113.3.2 Using ADM, 215

13.4 Numerical Results of Fuzzy Fractional Fokker–Planck Equation, 218Bibliography, 220

14 Fuzzy Fractional Bagley–Torvik Equations 223

14.1 Various Types of Fuzzy Fractional Bagley–Torvik Equations, 22314.2 Results and Discussions, 231

Bibliography, 241

APPENDIX A 243A.1 Fractionally Damped Spring–Mass System (Problem 1), 243

A.1.1 Response Analysis, 246A.1.2 Analytical Solution Using Fractional Green’s Function, 247

A.2 Fractionally Damped Beam (Problem 2), 248A.2.1 Response Analysis, 250A.2.2 Numerical Results, 251Bibliography, 255

INDEX 257

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PREFACE

Every physical problem is inherently biased by uncertainty. There is often a need tomodel, solve, and interpret the problems one encounters in the world of uncertainty.In general, science and engineering systems are governed by ordinary and partialdifferential equations, but the type of differential equation depends upon the applica-tion, domain, complicated environment, the effect of coupling, and so on. As such,the complicacy needs to be handled by recently developed arbitrary (fractional)-orderdifferential equations. The arbitrary-order differential equations are themselves noteasy to handle. In recent years, this subject has become an important area of researchdue to its wide range of applications in various disciplines, namely physics, chem-istry, applied mathematics, biology, economics, and in engineering systems such asfluid mechanics, viscoelasticity, civil, mechanical, aerospace, and chemical. In gen-eral, parameters, variables, and initial conditions involved in the model are consideredas crisp or defined exactly for easy computation. However, rather than the particularvalue, we may have only the vague, imprecise, and incomplete information about thevariables and parameters being a result of errors in measurement, observations, exper-iment, applying different operating conditions, or it may be maintenance-inducederrors, which are uncertain in nature. So, to overcome these uncertainties and vague-ness, one may use either stochastic and statistical approach or interval and fuzzy settheory, but stochastic and statistical uncertainty occurs due to the natural randomnessin the process. It is generally expressed by a probability density or frequency distribu-tion function. For the estimation of the distribution, it requires sufficient informationabout the variables and parameters involved in it. On the other hand, interval and fuzzyset theory refers to the uncertainty when we may have lack of knowledge or incom-plete information about the variables and parameters. As such, in this book, intervaland fuzzy set theory has been used for the uncertainty analysis. These uncertainties

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x PREFACE

are introduced in the general arbitrary (fractional)-order differential equations, whichare named as Fuzzy Arbitrary-Order Differential Equations. Due to the complexityin the fuzzy arithmetic, one may need reliable and efficient analytical and numericaltechniques for the solution of fuzzy arbitrary-order differential equations.

In view of the previous discussion, this book presents initially the basics of fuzzyand interval theory along with preliminaries of arbitrary (fractional)-order differentialequations. Then various methods to solve fuzzy arbitrary (fractional)-order differen-tial equations with fuzzy initial and/or boundary conditions are presented. The bookconsists of 14 chapters, and in order to understand the essence of fuzzy arbitrary-orderdifferential equations, the developed methods have been applied then to solve variousmathematical examples and application problems of engineering and sciences.

