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SOLUTION OF HIGHER ORDERDIFFERENTIAL EQUATIONS WITHFUZZY INITIAL AND BOUNDARY

CONDITIONS

A Thesis submitted to Gujarat Technological University

for the award of

Doctor of Philosophyin

Science - Mathematicsby

Patel Komalben RameshbhaiEnrollment No.: 149997673006

under supervision of

Dr. Narendrasinh B. Desai

GUJARAT TECHNOLOGICAL UNIVERSITY

AHMEDABAD

SEPTEMBER 2020

[email protected]

Faculty Web Site URL Here (include http://)

SOLUTION OF HIGHER ORDERDIFFERENTIAL EQUATIONS WITHFUZZY INITIAL AND BOUNDARY

CONDITIONS

A Thesis submitted to Gujarat Technological University

for the award of

Doctor of Philosophyin

Science - Mathematicsby

Patel Komalben RameshbhaiEnrollment No.: 149997673006

under supervision of


GUJARAT TECHNOLOGICAL UNIVERSITY

AHMEDABAD

SEPTEMBER 2020

c©Patel Komalben Rameshbhai

ii

DeclarationI declare that the thesis entitled "Solution of Higher Order Differential Equations

with Fuzzy Initial and Boundary Conditions" submitted by me for the degree of

Doctor of Philosophy is the record of research work carried out by me during the period

from March 2015 to September 2019 under the supervision of Dr. Narendrasinh B.

Desai, Head of Mathematics Department, A. D. Patel Institute of Technology, New V. V.

Nagar, Anand, and this has not formed the basis for the award of any degree, diploma,

associateship, fellowship, titles in this or any other university or other institution of

higher learning. I further declare that the material obtained from other sources has been

duly acknowledged in the thesis. I shall be solely responsible for any plagiarism or

other irregularities, if noticed in the thesis.

Signature of the Research Scholar:

Date: 23/09/2020

Name of Research Scholar: Patel Komalben RameshbhaiPlace: Vadodara.

iii

CertificateI certify that the work incorporated in the thesis "Solution of Higher Order

Differential Equations with Fuzzy Initial and Boundary Conditions" submitted by

Patel Komalben Rameshbhai was carried out by the candidate under my guidance.

To the best of my knowledge: (i) the candidate has not submitted the same research

work to any other institution for any degree, diploma, associateship, fellowship or

other similar titles; (ii) the thesis submitted is a record of original research work done

by Research Scholar during the period of study under my supervision; and (iii) the

thesis represents independent research work on the part of the Research Scholar.

Signature of Supervisor:

Date: 23/09/2020

Name of Supervisor: Dr. Narendrasinh B. DesaiPlace: New V. V. Nagar.

iv

Course-work Completion CertificateThis is to certify that Patel Komalben Rameshbhai, enrolment no. 149997673006 is

a PhD scholar enrolled for PhD program in the branch Science-Mathematics of

Gujarat Technological University, Ahmedabad.

(Please tick the relevant option(s))

f She has been exempted from the course-work (successfully completed during

M.Phil Course)

f She has been exempted from Research Methodology Course only (successfully

completed during M.Phil Course)

f! She has successfully completed the PhD course work for the partial requirement

for the award of PhD Degree. Her performance in the course work is as follows-

Grade Obtained in Research Methodology Grade Obtained in Self Study Course (Core Subject)

(PH001) (PH002)

BC AB

Supervisor’s Sign:

Name of Supervisor: Dr. Narendrasinh B. Desai

v

Originality Report CertificateIt is certified that PhD Thesis entitled "Solution of Higher Order Differential

Equations with Fuzzy Initial and Boundary Conditions" by Patel Komalben

Rameshbhai has been examined by us. We undertake the following:

a. Thesis has significant new work / knowledge as compared to already published or

are under consideration to be published elsewhere. No sentence, equation, diagram,

table, paragraph or section has been copied verbatim from previous work unless it is

placed under quotation marks and duly referenced.

b. The work presented is original and own work of the author (i.e. there is no

plagiarism). No ideas, processes, results or words of others have been presented as

author’s own work.

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d. There is no falsification by manipulating research materials, equipment or

processes, or changing or omitting data or results such that the research is not

accurately represented in the research record.

e. The thesis has been checked using Turnitin (copy of originality report attached)

and found within limits as per GTU Plagiarism Policy and instructions issued from

time to time (i.e. permitted similarity index ≤ 10%).

Signature of the Research Scholar: Date: 23/09/2020

Name of Research Scholar: Patel Komalben RameshbhaiPlace: Vadodara.

Signature of Supervisor: Date: 23/09/2020

Name of Supervisor: Dr. Narendrasinh B. DesaiPlace: New V. V. Nagar.

vi

PhD THESIS Non-Exclusive License to

GUJARAT TECHNOLOGICAL

UNIVERSITYIn consideration of being a PhD Research Scholar at GTU and in the interests of the

facilitation of research at GTU and elsewhere, I, Patel Komalben Rameshbhai, having

enrolment no. 149997673006, hereby grant a non-exclusive, royalty free and perpetual

license to GTU on the following terms:

a). GTU is permitted to archive, reproduce and distribute my thesis, in whole or in

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anywhere in the world, for non-commercial purposes, in all forms of media;

b). GTU is permitted to authorize, sub-lease, sub-contract or procure any of the acts

mentioned in paragraph (a);

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the authority of their "Thesis Non-Exclusive License";

d). The Universal Copyright Notice ( c©) shall appear on all copies made under the

authority of this license;

e). I undertake to submit my thesis, through my University, to any Library and

Archives. Any abstract submitted with the thesis will be considered to form part of

the thesis.

f). I represent that my thesis is my original work, does not infringe any rights of others,

including privacy rights, and that I have the right to make the grant conferred by this

non-exclusive license.

g). If third party copyrighted material was included in my thesis for which, under the

terms of the Copyright Act, written permission from the copyright owners is

required, I have obtained such permission from the copyright owners to do the acts

mentioned in paragraph (a) above for the full term of copyright protection.

viii

h). I retain copyright ownership and moral rights in my thesis, and may deal with the

copyright in my thesis, in any way consistent with rights granted by me to my

University in this non-exclusive license.

i). I further promise to inform any person to whom I may hereafter assign or license

my copyright in my thesis of the rights granted by me to my University in this

non-exclusive license.

j). I am aware of and agree to accept the conditions and regulations of PhD including

all policy matters related to authorship and plagiarism.

Signature of the Research Scholar:

Name of Research Scholar:Patel Komalben RameshbhaiDate: 23/09/2020 Place: Vadodara.

Signature of Supervisor:

Name of Supervisor: Dr. Narendrasinh B. DesaiDate: 23/09/2020 Place: New V. V. Nagar.

Seal:

ix

Thesis Approval FormThe viva-voce of the PhD Thesis submitted by Patel Komalben Rameshbhai

(enrolment no.149997673006) entitled "Solution of Higher Order Differential

Equations with Fuzzy Initial and Boundary Conditions" was conducted on

Wednesday, 23 September 2020 at Gujarat Technological University.

(Please tick any one of the following option)

f! The performance of the candidate was satisfactory. We recommend that she be

awarded the PhD degree.

f Any further modifications in research work recommended by the panel after 3

months from the date of first viva-voce upon request of the Supervisor or request

of Independent Research Scholar after which viva-voce can be re-conducted by

the same panel again.