Accordingly, Chapter 1 addresses the preliminaries on fuzzy set theory, andChapter 2 recalls the basics of fractional and fuzzy fractional differential equations.Chapter 3 deals with the analytical methods for the solution of n-term fuzzyfractional differential equations. The concept of n-term fuzzy fractional lineardifferential equations is briefly discussed here. As the sign of the coefficients inthe fuzzy fractional-order differential equations plays a very important role, threepossible cases, namely when all the coefficients are positive, when all the coefficientsare negative, and when the coefficients are combinations of positive and negative,are all discussed. Methods based on fuzzy center, addition and subtraction of fuzzynumbers, and double parametric form of fuzzy numbers are also included here.In Chapter 4, numerical schemes, namely homotopy perturbation method (HPM),Adomian decomposition method (ADM), and variational iteration method (VIM)have been presented for fuzzy fractional differential equations. Chapters 3 and 4 alsocontain simple mathematical examples for better understanding of these methods.Solution of fuzzy arbitrary-order heat equations using HPM has been addressed inChapter 5. Fuzziness in the initial conditions is taken in terms of triangular fuzzynumber. Chapter 6 presents the solution of fuzzy arbitrary-order predator–preyequation. In the predator–prey equation, fuzziness in the initial conditions, whichis again taken in the form of triangular fuzzy number and solution, is obtainedby HPM. Comparisons are also made with crisp solutions. Numerical solution ofuncertain arbitrary-order Rossler’s system has been analyzed in Chapter 7. It isworth mentioning that Rossler’s system was found to be useful in the modelingof equilibrium in chemical reactions. Chapter 8 describes the numerical solutionof imprecisely defined fractionally damped structural systems. In this regard, bothdiscrete and continuous systems have been taken into consideration subjected to unitimpulse and step loads. First, a mechanical spring–mass system having fractionaldamping of order 1/2 with fuzzy initial condition has been taken to analyze thediscrete system. Fuzziness in the initial conditions is modeled through differenttypes of convex, normalized fuzzy sets, namely triangular, trapezoidal, and Gaussianfuzzy numbers. HPM is used with fuzzy-based approach to obtain the uncertainimpulse response. Next, this chapter includes the study of fuzzy fractionally dampedcontinuous system that is a beam using the double parametric form of fuzzy numberssubject to unit step and impulse loads. HPM is used for obtaining the fuzzy response

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PREFACE xi

and various numerical examples are solved. Chapter 9 gives the double parametricform of fuzzy numbers to solve fuzzy fractional diffusion equation with initialconditions as triangular and Gaussian fuzzy numbers. In the solution process, HPMand ADM are used. Lastly, Chapter 10 presents a type of traveling-wave problem,namely the nonlinear interval fractional Fornberg–Whitham equations subject tointerval initial conditions. VIM has been applied to obtain the uncertain solution.

Further, double-parametric-based method has also been used to solve variousother fuzzy fractional differential equations, namely vibration equation of largemembrane, telegraph equation, and Fokker–Planck equation in Chapters 11–13,respectively. Finally, Chapter 14 addresses the solution of fuzzy fractionalBagley–Torvik equations using HPM following concepts of Hukuhara derivative.It is worth mentioning that an appendix has also been included for handling crispdifferential equations with fractionally damped spring–mass and beam problems forthe sake of completeness so that readers may go through the same to have readyreference of crisp cases.

This book aims to provide basic concepts of fuzzy arbitrary (/fractional)-orderdifferential equations with various important applications in science and engineeringsystems in a systematic manner along with the recent trends, usefulness, and develop-ments. The book will certainly find an important source for graduate and postgraduatestudents, teachers, and researchers in colleges, universities/institutes, and industriesin various science and engineering fields, wherever one wants to model and analyzetheir uncertain physical problems. It is known that uncertainty is a must in every fieldof science and engineering, so this work will prove to be a handy and important bookto handle their problems.

Finally, we do believe that the book may represent a new vista because it demon-strates how the most current, advanced, and revolutionary mathematical and compu-tational techniques can be put to effective use of fuzzy and interval analysis in theuncertain arbitrary-order differential equations.

S. Chakraverty, RourkelaSmita Tapaswini, ChongqingDiptiranjan Behera, Chengdu

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ACKNOWLEDGMENTS

Writing this book was an amazing journey that would not have been possible withoutthe continuous support, encouragement, and motivation received from many out-standing people around us who not only guided us through our hardships, but alsomade us believe that we could achieve what we wanted to.

As such, the first author would firstly like to thank his parents for being his pillarsof motivation. Next, he would like to thank his wife, Mrs. Shewli Chakraborty, for herimmense love and support. He would also like to thank his daughters, Shreyati andSusprihaa, for their love and source of inspiration during the course of writing thisbook. The support of all the Ph.D. students of the first author and the NIT Rourkelafacilities are also gratefully acknowledged.

The second author would like to thank her family members for their continuoussource of encouragement and motivation. Writing this book would not have been pos-sible without the love, concern, and motivation of her father Mr. Kedarnath Behera,mother Mrs. Sandhyarani Behera, and brother Mr. Deepak Behera.

The third author would like to express his sincere gratitude to his parents, Mr.Muralidhar Behera and Mrs. Sachala Behera, and elder brothers, Manoranjan andSrutiranjan, for their unwavering support and invariable source of motivation. Hefirmly believes that without their support he would not have been able to fly high andachieve success.

The second and third authors, namely Dr. Diptiranjan Behera and Dr. SmitaTapaswini of this book, are highly obliged to the family members of the first author,especially to his wife Mrs. Shewli Chakraborty and daughters Shreyati and Susprihaafor their continuous love, support, and source of inspiration at all the time.

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xiv ACKNOWLEDGMENTS

We are grateful to all the contributors and to the authors of the books and jour-nal/conference papers listed in the book for providing us with valuable information.Finally, we heartily acknowledge the support and help of the editorial team of thepublisher throughout this project to complete it in the targeted time.

S. ChakravertySmita Tapaswini

Diptiranjan Behera

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1PRELIMINARIES OF FUZZY SETTHEORY

This chapter presents the notations, definitions of fuzzy numbers (namely trian-gular, trapezoidal, Gaussian, double parametric form), type of differentiability,theorems/lemma related to fuzzy/fuzzy fractional differential equations, andfuzzy/interval arithmetic, which are relevant to the current investigation. Severalexcellent books related to this have also been written by different authors represent-ing the scope and various aspects of fuzzy set theory such as in Zimmermann (2001),Jaulin et al. (2001), Ross (2004), Hanss (2005), Moore (1966), and Chakraverty(2014). These books also give an extensive review on fuzzy set theory and itsapplications, which may help the reader in understating the basic concepts of fuzzyset theory and its application.

Definition 1.1 Interval An interval x is denoted by [x, x ] on the set of real numbersR given by

x = [x, x ] = {x ∈ R ∶ x ≤ x ≤ x}. (1.1)

We have only considered closed intervals throughout this thesis, although there existvarious other types of intervals such as open and half-open intervals. x and x areknown as the left and right end points, respectively, of the interval x in Eq. (1.1).

Fuzzy Arbitrary Order System: Fuzzy Fractional Differential Equations and Applications, First Edition.Snehashish Chakraverty, Smita Tapaswini, and Diptiranjan Behera.© 2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.

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2 FUZZY ARBITRARY ORDER SYSTEM

Let us now consider two arbitrary intervals x = [x, x ] and y = [y, y ]. These twointervals are said to be equal if they are in the same set. Mathematically, it onlyhappens when corresponding end points are equal. Hence, one may write

x = y if and only if x = y and x = y. (1.2)

For the given two arbitrary intervals x = [ x, x ] and y = [ y, y ], interval arithmeticoperations such as addition (+), subtraction (−), multiplication (×), and division (/)are defined as follows:

x + y = [x + y, x + y ], (1.3)

x − y = [ x − y, x − y ], (1.4)

x × y = [min S,max S], where S = {x × y, x × y, x × y, x × y}, (1.5)

and

x∕ y = [x, x ] ×

[1y,

1y

]if 0 ∉ y. (1.6)

Now if k is a real number and x = [x, x] is an interval, then the multiplication of themis given by

kx =

{[kx, kx], k < 0,

[kx, kx], k ≥ 0.(1.7)

Definition 1.2 Fuzzy Number A fuzzy number U is convex, normalized fuzzy setU of the real line R such that

{𝜇U(x) ∶ R → [0, 1], ∀x ∈ R},

where, 𝜇U is called the membership function of the fuzzy set, and it is piecewisecontinuous. There exists a variety of fuzzy numbers. But in this study, we have usedonly the triangular, trapezoidal, and Gaussian fuzzy numbers. So, we define thesethree fuzzy numbers as follows.

Definition 1.3 Triangular Fuzzy Number (TFN) A triangular fuzzy number(TFN) U is a convex, normalized fuzzy set U of the real line R such that

1. There exists exactly one x0 ∈ R with 𝜇U(x0) = 1 (x0 is called the mean valueof U), where 𝜇U is called the membership function of the fuzzy set.