(briefly specify the modifications suggested by the panel)

f The performance of the candidate was unsatisfactory. We recommend that he/she

should not be awarded the PhD degree.

(The panel must give justifications for rejecting the research work)


Dr. Pragnesh Gajjar

Name and Signature of Supervisor with Seal 1)(External Examiner:1) Name and Signature

Dr. Shrikant Chaudhari2)(External Examiner:2) Name and Signature 3)(External Examiner:3) Name and Signature

x

AbstractThe study of the solution of higher-order differential equations(DEs) with fuzzy initial

and boundary conditions(FIBCs) and the development of the solution methods for

higher-order differential equations are the best practical solutions for real-life

problems. The solution of higher-order DEs with FIBCs is one of the most important

concepts in many areas of applied sciences and engineering such as, mechanical

engineering, civil engineering, electrical engineering, electrical and communication

engineering, environmental engineering, chemical engineering, medical science, etc.

In recent years, extensive research have been carried out to study the solution of

higher-order DEs with FIBCs. This field has achieved extensive attention due to its

broad range of applications in the field of science, engineering and real-life practical

problems. In particular, finding the solution of higher-order differential equation with

fuzzy initial and boundary conditions is a central problem in the various fields of

engineering and sciences. The scope of the present study lies in increasing importance

of the solution of higher-order fuzzy differential equations(FDEs) as well as fuzziness

in the forcing function. Thus, an investigation of such research area leads directly or

indirectly to studying another research area far away from the first in terms of physical

context, yet related through the fundamental principles of the ordinary differential

equation(ODE) that involved fuzziness in initial and boundary conditions. Due to the

vast scope of application of DEs in the fields of engineering and sciences, the specific

problems are almost unlimited in fields that involved fuzziness and therefore it is

reasonable to select such types of problems for discussion here. Accordingly, a

selection of more interesting problems of current interest has been made for

mathematical treatment in the thesis. The problems investigated in the present study

are concerned with the application of a solution of higher-order DEs that contain

fuzziness in initial and boundary conditions as well as in forcing function that are

occurring in different fields of engineering and sciences.

This work has been devoted to the study of higher-order DEs with FIBCs by using

suitable methods. Here, we have discussed fuzzy dynamic problems like the bending of

an elastic beam under the constant load in civil engineering, Ariy’s non-homogeneous

xi

differential equation in optics and the simple pendulum problem in physics, the mass-

spring system in mechanical engineering, etc. The mathematical problem of different

physical phenomenon gives us one-dimensional differential equations with fuzzy initial

and boundary conditions. This Ph.D. work would be useful for solving fuzzy dynamic

problems arising in different fields of engineering and sciences.

xii

Acknowledgement

It is a matter of tremendous pleasure for me to submit my Ph.D. thesis entitled "Solutionof Higher Order Differential Equations with Fuzzy Initial and Boundary Conditions" inthe subject of Mathematics. My registration was done in the year 2015 for carrying outthe work related to the subject which seemed to me to be herculean task ab initio butwith the passing of time, everything seemed to be within the reach by god’s grace.

I feel highly indebted to my guide Dr. Narendrasinh B. Desai, Head, Departmentof Mathematics, A. D. Patel Institute of Technology, New V. V. Nagar, Anand. He hasbeen a consistent supporter during the course of my work. He has been the real guidingforce behind this work who inspired me to go on with undaunted courage. My DoctorateProgress Committee members also guided me through all these years. I would like tothank Dr. J. M. Dhodiya, Associate Professor, SVNIT, Surat, and Dr. M. S. Joshi,Associate Professor, C. K. Pithawala College of Engineering & Technology, Surat, whohave reviewed my research work time to time and given the valuable suggestion in myresearch work.

During the process of preparation of my thesis, I have referred many books andresearch papers for which I personally thank the authors who provided me a deep insightand inspiration for the solution of problems.

I sincerely thank my Director Dr. K baba Pai and my Dean Academics Dr. S. K. Vijand Dr.Som Sahani Head, Mathematics Department, who have allowed and cooperatedto do this research work as a part-time candidate along with my job responsibilities. Iwould like to thanks to my Ph.D. colleagues like Dr. D. J. Prajapati, Dr. M. A. Pateland Mr. Ujjaval Trivedi, AHOD of Mechanical department as well as all the facultymembers of the Mathematics department of ITM(SLS) Baroda University, Vadodara.

No value is more precious than the value of the family hence, I sincerely thank myin-laws, my husband Hiren Patel and my son Kavya Patel for the great help and constantinspiration throughout my work. They have done everything they could to make me feelrelaxed and inspired.

At the end, I would like to appreciate all the help and support extended to meduring this journey.

Komal R. Patel

xiii

Contents

Abstract xi

Acknowledgement xiii

List of Abbreviations xvii

List of Symbols xviii

List of Figures xx

List of Tables xxii

1 General Introduction 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Brief description on the state of the art of the research topic . . . . . . . 31.4 Definition of the problem . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Objective and future scope of work . . . . . . . . . . . . . . . . . . . . 51.6 Original contribution by the thesis . . . . . . . . . . . . . . . . . . . . 61.7 Methodology of Research and Results/Comparisons . . . . . . . . . . . 61.8 Achievements with respect to objectives . . . . . . . . . . . . . . . . . 71.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Fundamentals of Fuzzy Set Theory and Basic Concepts 92.1 History of development of fuzzy set theory . . . . . . . . . . . . . . . . 92.2 Crisp set vs. Fuzzy set . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Crisp set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 Fuzzy set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Standard operations on fuzzy sets: . . . . . . . . . . . . . . . . . . . . 112.3.1 Logical connectives . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Fuzzy set and Membership function . . . . . . . . . . . . . . . . . . . 132.4.1 Membership function . . . . . . . . . . . . . . . . . . . . . . . 13

2.5 Properties of Membership function . . . . . . . . . . . . . . . . . . . . 132.5.1 Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.5.2 Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.5.3 Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.5.4 Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.5.5 Normal fuzzy set . . . . . . . . . . . . . . . . . . . . . . . . . 152.5.6 Subnormal fuzzy set . . . . . . . . . . . . . . . . . . . . . . . 162.5.7 Convex fuzzy set . . . . . . . . . . . . . . . . . . . . . . . . . 16

xiv

Contents xv

2.5.8 r-cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.6 Fuzzy number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.6.1 Triangular fuzzy number . . . . . . . . . . . . . . . . . . . . . 182.6.2 Trapezoidal fuzzy number . . . . . . . . . . . . . . . . . . . . 192.6.3 Gaussian fuzzy number . . . . . . . . . . . . . . . . . . . . . . 20

2.7 Fuzzy arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.8 Zadeh’s extension principle . . . . . . . . . . . . . . . . . . . . . . . . 212.9 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.10 Hukuhara difference . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.11 Hukuhara differential . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.12 Generalized Hukuhara differentiability . . . . . . . . . . . . . . . . . . 232.13 Fuzzy Reimann integration . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Solutions of nth order Differential Equations with Fuzzy Initial Conditionsby using the Properties of Linear Transformations 253.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Linear transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.1 Wronskain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.4 Method of solving nth order differential equation with fuzzy initial

conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.5 Solution of a higher order differential equation with fuzzy initial condition 30