2. 𝜇U(x) is piecewise continuous.

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PRELIMINARIES OF FUZZY SET THEORY 3

a b cx

Mem

bers

hip

valu

e

0.1

1

Figure 1.1 Triangular fuzzy number

Let us consider an arbitrary TFN U = (a, b, c). The membership function 𝜇U of Uis defined as follows:

𝜇U(x) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

0, x ≤ a,

x − ab − a

, a ≤ x ≤ b

c − xc − b

, b ≤ x ≤ c

0, x ≥ c.

The TFN U = (a, b, c) can be represented by an ordered pair of functions throughr-cut approach, namely [u(r), u(r)] = [(b − a)r + a,−(c − b)r + c] where, r ∈ [0, 1](Fig. 1.1).

Definition 1.4 Trapezoidal Fuzzy Number (TrFN) We consider an arbitrary trape-zoidal fuzzy number (TrFN) U = (a, b, c, d). The membership function 𝜇U of U isgiven as

𝜇U(x) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

0, x ≤ ax − ab − a

, a ≤ x ≤ b

1, b ≤ x ≤ c

d − xd − c

, c ≤ x ≤ d

0, x ≥ d.

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4 FUZZY ARBITRARY ORDER SYSTEM

a b

x

c d

Mem

bers

hip

valu

e

0

1

Figure 1.2 Trapezoidal fuzzy number

The TrFN U = (a, b, c, d) can be represented with an ordered pair of functionsthrough r-cut approach that is [u(r), u(r)] = [(b − a)r + a, −(d − c)r + d] where,r ∈ [0, 1] (Fig. 1.2).

Definition 1.5 Gaussian Fuzzy Number (GFN) Let us now define an arbitraryasymmetrical Gaussian fuzzy number, U = (𝛿, 𝜎l, 𝜎r). The membership function 𝜇Uof U will be as follows:

𝜇U(x) =⎧⎪⎨⎪⎩

exp

[−(x−𝛿)2

2𝜎2l

]for x ≤ 𝛿

exp[−(x−𝛿)2

2𝜎2r

]for x ≥ 𝛿

∀x ∈ R,

where, the modal value is denoted as 𝛿 and 𝜎l, 𝜎r denote the left-hand and right-handspreads (fuzziness), respectively, corresponding to the Gaussian distribution. Forsymmetric Gaussian fuzzy number, the left-hand and right-hand spreads are equal,that is, 𝜎l = 𝜎r = 𝜎. So the symmetric Gaussian fuzzy number may be writtenas U = (𝛿, 𝜎, 𝜎) and the corresponding membership function may be defined as𝜇U(x) = exp{−𝛽(x − 𝛿)2} ∀x ∈ R where 𝜆 = 1∕2𝜎2. The symmetric Gaussian fuzzynumber (Fig. 1.3) in parametric can be represented as

U = [u(r), u(r)] =

[𝛿 −

√−(loger)

𝜆, 𝛿 +

√−(loger)

𝜆

], where r ∈ [0, 1].

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PRELIMINARIES OF FUZZY SET THEORY 5

Mem

bers

hip

valu

e

0

1

q

x

σ σ

Figure 1.3 Gaussian fuzzy number

For all the aforementioned type of fuzzy numbers, the lower and upper bounds of thefuzzy numbers satisfy the following requirements:

(i) u(r) is a bounded left-continuous nondecreasing function over [0, 1];(ii) u(r) is a bounded right-continuous nonincreasing function over [0, 1];

(iii) u(r) ≤ u(r), 0 ≤ r ≤ 1.

Definition 1.6 Double Parametric Form of Fuzzy Number Using the r-cutapproach as discussed in Definitions 1.2–1.5 for all the fuzzy numbers, we haveU = [u(r), u(r)]. Now one may write this as crisp number with double parametricform as U(r, 𝛽) = 𝛽(u(r) − u(r)) + u(r) where r and 𝛽 ∈ [0, 1]. To obtain the lowerand upper bounds of the solution in single parametric form, we may use 𝛽 = 0 and1, respectively. This may be represented as U(r, 0) = u(r) and U(r, 1) = u(r).