3.5.1 Solution of a higher order constant coefficient differentialequation with fuzzy initial condition . . . . . . . . . . . . . . . 30

3.5.2 Solution of higher order Cauchy-Euler’s variable coefficientdifferential equations with fuzzy initial conditions . . . . . . . 37

3.6 Result and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4 Solutions of the nth order Differential Equations with Fuzzy Initial andBoundary Conditions by using the Gauss Elimination Method 454.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.3 Method for solving the nth order differential equation with FBCs . . . . 464.4 Solutions of higher order differential equations with fuzzy initial and

boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.4.1 Solutions of a second order differential equation with fuzzy

boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 554.4.2 Solution of fourth order differential equation with fuzzy initial

and boundary conditions . . . . . . . . . . . . . . . . . . . . . 594.4.2.1 Example based on the application of differential

equation in civil engineering . . . . . . . . . . . . . 594.5 Result and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Contents xvi

5 Solutions of Higher Order Differential Equations with Fuzzy BoundaryConditions by Finite Difference Method 665.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.3 Method for solving higher order differential equation with FBCs . . . . 705.4 Solution of the higher order differential equation with fuzzy boundary

conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.4.1 Solution of the second order differential equation with fuzzy

boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 725.4.2 Airy’s functions . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.4.2.1 Solution of Airy’s non-homogeneous differentialequation with fuzzy boundary conditions . . . . . . . 77

5.5 Result and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6 Methods for solving Higher Order Differential Equations with Fuzzy Initialand Boundary Conditions by using Laplace Transforms 856.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.2 Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.2.1 Conditions for the existence of Laplace transforms . . . . . . . 866.3 Strongly generalized Hukuhara differentiability . . . . . . . . . . . . . 866.4 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.5 Method for solving higher order differential equations with fuzzy initial

conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.6 Solution of higher order differential equations with fuzzy initial and

boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.6.1 Solution of second order constant coefficient differential

equations with fuzzy initial conditions . . . . . . . . . . . . . . 906.6.1.1 An example based on the application of differential

equation in mechanical engineering . . . . . . . . . . 906.6.1.2 Example based on the application of differential

equation in physics . . . . . . . . . . . . . . . . . . 946.6.2 Solution of second order variable coefficient differential

equations with fuzzy initial and boundary conditions . . . . . . 986.7 Result and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

References 104

List of Publications 109

List of Abbreviations

Acronym What (it) Stands ForDE Differential EquationODE Ordinary Differential EquationFDE Fuzzy Differential EquationMF Membership FunctionTMF Triangular Membership FunctionTraMF Trarapezoidal Membership FunctionGMF Gaussian Membership FunctionTFN Triangular Fuzzy NumberTraFN Trarapezoidal Fuzzy NumberGFN Gaussian Fuzzy NumberHD Hukuhara DifferentialGHD Generalized Hukuhara DifferentiabilitySGHD Strongly Generalized Hukuhara DifferentiabilityIVP Initial Value ProblemBVP Boundary Value ProblemFIVP Fuzzy Initial Value ProblemFBVP Fuzzy Boundary Value ProblemFIC Fuzzy Initial ConditionFBC Fuzzy Boundary ConditionFIBC Fuzzy Initial and Boundary ConditionLT Laplace TransformFLT Fuzzy Laplace TransformFRI Fuzzy Reimann IntegrableGEM Gauss Elimination MethodFDM Finite Difference Method

xvii

List of Symbols

symbol nameA crisp setχ characteristic function for crisp setIR set of real numberA fuzzy setE set of real fuzzy numberµ membership function for fuzzy set∨ or ∪ Disjunction or Union∧ or ∩ Conjunction or IntersectionAr r - cutAr+ strong r - cut Hukuhara differenceα Modal valueσl left fuzzinessσr right fuzziness∃ there exist at least∈ member of6= not equal to∀ all elements of the setW WronskainC(n−1)(−∞,∞) the set of n−1 time continuous functionT Linear Transformationy lower-bound of solutiony upper-bound of solutionn number of equal size sub-intervalh step sizeL Laplace transformL length of bridgeE Younge’s modulus of elasticity of material of bridgeI beam cross sectional moment of inertia about horizontal axisEI measurement of beam stiffnessw0 constant loadm mass of springb damping coefficientk spring constant

xviii

List of Symbols xix

θ displacement functionl length of pendulumm1 mass of bobλ constantY exact solutiony approximate solutiony′

first order derivativey′′

second order derivativey′′′

third order derivativeyiv fourth order derivativeyn nth order derivativeα, β fuzzy boundary valuesγk,σak,σbk,δk,ζk,ξk parameters used for matrices entriesζ+,ζ−,ζ ,η parameters used in Taylor’s series

List of Figures

2.1 Lotfi Zadeh [https://www.slideserve.com/tambre/lotfi-zadeh] . . . . . . 92.2 Crisp set vs. Fuzzy set . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 A∧ B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.6 A∨ B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.7 A

′. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.8 Core, support, and boundaries of a fuzzy set . . . . . . . . . . . . . . . 152.9 Normal fuzzy set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.10 Subnormal fuzzy set . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.11 Convex fuzzy set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.12 Triangular fuzzy number . . . . . . . . . . . . . . . . . . . . . . . . . 192.13 Trapezoidal fuzzy number . . . . . . . . . . . . . . . . . . . . . . . . 202.14 Gaussian fuzzy number . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1 The fuzzy solution obtained by the properties of linear transformations . 313.2 The fuzzy solution obtained by the properties of linear transformations

and the red line represents the crisp solution . . . . . . . . . . . . . . . 323.3 The fuzzy solution obtained by the properties of linear transformation . 343.4 The fuzzy solution obtained by the properties of linear transformations,

and the red line represents the crisp solution . . . . . . . . . . . . . . . 363.5 If t increases, fuzziness disappears . . . . . . . . . . . . . . . . . . . . 363.6 The fuzzy solution obtained by the properties of linear transformations . 383.7 The fuzzy solution obtained by the properties of linear transformations,

The red line represents the crisp solution . . . . . . . . . . . . . . . . . 383.8 The fuzzy solution obtained by properties of linear transformation . . . 403.9 The fuzzy solution obtained by properties of linear transformation, The

red line represents the crisp solution . . . . . . . . . . . . . . . . . . . 413.10 If we increase t fuzziness increases . . . . . . . . . . . . . . . . . . . . 413.11 Mass tapped periodically with a hammer . . . . . . . . . . . . . . . . . 423.12 The fuzzy solution obtained by the properties of linear transformation,

The red line represents the crisp solution . . . . . . . . . . . . . . . . . 43

4.1 The lower-bound and upper-bound for exact solution . . . . . . . . . . 564.2 The lower-bound and upper-bound for approximate solution . . . . . . 594.3 Elastic or Deflection curve . . . . . . . . . . . . . . . . . . . . . . . . 604.4 The lower-bound and upper-bound for exact solution . . . . . . . . . . 614.5 The lower-bound and upper-bound for approximate solution . . . . . . 64