Definition 1.7 Fuzzy Center Fuzzy center of an arbitrary fuzzy number u = [u(r),u(r)] is defined as u c= u(r)+u(r)

2, for all 0 ≤ r ≤ 1.

Definition 1.8 Fuzzy Radius Fuzzy radius of an arbitrary fuzzy number u = [u(r),u(r)] is defined as Δu = u(r)−u(r)

2for all 0 ≤ r ≤ 1.

Definition 1.9 Fuzzy Width Fuzzy space or width of an arbitrary fuzzy numberu = [u(r), u(r)] is defined as |u(r) − u(r)|, for all 0 ≤ r ≤ 1.

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6 FUZZY ARBITRARY ORDER SYSTEM

Definition 1.10 Fuzzy Arithmetic For any two arbitrary fuzzy numbers x = [x(r),x(r)], y = [y(r), y(r)] and scalar k, the fuzzy arithmetic is similar to the interval arith-metic defined as follows:

(i) x = y if and only if x(r) = y(r) and x(r) = y(r)(ii) x + y = [x(r) + y(r), x(r) + y(r)]

(iii) x × y = [min(S),max(S)] where

S = {x(r) × y(r), x(r) × y(r), x(r) × y(r), x(r) × y(r)}

(iv) kx ={[kx(r), kx(r)], k < 0[kx(r), kx(r)], k ≥ 0

(v) xy= [x(r), x(r)] ×

[1

y(r) ,1

y(r)

], where 0 ∉ y, where 0 ∉ y.

Definition 1.11 Let F ∶ (a, b) → RF and t0 = (a, b) (Khastan et al., 2011; Chalco-Cano and Roman-Flores, 2008). X is called differentiable at t0, if there existsF′(t0) ∈ RF such that

(i) for all h > 0 sufficiently close to 0, the Hukuhara difference F(t0 + h)ΘF(t0)and F(t0)ΘF(t0 − h) exists and (in metric D)

limh→0+

F(t0 + h)ΘF(t0)h

= limh→0+

F(t0)ΘF(t0 − h)h

= F′(t0),

or

(ii) for all h > 0 sufficiently close to 0, the Hukuhara difference F(t0)ΘF(t0 + h)and F(t0 − h)ΘF(t0) exists and (in metric D)

limh→0+

F(t0)ΘF(t0 + h)−h

= limh→0+

F(t0 − h)ΘF(t0)−h

= F′(t0)

Chalco-Cano and Roman-Flores (2008) used Definition 1.11 to obtain the follow-ing results.

Theorem 1.1 Let F ∶ (a, b) → RF and denote [F(t; r)] = [ f (t; r), f (t; r)] for eachr ∈ [0, 1].

(i) If F is differentiable of the first type (I), then f (t; r) and f (t; r) are differentiable

functions, and we have [F′(t; r)] = [ f ′(t; r), f ′(t; r)].

(ii) If F is differentiable of the second type (II), then f (t; r) and f (t; r) are differ-

entiable functions, and we have [F′(t; r)] = [ f ′(t; r), f ′(t; r)].

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PRELIMINARIES OF FUZZY SET THEORY 7

Proof The proof of the theorem is given in Chalco-Cano and Roman-Flores(2008).

BIBLIOGRAPHY

Chakraverty S. Mathematics of Uncertainty Modeling in the Analysis of Engineering and Sci-ence Problems. USA: IGI Global Publication; 2014.

Chalco-Cano Y, Roman-Flores H. On new solutions of fuzzy differential equations. ChaosSolitons Fract 2008;38:112–119.

Hanss M. Applied Fuzzy Arithmetic: An Introduction with Engineering Applications. Berlin:Springer-Verlag; 2005.

Jaulin L, Kieffer M, Didri OT, Walter E. Applied Interval Analysis. London: Springer; 2001.

Khastan A, Nieto JJ, Rodriguez-Lopez R. Variation of constant formula for first order fuzzydifferential equations. Fuzzy Sets Syst 2011;177:20–33.

Moore RE. Interval Analysis. Englewood Cliffs: Prentice Hall; 1966.

Ross TJ. Fuzzy Logic with Engineering Applications. New York: John Wiley & Sons; 2004.

Zimmermann HJ. Fuzzy Set Theory and Its Application. Boston/Dordrecht/London: KluwerAcademic Publishers; 2001.

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2BASICS OF FRACTIONAL AND FUZZYFRACTIONAL DIFFERENTIALEQUATIONS

We discussed in the previous chapter about fuzzy set theory and related definitionsand notations, which will be helpful for understanding the fuzzy and fuzzy fractionaldifferential equations. In this chapter, we present some preliminaries related to fuzzydifferential equations and fractional and fuzzy fractional differential equations, whichwill be used further in this book.

Definition 2.1 Fuzzy Initial Value Problem (FIVP) Let us consider the nth orderfuzzy initial value problem (FIVP)

y (n)(t) + an−1(t)y (n−1)(t) + · · · + a1(t)y ′(t) + a0(t)y(t) = g(t), (2.1)

where y is a fuzzy function of t, y (n)(t), y (n−1)(t),… , y ′(t), y(t) are the Hukahara fuzzyderivatives, ai(x), 0 ≤ i ≤ n − 1, continuous on some interval I, subject to fuzzy initialconditions

y(0) = b0, y′(0) = b1,… , y(n−1)(0) = bn−1.

Definition 2.2 Fuzzy Boundary Value Problem (FBVP) Accordingly, let us con-sider the n-th order FBVP as

y (n)(t; r) + an−1(t)y (n−1)(t; r) + · · · + a1(t)y′(t; r) + a0(t)y(t; r) = g(t; r), (2.2)

Fuzzy Arbitrary Order System: Fuzzy Fractional Differential Equations and Applications, First Edition.Snehashish Chakraverty, Smita Tapaswini, and Diptiranjan Behera.© 2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.

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10 FUZZY ARBITRARY ORDER SYSTEM

where ai(t), 0 ≤ i ≤ n − 1, is continuous on some interval I, subject to fuzzy boundaryconditions

y(a; r) =[𝛽(r), 𝛽(r)

], y(b; r) =

[𝛾(r), 𝛾(r)

],

and y(t; r) is the solution to be determined.

Lemma 2.1 If u(t) = (x(t), y(t), z(t)) is a fuzzy triangular number valued functionand if u is Hukuhara differentiable (Bede, 2008), then u′ = (x′, y′, z′).

By using Hukuhara differentiable, we intend to solve the FIVP

x ′ = f (t, x), (2.3)

subject to triangular fuzzy initial condition

x(t0) = x0,

where, x0 = (x0, xc0, x0) ∈ R, x(t) = (u, uc, u) ∈ R, and

f ∶[t0, t0 + a

]× R → R, f

(t,(u, uc, u

))=(

f(t, u, uc, u

), f c

(t, u, uc, u

), f

(t, u, uc, u

)).

We can translate this into the following system of ordinary differential equations:

⎧⎪⎪⎨⎪⎪⎩

u = f (t, u, uc, u)uc = f c(t, u, uc, u)u = f (t, u, uc, u)u(0) = x0, u

c(0) = xc0, u(0) = x0.

(2.4)

Definition 2.3 Riemann–Liouville Fractional Integral (Podlubny, 1999) TheRiemann–Liouville integral operator J𝛼 of order 𝛼 > 0 is defined as

J𝛼f (t) = 1Γ(𝛼)

t

∫0

(t − 𝜏)𝛼−1f (𝜏)d𝜏, t > 0. (2.5)

Definition 2.4 Fuzzy Riemann–Liouville Fractional Integral (Mazandarani andKamyad, 2013) The Riemann–Liouville fractional integral of order 𝛼 of the fuzzynumber valued function f , based on its r-cut representations, can be expressed as[

J𝛼 f (t; r)]=[J𝛼f (t; r), J𝛼f (t; r)

], t > 0,

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BASICS OF FRACTIONAL AND FUZZY FRACTIONAL DIFFERENTIAL EQUATIONS 11

where

J𝛼f (t) = 1Γ(𝛼)

t

∫0

(t − 𝜏)𝛼−1f (𝜏)d𝜏, t > 0,

J𝛼f (t) = 1Γ(𝛼)

t

∫0

(t − 𝜏)𝛼−1f (𝜏)d𝜏, t > 0.