5.1 The lower-bounds and the upper-bounds for an approximate solution . . 765.2 The lower-bounds and the upper-bounds for an approximate solution . . 83

xx

List of Figures xxi

6.1 The mass-spring system . . . . . . . . . . . . . . . . . . . . . . . . . . 916.2 The lower-bound and upper-bound of solution at r = 0 . . . . . . . . . . 926.3 The lower-bound and upper-bound of solution at r = 0.5 . . . . . . . . . 936.4 The lower-bound and upper-bound of solution at r = 0.8 . . . . . . . . . 936.5 The lower-bound and upper-bound of solution at r = 0.9 . . . . . . . . . 936.6 The lower-bound and upper-bound of solution at r = 1 . . . . . . . . . . 936.7 The simple pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . 956.8 The lower-bound and upper-bound of solution at r = 0 . . . . . . . . . . 966.9 The lower-bound and upper-bound of solution at r = 0.5 . . . . . . . . . 976.10 The lower-bound and upper-bound of solution at r = 0.8 . . . . . . . . . 976.11 The lower-bound and upper-bound of solution at r = 0.9 . . . . . . . . . 976.12 The lower-bound and upper-bound of solution at r = 1 . . . . . . . . . . 976.13 Solution by considering parameters TFN . . . . . . . . . . . . . . . . . 986.14 Solution by considering parameters TraFN . . . . . . . . . . . . . . . . 996.15 Solution by considering parameters GFN . . . . . . . . . . . . . . . . . 1006.16 Solution by considering parameters TFN . . . . . . . . . . . . . . . . . 1016.17 Solution by considering parameters TraFN . . . . . . . . . . . . . . . . 1026.18 Solution by considering parameters GFN . . . . . . . . . . . . . . . . . 102

List of Tables

4.1 The exact values for lower-bounds. . . . . . . . . . . . . . . . . . . . . 564.2 The exact values for upper-bounds. . . . . . . . . . . . . . . . . . . . . 564.3 The approximate values for lower-bounds. . . . . . . . . . . . . . . . . 584.4 The approximate values for upper-bounds. . . . . . . . . . . . . . . . . 584.5 The exact values for lower-bounds. . . . . . . . . . . . . . . . . . . . . 614.6 The exact values for upper-bounds. . . . . . . . . . . . . . . . . . . . . 614.7 The approximate values for lower-bounds. . . . . . . . . . . . . . . . . 634.8 The approximate values for upper-bounds. . . . . . . . . . . . . . . . . 64

5.1 Approximate values for the lower-bounds. . . . . . . . . . . . . . . . . 755.2 Approximate values for the upper-bounds. . . . . . . . . . . . . . . . . 765.3 Approximate values for the lower-bounds. . . . . . . . . . . . . . . . . 825.4 Approximate values for the upper-bounds. . . . . . . . . . . . . . . . . 82

xxii

Dedicated to my son Kavya .........

xxiii

CHAPTER 1

General Introduction

1.1 Introduction

This chapter deals with the introductory nature and general introduction of the subject

matter of the thesis viz, the structure of the thesis, problem definition, objective and

future scope of work, original contribution by the thesis, methodology of research and

results, achievements with respect to objectives and conclusion.

Due to the vast scope of applications of the differential equations in the fields of

science, engineering, and technology, the specific problems are almost unlimited in a

different field that involves fuzziness and therefore it is reasonable to select such types

of problems for discussion here. Accordingly, the most interesting problems of current

interest have been used for mathematical treatment in the thesis. The explored problems

of the present study are related to the application of the solution of higher-order DEs that

contain fuzziness in initial and boundary conditions as well as in the forcing function.

Here, we are going to discuss fuzzy dynamic problems like the mass-spring system in

mechanical engineering, the bending of an elastic beam under the constant load in civil

engineering, the simple pendulum and Ariy’s non-homogeneous DE in physics(i.e.in

optics), etc. The mathematical problem of different physical phenomenon gives us a

one-dimensional DE with FIBCs. This Ph.D. work would be useful for solving fuzzy

dynamic problems arising in different fields of science, engineering, and technology.

1

General Introduction 2

1.2 Structure of the thesis

The thesis consists of 6 main chapters. After these introductory words, Chapter TWO

includes a brief discussion on certain relevant topics, like fuzzy set, membership

function, fuzzy number, fuzzy arithmetic, Zadeh’s extension principle, continuity,

Hukuhara difference, Hukuhara differential, generalized Hukuhara differentiability,

fuzzy Reimann integration and related literature review.

Chapter THREE deals with the solution of nth order DEs with constant coefficients

and Cauchy-Euler’s DEs with variable coefficients with fuzzy initial conditions(FICs)

by using properties of the linear transformation. We also solved one application-level

problem related to physics i.e. the study of motion of mass when the mass tapped

periodically with a hammer.

Chapter FOUR deals with the solution of nth order DEs with FIBCs by Gauss

elimination method(GEM) by using MATLAB and in comparing the exact and

approximate solutions. In this chapter, we solved a problem-related to civil

engineering - the bending of an elastic beam under the constant load.

Chapter FIVE deals with the solution of higher-order DEs with fuzzy boundary

conditions(FBCs) whose exact solution is difficult to find and interpret, that type of DEs

are solved by using the finite difference method(FDM). Here, we solve the differential

equation whose solution contains Airy’s function. The problems have been studied

numerically and graphically with the help of MATLAB.

Chapter SIX deals with the solution of some higher-order DEs with

constant-coefficients and variable coefficients (not a Cauchy-Euler’s) DEs with FIBCs

by using Laplace transform(LT). In this chapter, we solved problems related to the

mass-spring system in mechanical engineering branch and the simple pendulum

problem related to physics.

Some of the results/solutions reported here are also published in scholarly, peer-

reviewed and indexed journals as well as presented in the conference.

Brief description on the state of the art of the research topic 3

The references are given alphabetically at the end.

1.3 Brief description on the state of the art of the

research topic

Lotfi Zadeh introduced the concept of fuzzy set theory and pointed out ambiguities and

impreciseness in his seminal paper entitled "Fuzzy Sets" [80] in 1965. Here, we are

going to study the dynamic problems with fuzzy parameters which occurs in different

fields like mechanical engineering, civil engineering, physics, etc. All these branches

of sciences and engineering have contributed to a vast amount of literature on this topic.

The notion of a fuzzy derivative was first introduced by Chang and Zadeh [25], it

was reviewed by Dubois and Prade [31] who used the extension principle [79] in their

approach. Other fuzzy derivative concepts were introduced by Puri and Ralescu [69]

and Goetschel and Voxman [40] as an extension of HD for multivalued functions.

Kandel and Byatt [48] used the concept of FDE for the analysis of fuzzy dynamical

problems. The FDEs were discussed by Kaleva in [44, 46] and "The Cauchy problem

for fuzzy differential equations" was studied by Kaleva in [45]. Seikkala [74]

discussed "On the fuzzy initial value problems". Buckley and Feuring [22] approached

to "Fuzzy initial value problem for nth order linear differential equations". "The

Cauchy problem for continuous fuzzy differential equations" was discussed by Nieto

[59]. Mondal et al. [58] presented the solution technique for the "First order linear non

homogeneous ordinary differential equation in fuzzy environment based on Laplace

transform". "Existence of the solutions of fuzzy differential equations with

parameters" was discussed by Ding et al. [30]. Existence and uniqueness of solution to

FBVP have been proved by Esfahani et al. [32]. "Existence and uniqueness of

solutions to Cauchy problem of fuzzy differential equations" have been discussed by

Song et al. [75]. Lakshikantham et al. [55] worked on "Interconnection between set

and FDEs" and on the solution of "Two-point boundary value problem associated with

the non-linear fuzzy differential equation" [53].