Definition 2.5 Caputo Derivative (Podlubny, 1999) The fractional derivative off (t) in the Caputo sense is defined as follows:

D𝛼f (t) = Jm−𝛼Dmf (t) =

⎧⎪⎪⎨⎪⎪⎩

1Γ(m − 𝛼)

t

∫0

f (m)(𝜏)d𝜏(t − 𝜏)𝛼+1−m

, m − 1 < 𝛼 < m, m ∈ N

dm

dtmf (t), 𝛼 = m, m ∈ N.

(2.6)Some basic properties of the fractional operator are as follows:

(i) J𝛼J𝛾 f (t) = J𝛼+𝛾 f (t), 𝛼, 𝛾 ≥ 0

(ii) J𝛼(t𝛾 ) ={

Γ(𝛾+1)t𝛼+𝛾Γ(𝛼+𝛾+1) , 𝛼 > 0, 𝛾 > −1, t > 0.

Definition 2.6 Caputo-Type Fuzzy Fractional Derivatives (Mazandarani andKamyad, 2013) Let f (t; r) be a fuzzy valued function and [ f (t; r)] = [ f (t; r), f (t; r)],for r ∈ [0, 1], 0 < 𝛼 < 1, and t ∈ (a, b).

(a) If f (t; r) is a Caputo-type fuzzy fractional differentiable function in the firstform, then [

D𝛼 f (t; r)]=[D𝛼f (t; r),D𝛼f (t; r)

].

(b) If f (t; r) is a Caputo-type fuzzy fractional differentiable function in the secondform, then [

D𝛼 f (t; r)]=[D𝛼f (t; r),D𝛼f (t; r)

].

where,

D𝛼f (t; r) = 1Γ(m − 𝛼)

t

∫0

f (m)(𝜏)d𝜏

(t − 𝜏)𝛼+1−m, m − 1 < 𝛼 < m, m ∈ N,

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12 FUZZY ARBITRARY ORDER SYSTEM

D𝛼f (t; r) = 1Γ(m − 𝛼)

t

∫0

f(m)

(𝜏)d𝜏(t − 𝜏)𝛼+1−m

, m − 1 < 𝛼 < m, m ∈ N,

dm

dtmf (t), 𝛼 = m, m ∈ N.

Definition 2.7 Fractional Initial Value Problem (Podlubny, 1999) Let us considerthe following fractional initial value problem (FrIVP):

D𝛼y(t) = f (t, y), (2.7)

subject to the initial condition

y(0) = y0, t ∈ [a, b] , 𝛼 ∈ (0, 1),

where D𝛼 denotes the Caputo fractional differential operator.Next, we combine differential equations of fractional order and with uncertainty,

to consider a new type of dynamical system, that is, fuzzy differential equations offractional order.

Definition 2.8 Fuzzy Fractional Initial Value Problem (Mazandarani andKamyad, 2013) Let us consider the following fuzzy fractional initial value problem(FFIVP):

D𝛼 y(t) = f (t, y), (2.8)

subject to the fuzzy initial condition

y(0) = y0, t ∈ [a, b], 𝛼 ∈ (0, 1).

The FFIVP (2.8) can be considered equivalent by the following initial value problems:

[D𝛼y(t),D𝛼y(t)

]=[f (t, y), f (t, y)

],

subject to the fuzzy initial condition

[y(0), y(0)

]=[y

0, y0

].

BIBLIOGRAPHY

Agarwal RP, Lakshmikantham V, Nieto JJ. On the concept of solution for fractional differentialequations with uncertainty. Nonlinear Anal 2010;72:2859–2862.

Allahviranloo T, Salahshour S, Abbasbandy S. Explicit solutions of fractional differentialequations with uncertainty. Soft Comput 2012;16:297–302.