The idea of GHD was employed by Bede and Gal [18, 20] to investigate

first-order linear FDEs. The "Remark on the new solution of FDEs" was given by Bede

and Gal [19]. GHD was recommended by Chalco-Cano and Romon Flores [24] for

solving FDEs. "First order linear FDEs under generalized differentiability" was

discussed by Bede et al. [21]. Allahviranloo et al. [10] found solution of "nth order

fuzzy linear differential equations". Allahviranlloo et al. [8] obtained "A new method

for solving fuzzy linear differential equations". The existence and uniqueness of

solutions of second-order FDEs has proposed by Allahviranloo et al. [12]. The

existence and uniqueness theorem of the solution of nth order FDE and the integral

form of nth order FDE were studied by Salahshour [71] under GHD. Salahshour and

Allahviranloo [72] worked on "Application of fuzzy Laplace Transform". Xiaobin et

al. [78] found "Fuzzy approximate solutions of second-order fuzzy linear boundary

value problems". "A boundary value problem for second-order fuzzy differential

equations" was studied by Khastan et al. [50]. "A new result on multiple solutions for

nth order fuzzy differential equation under generalized differentiability" has been

suggested by Bahrami et al. [15]. Ahmad et al. [4] studied analytic and numerical

solution of the FDEs based on the extension principle. Ivaz et al. [42] investigated "A

numerical algorithm for the solution of first-order fuzzy differential equations and

hybrid fuzzy differential equations". "Variation of constant formula for first-order

FDEs" has been presented by Khastan et al. [49]. Akin et al. [6] developed "An

algorithm for the solution of second-order FIVPs".

Mansouri et al. [57] offered "A numerical method for solving nth order FDEs by

using characterization theorem". "Numerical solutions of fuzzy differential and

integral equations" were proposed by Friedman et al. [33]. "A new fuzzy version of

Euler’s method for solving differential equations with fuzzy initial values" were

introduced by Ahmad and Hasan [3]. "Homotopy perturbation method for solving nth

order fuzzy linear differential equations" was implemented by Tapaswini and

Chakraverty [76]. "The variational iteration method for solving nth order FDEs." was

discussed by Jafari et al. [43]. The different methods for solving nth order FDEs were

discussed in [5, 10, 11, 15, 66, 71].

Definition of the problem 5

1.4 Definition of the problem

In the present study different dynamic problems with FIBCs that occur in many fields

of engineering, science and technology are discussed. Here, we discuss dynamic

problems with fuzzy parameters related to mechanical engineering, civil engineering,

and physics, etc by using a different method and compare the exact and approximate

solutions by using graphical representation. The aim of the present study was to

investigate the behaviour of the solution of the DEs with FIBCs which are arising in

the different fields and also approximated the solution of the Airy’s non-homogeneous

DE by considering fuzzy boundaries. The objective of the work is to investigate the

behaviour of the solution of higher-order ordinary differential equations with fuzzy

parameters and study the fuzziness involved.

1.5 Objective and future scope of work

The main goal of the present work is to study the DEs with fuzzy parameters that are

arising in many fields of engineering and sciences. The objectives of our study are as

under:

• To find the fuzziness in the solution to dynamic problems due to the fuzziness in

the initial conditions and the forcing function,


the boundary conditions and the forcing function,


the initial and the boundary conditions as well as the forcing function.

Different methods are used to study the effect of FIBCs in the final solution of DEs. The

future scope of the current work is to study the problems related to partial differential

equations with FIBCs as well as fuzziness in the forcing function.


1.6 Original contribution by the thesis

The original contribution made by the study is the solution of nth order DE with

constant coefficients and Cauchy-Euler’s with variable coefficients with FICs by using

the properties of a linear transformations. The GEM is used to solve nth order DEs

with FIBCs using MATLAB, and the exact and approximate solutions are compared.

In chapter FOUR, we have solved a problem related to the civil engineering viz., the

bending of an elastic beam under the constant load.

The thesis includes the solution of higher-order DEs with FBCs whose exact

solution is difficult to find and interpret that type of DEs are solved by FDM. Here, we

also solved the differential equation that contains the Airy’s function. The solutions of

the problems has been studied numerically and graphically with the help of MATLAB.

The solutions of some constant-coefficient and variable coefficient(not a

Cauchy-Euler’s) DEs with FIBCs is obtained by using Laplace transform(LT). In last

chapter, we solved a problem-related to mechanical engineering viz., the mass-spring

system and problem-related to physics viz., the simple pendulum. The solution of

higher-order DEs with FIBCs that occur in different fields of engineering and sciences

are discussed in this thesis.

1.7 Methodology of Research and Results/Comparisons

We have studied various kinds of literature related to the solution of different types of

higher-order DEs with FIBCs and did a comparative analysis to find the research gap

and the problem statement. The literature survey helped us to define an objective of the

research.

We have used MATLAB coding to solve various DEs with fuzzy parameters. We

have also used Mathematica and MATLAB software for graphical representation of

different types of solution bounds. It is a combination of different solution methods to

solve DEs with FIBCs that provides a convenient analytic and numerical tool to solve

Achievements with respect to objectives 7

many types of ordinary differential equations. We have studied various problems

related to engineering and science and according to research gap, we have created the

mathematical models for different physical phenomenon arising in the fields of

mechanical engineering, civil engineering, and physics.

An approximate analytical solution of nth order ordinary differential

equations(ODEs) with FICs and the study of the motion of mass arising in physics are

obtained by the properties of the linear transformation. The GEM has been used to find

the solution of nth order DEs with appropriate FIBCs. The solution is interpreted

numerically and graphically using MATLAB and Mathematica software and it is

compared with the exact solution. The FDM is used to approximate the solution of

higher-order DEs with FBCs and also solved the Airy’s non-homogeneous DE by the

help of MATLAB coding and plotting. The variable coefficient(not a Cauchy-Euler’s)

DEs with FIBCs are solved by using the method of LT. Here, we solved the simple

pendulum problem in physics and the mass-spring problem in mechanical engineering

branch by using LT. The solution is interpreted numerically and graphically by using

Mathematica software and it is compared with the exact solution.

1.8 Achievements with respect to objectives

A mathematical model is developed for the problem related to the mass-spring system in

the mechanical engineering with FICs, the bending of an elastic beam under the constant

load in civil engineering with FIBCs, the second-order linear differential equation viz.,

Airy’s non-homogeneous DE with FBCs, and the simple pendulum problem in physics

are solved by various methods.

Here, we discuss different solution methods to solve different type of DEs with

FIBCs as well as fuzziness in the forcing function. We find the solution of nth order

DEs with FICs by using properties of linear transformation and discuss the application

related to the study of the motion of the mass in physics. The solution of nth order DEs

with FIBCs is obtained using the GEM and an application related to civil engineering

viz., the bending of an elastic beam under the constant load is studied. We also find the


solution of higher-order DEs with FBCs by using the FDM and solved the Airy’s

non-homogeneous DE with FBCs. In last, we have solved higher-order variable

coefficient(not a Cauchy-Euler’s) DEs with FIBCs by using LT method and the

mass-spring system and the simple pendulum problem in mechanical engineering and

physics respectively.

1.9 Conclusion

We have studied different methods of solving DEs with FIBCs arising in different filed

of engineering such as mechanical, civil, and the Airy’s non-homogeneous DE and the

simple pendulum problem in physics. Here, we have discussed the analytic and the

numerical methods to solve different types of DEs with FIBCs. We have solved DEs

with FIBCs whose exact solution is known then we compare the exact and

approximate solutions of higher-order DEs. We have also approximated the solution of

DEs with FBCs whose exact solution is not known, the solution of such type of DEs is

approximated by numerical method i.e. using FDM. The mass-spring problem in

mechanical engineering and the simple pendulum problem in physics were solved

using the Laplace transform.

CHAPTER 2

Fundamentals of Fuzzy Set Theory and

Basic Concepts

2.1 History of development of fuzzy set theory

Lotfi Zadeh was best known for proposing fuzzy Mathematics. Fuzzy set theory is a

process to deal with imprecise data. Fuzzy logic is a form of many-valued logic; it deals

with reasoning that is approximate rather than fixed and exact. Compared to traditional

binary sets(true or false values) fuzzy logic variables may have truth value that ranges

in degree between 0 and 1(no and yes).[https://www.igi-global.com/dictionary/fuzzy-

logic/11740]

FIGURE 2.1: Lotfi Zadeh [https://www.slideserve.com/tambre/lotfi-zadeh]

9

Fundamentals of Fuzzy Set Theory and Basic Concepts 10

2.2 Crisp set vs. Fuzzy set

Let U be universe of discourse or a universal set.

2.2.1 Crisp set

Crisp set A is defined by its characteristic function. The characteristic function is

denoted by χA and it is defined by

χA(x) =

1, if x ∈ A

0, if x 6∈ A

It takes value either 1 or 0.

2.2.2 Fuzzy set

Let U 6= φ . The fuzzy set A ∈U is defined by its membership function.

µA : U → [0,1]

It takes values in the range [0,1].

U

x

µA(x)

Fuzziness Fuzziness

1

o

Crisp Subset A

Fuzzy Subset A

FIGURE 2.2: Crisp set vs. Fuzzy set

Standard operations on fuzzy sets: 11

2.3 Standard operations on fuzzy sets:

The conjunction, dejection and complement are three standard properties for fuzzy sets.

These are true for both fuzzy set and crisp set. The fuzzy set theory defines fuzzy

operations on fuzzy sets. The AND, OR, NOT of Boolean logic exist in fuzzy logic

usually defined as minimum, maximum, and complement, respectively [70].

2.3.1 Logical connectives

Let A and B are fuzzy sets

• Conjunction or Intersection:

A∧ B = A∩ B = min(µA(x),µB(x))

• Disjunction or Union:

A∨ B = A∪ B = max(µA(x),µB(x))

• Complementary:

A→ µA′ (x) = 1−µA(x)

x

µA

5 8

1

0

A

FIGURE 2.3: A


x

µB

4

1

0

B

FIGURE 2.4: B

x

µA∧B

4

1

0 5 8

1B A

FIGURE 2.5: A∧ B

x

µA∨B

4

1

0 5 8

1B A

FIGURE 2.6: A∨ B

x

µA′

5 8

1

0

A

FIGURE 2.7: A′

Fuzzy set and Membership function 13

The below two laws do not hold for fuzzy sets because of the fact that, since fuzzy sets

can overlap, a set and its complement can also overlap.

Axiom of the contradiction A∧ A′= A∩ A

′ 6= φ

Axiom of the excluded-middle A∨ A′= A∪ A

′ 6=U

2.4 Fuzzy set and Membership function

The membership function (MF) [70] incorporate the mathematical representation of

membership in a set, and the notation used throughout the thesis for a fuzzy set is a

set symbol with a tilde over-score, say A, where the functional mapping is given by

µA(x) ∈ [0,1] and the symbol µA is the degree of membership of element x in fuzzy set

A. Therefore, µA(x) is a value on the unit interval that measures the degree to which

element x belongs to fuzzy set A ; equivalently,

µA(x) = degree to which x ∈ A.

2.4.1 Membership function

A fuzzy set A can be defined as a pair of the universal set U and MF µA : U → [0,1] for

each x ∈ U , the number µA is called the membership degree of x in A. It can be

represented mathematically as

A = {(x,µA(x))|x ∈U}, µA(x) ∈ [0,1].

2.5 Properties of Membership function

All the information contained in a fuzzy set is described by its membership function.


2.5.1 Core

The core [70] of a MF for some fuzzy set A is defined as the region of the universe U

that is characterized by complete and full membership in the set A. That is, the core

comprises those elements x of the universe such that µA(x) = 1.

2.5.2 Support

The support [70] of a MF for some fuzzy set A is defined as the region of the universe

U that is characterized by a non-zero membership in the set A. That is, the support

comprises those elements x of the universe such that µA(x) > 0. The support of A is a

crisp set and it is defined as

supp (A) = {x ∈U |µA(x)> 0}

2.5.3 Boundary

The boundaries of a MF for some fuzzy set A is defined as the region of the universe U

that have a non-zero membership but not a complete membership. That is, the

boundaries contains those elements x of the universe such that

0 < µA(x)< 1.

These are the elements of the universe that have some degree of fuzziness or only

partial membership in the fuzzy set A. Fig.2.8 [70] shows the regions in the universe

comprising the core, support, and boundaries of a typical fuzzy set.

Properties of Membership function 15

x

µA(x)

Core

Support

Boundary Boundary

1

o

FIGURE 2.8: Core, support, and boundaries of a fuzzy set

2.5.4 Height

The height [70] of a fuzzy set A is the largest membership grade of any element in A.

h(A) = height(A) = max {µA(x),x ∈ A}.

2.5.5 Normal fuzzy set

A normal fuzzy [70] set is the one whose MF has at least one element x in the universe

whose membership value is unity. i.e. h(A) = 1

x

µA(x)

1A

0

FIGURE 2.9: Normal fuzzy set


2.5.6 Subnormal fuzzy set

If h(A)< 1,∀x ∈U , then the fuzzy set is called a subnormal fuzzy set [70].

x

µA(x)

1

A

0

FIGURE 2.10: Subnormal fuzzy set

2.5.7 Convex fuzzy set

A convex fuzzy set is described by a MF whose membership values are strictly

monotonically increasing, or whose membership values are strictly monotonically

decreasing, or whose membership values are strictly monotonically increasing then

strictly monotonically decreasing with increasing values for elements in the universe.

In notations, if for any elements x,y and z in a fuzzy set A,

x < y < z⇒ µA(y)≥ min[µA(x),µA(z)]

then A is said to be a convex fuzzy set [70].

Fuzzy number 17

x

µA(x)

1

0 x zy

A

FIGURE 2.11: Convex fuzzy set

2.5.8 r-cut

It is one of the most important concepts of fuzzy sets.

The r-cut [70] of A is a crisp set and it is denoted as Ar and defined as

Ar = {x ∈U |µA(x)≥ r}

The strong r-cut [70] of A is also a crisp set and it is denoted as Ar+ and defined as

Ar+ = {x ∈U |µA(x)> r}

For r = 0, A0 = closure(supp(A))

2.6 Fuzzy number

A fuzzy number is a fuzzy set defined on the set of real numbers IR like µ : IR→ [0,1].

which satisfies:

• µ is upper semi-continuous i.e. ∀ ε > 0, ∃ δ > 0, 3 µ(x)−µ(x0)< ε whenever

|x− x0|< δ ,


• µ is fuzzy convex i.e µ(λx+(1−λ )y)≥min{µ(x),µ(y)} ∀ x,y∈ IR, λ ∈ [0,1],

• µ is normal i.e ∃ x0 ∈ IR for which µ(x0) = 1,

• supp µ = {x ∈ IR | µ(x) > 0} is support of µ , and its closure cl(supp µ) is

compact.

Let E be set of all real fuzzy numbers [44] which are normal, upper semi-continuous,

convex and compactly supported fuzzy sets. The parametric form a fuzzy number µ is

a order pair (µ,µ) of functions (µ(r),µ(r)),0 ≤ r ≤ 1, which satisfies the following

requirements:

• µ(r) is a bounded monotonic increasing left continuous function over [0,1];

• µ(r) is a bounded monotonic decreasing right continuous function over [0,1];

• µ(r)≤ µ(r),0≤ r ≤ 1.

2.6.1 Triangular fuzzy number

Consider triangular fuzzy number (TFN) [76] A = (a,b,c) as seen in Fig.2.12. The

TMF µA of A is defined as follows.

µA(x) =

0 , if x < ax−ab−a , if a≤ x≤ bc−xc−b , if b≤ x≤ c

0 , if x > c

The TFN A = (a,b,c) can be written as an order pair of function of r-cut approach

i.e. [µ(r), µ(r)] = [a+(b−a)r, c− (c−b)r], where r ∈ [0,1].

Fuzzy number 19

x

µA(x)

a cb

1

0

FIGURE 2.12: Triangular fuzzy number

2.6.2 Trapezoidal fuzzy number

Consider trapezoidal fuzzy number (TraFN) [82] A = (a,b,c,d) as presented in

Fig.2.13. The TraMF µA of A is defined as follows.

µA(x) =

0 , if x < ax−ab−a , if a≤ x≤ b

1 , if b≤ x≤ cd−xd−c , if c≤ x≤ d

0 , if x≥ d

The TraFN A = (a,b,c,d) can be written as an order pair of function of r-cut approach

i.e. [µ(r), µ(r)] = [a+(b−a)r, d− (d− c)r], where r ∈ [0,1]


x

µA(x)

a cb d

1

o

FIGURE 2.13: Trapezoidal fuzzy number

2.6.3 Gaussian fuzzy number

The Gaussian membership function (GMF) [82] µA of an asymmetric Gaussian fuzzy

number(GFN) A = (α,σl,σr) is defined as follows.

µA(x) =

e− (x−α)2

2σ2l , if x≤ α

e− (x−α)2

2σ2r , if x≥ α

where, α denotes the modal value and σl,σr denote left and right fuzziness,

respectively, corresponding to the Gaussian distribution.

For symmetric GFN the left and right fuzziness are equal i.e. σl = σr = σ . So, the

symmetric GFN is denoted as A = (α,σ ,σ) and corresponding GMF is defined as

µA(x) = e−β (x−α)2,∀x ∈ IR where β = 1

2σ2 . The symmetric GFN in parametric form

can be written as A = [µ(r),µ(r)] =[α−

√− (loger)

β, α +

√− (loger)

β

]where r ∈ [0,1].

Fuzzy arithmetic 21

0.368

x0

1

µA(x)

FIGURE 2.14: Gaussian fuzzy number

2.7 Fuzzy arithmetic

Let u =(u(r),u(r)),v = (v(r), v(r)) ∈ E, r ∈ [0,1] and arbitrary k ∈ IR [76].

Then

u = v iff u(r) = v(r) and u(r) = v(r),

u+ v = (u(r)+ v(r), u(r)+ v(r)),

u− v = (u(r)− v(r), u(r)− v(r)),

ku =

(ku(r),ku(r)), k ≥ 0

(ku(r),ku(r)), k < 0

u · v = (min{u(r)v(r), u(r)v(r), u(r)v(r), u(r)v(r)},max{u(r)v(r), u(r)v(r), u(r)v(r), u(r)v(r)})

2.8 Zadeh’s extension principle

Zadeh’s extension principle [79] provides a mathematical approach for extending

classical functions to fuzzy mappings. Let X be a set, P(X) be the power set of X , and


F(X) be fuzzy power set of X . That is,

P(X) = {U |U ⊂ X}= {U |U : X →{0,1}}

and

F(X) = {U |U : X → [0,1]}

Let f be a mapping from a set X to another set Y and fmax be mapping from F(X) to

F(Y ) which can be induced by f as following:

fmax : F(X)→ F(Y ),u→ fmax(u)

where ∀y ∈ Y ,

fmax(u)(y) =

∨f (x)=y u(x), i f

{x| f (x) = y,x ∈ X

}6= φ

0, otherwise

2.9 Continuity

Let f : IR→ E, f is called continuous [33], if for every t0 ∈ (a,b) and every ε > 0,

there exists δ > 0 such that if |t− t0|< δ then D( f (t), f (t0))< ε .

A mapping f : IR×E→ E is called continuous [75] at point (t0,x0) ∈ IR×E provided

for any fixed r ∈ [0,1] and arbitrary ε > 0, there exist a δ (ε,r) such that

D([ f (t,x)]r, [ f (t0,x0)]r)< ε whenever | t− t0 |< δ (ε,r).

2.10 Hukuhara difference

Let u,v ∈ E. If there exists w ∈ E such that u = v+w, then w is called the Hakuhara-

difference [18] of fuzzy numbers u and v, and it is denoted by w = u v.

The sign stands for Hakuhara-difference, and u v 6= u+(−1)v.

Hukuhara differential 23

2.11 Hukuhara differential

Let f : (a,b)→ E and t0 ∈ (a,b). We say that f is Hukuhara differentiable(HD) [18] at

t0, if there exists an element f′(t0) ∈ E such that for all h > 0 sufficiently small,

∃ f (t0 +h) f (t0), f (t0) f (t0−h) and the limits holds (in the metric D)

limh→0

f (t0 +h) f (t0)h

= limh→0

f (t0) f (t0−h)h

= f′(t0).

2.12 Generalized Hukuhara differentiability

Let f : (a,b)→ E and t0 ∈ (a,b). We say that f is (1)-differential at t0, if there exists

an element f′(t0) ∈ E such that for all h > 0 sufficiently small,

∃ f (t0 +h) f (t0), f (t0) f (t0−h) and the limits holds (in the metric D)

limh→0

f (t0 +h) f (t0)h

= limh→0

f (t0) f (t0−h)h

= f′(t0).

and f is (2)-differentiable if for all h > 0 sufficiently small,

∃ f (t0) f (t0 +h), f (t0−h) f (t0) and the limits holds (in the metric D)

limh→0

f (t0) f (t0 +h)−h

= limh→0

f (t0−h) f (t0)−h

= f′(t0).

If f′(t0) exist in above two cases then i.e called generalized Hukuhara

derivative(GHD)[24] of f (t).

2.13 Fuzzy Reimann integration

Let f (t) be fuzzy-valued function on [0,∞) and it is represented by ( f (t,r), f (t,r)).

For any fixed r ∈ [0,1], assume f (t,r) and f (t,r) are Reimann-integrable [77], on [a,b]

for every b ≥ a and assume there are two positive M(r) and M(r) such that


∫ ba | f (t,r) | dt ≤ M(r) and

∫ ba | f (t,r) | dt ≤ M(r) for every b ≥ a, then f (t) is

improper fuzzy Reimann integral(FRI) on [a,∞) and the improper FRI is a fuzzy

number. Furthermore, we have

∫∞

a f (t)dt =(∫

∞

a f (t,r)dt,∫

∞

a f (t,r)dt)

CHAPTER 3

Solutions of nth order Differential

Equations with Fuzzy Initial

Conditions by using the Properties of

Linear Transformations

3.1 Introduction

These days, the DEs with FICs is a prominent theme examined by numerous specialists

since it is widely used for the purpose of modelling in science and engineering. The

DEs with FICs emerge normally during the modelling of dynamic problems. The FDE

was studied by Kaleva [44, 46]. FIVPs were studied by Seikkala [74]. In this chapter,

the derivatives are considered as Hukuhara derivatives.

Bede et al. [17] used the notions of Hukuhara differential, generalized

differentiability, differential inclusions to solve FDEs and did the interpretation of

FDEs by using Zadeh’s extension principle on the classical solution. A new approach

to fuzzy initial value problem was studied by Gasilov et al. [37]. They proposed a new

approach to non-homogeneous fuzzy initial value problem [38]. Georgiou et al. [39]

solved IVPs for higher-order FDEs. Analytical solutions of FIVPs by homotopy

analysis method were studied by Abu-Arqub et al. [1]. Solution of linear differential

equations with fuzzy boundary values was put forwarded by Gasilov et al. [34, 35].

Gasilov et al. [36] also found solution method for a BVP with fuzzy forcing function.

25

Solutions of nth order Differential Equations with Fuzzy Initial Conditions by using theProperties of Linear Transformations 26

3.2 Linear transformation

Linear transformation is rule that converts one data into another. It is a function (say T )

that takes each element of the domain and associate it with exactly one element of the

co-domain.

3.2.1 Wronskain

If the functions f1 = f1(x), f2 = f2(x), ...., fn = fn(x) have n−1 continuous derivatives

on the interval (−∞,∞), and if the Wronskian of these functions

W (x) =

∣∣∣∣∣∣∣∣∣∣∣∣

f1(x) f2(x) . . . fn(x)

f1′(x) f2

′(x) . . . fn

′(x)

...... . . . ...

f1n−1(x) f2

n−1(x) . . . fnn−1(x)

∣∣∣∣∣∣∣∣∣∣∣∣

is not identically zero on (−∞,∞), then these functions form a linearly independent set

of vectors in C(n−1)(−∞,∞).

3.3 Problem formulation

In real life practical problems, the behaviour of an entity is obtained with the help of

physics and it is crisp. But the initial conditions are obtained from calculations, these

values can be vague and often it is convenient to model them using different types of

fuzzy numbers. This gives rise to IVPs with crisp dynamics but fuzzy conditions.

In the field of DEs, an IVP is an ODE together with a designated value, called

the initial condition, of the unknown function at a prescribed point in the domain of

the solution. Here we are going to solve nth order DEs by considering fuzzy initial

conditions. For same value of the independent variable, there should be a range of values

of the dependent variable since we are considering values of the dependent variable i.e.

initial conditions to be fuzzy. We show that if the solution of the corresponding crisp

Method of solving nth order differential equation with fuzzy initial conditions 27

problem exists and is unique then the fuzzy problem also has a unique solution. Here

we are going to solve homogeneous and non-homogeneous constant coefficient as well

as Cauchy-Euler’s variable coefficient DEs with FICs.

3.4 Method of solving nth order differential equation

with fuzzy initial conditions

The notation u = (uL(r), uR(r)),0 ≤ r ≤ 1 will be used to represent fuzzy number in

parametric form.

Let u = uL(0) and u = uR(0) are used to indicate the left and the right end points of u

respectively.

An r-cut of u is an interval [uL(r), uR(r)], which is denoted as

ur = [ur, ur]

Consider the triangular fuzzy number u = (a,c,b) for which

uL(r) = a+ r(c−a), uR(r) = b− r(b− c)

Its geometric interpretation is u = a,u = b are end points and c is the vertex. Let us

consider the TFN whose vertex is zero i.e. u = (a,0,b) (note that a < 0 and b > 0).

Then,

u = [uL(r), uR(r)] = [(1− r)a, (1− r)b] = (1− r)[a,b], r ∈ [0,1].

From the above result, we can say that r-cut is similar to interval [a,b] (i.e. to 0-cut)

with similarity coefficient (1− r).

A fuzzy number u can be express as

u = ucr + uun


Here ucr is a fuzzy number with membership degree 1 and represent the crisp part(i.e.

vertex) of u; whereas uun is uncertain part with vertex at origin.

For triangular fuzzy number u = (a,c,b)

ucr = c and uun = (a− c,0,b− c).

Consider the nth order DE with FICs:

bn(t)xn +bn−1(t)xn−1 + ....+b1(t)x′+b0(t)x = f (t)

x(l) = A1

x′(l) = A2

... =...

xn−1(l) = An

(3.1)

where A1, A2, ....., An are fuzzy numbers and b0(t),b1(t),b2(t), ....,bn(t) and f (t) are

continuous crisp functions or constants and l is any integer.

Let us represent the initial values as

A1 = a1 + a1

A2 = a2 + a2

... =...

...

An = an + an

where a1,a2, ....an are crisp numbers, while a1, a2, ..., an are fuzzy numbers. Let us

split the DE with FICs in (3.1) to following problems:

The homogeneous DE with FICs

bn(t)xn +bn−1(t)xn−1 + ....+b1(t)x′+b0(t)x = 0

x(l) = a1

x′(l) = a2

... =...

xn−1(l) = an

(3.2)

Method of solving nth order differential equation with fuzzy initial conditions 29

and a non-homogeneous DE with crisp conditions

bn(t)xn +bn−1(t)xn−1 + ....+b1(t)x′+b0(t)x = f (t)

x(l) = a1

x′(l) = a2

... =...

xn−1(l) = an

(3.3)

It is easy to see that if xun(t) and xcr(t) are solutions of (3.2) and (3.3) respectively, then

x(t) = xcr(t)+ xun(t) is a solution of given problem (3.1).

To determine x(t) we consider the transformation T : IRn→ IR,T (u) = v · u is a linear

transformation,

where v is fixed n×n determinant and u = [a1,a2, . . . ,an]T .

The solution algorithm consist four steps:

1. Represent the initial values as A1 = a1 + a1, A2 = a2 + a2,..., An = an + an.

2. Find linearly independent solutions x1(t),x2(t), ...,xn(t) of the crisp differential

equation bn(t)xn +bn−1(t)xn−1 + ....+b1(t)x′+b0(t)x = 0. Constitute the vector

function s(t) = (x1(t),x2(t), ....,xn(t)), the determinant W by using the definition

of Wronskain and calculate the vector function

y(t) = s(t)W−1 = (y1(t),y2(t), ....,yn(t)).

3. Find solution xcr(t) of the non-homogeneous crisp problem.

4. Find the solutions of the given problems

(a) For homogeneous DE with FICs

x(t) = y1(t)a1 + y2(t)a2 + ......+ yn(t)an

(b) For non-homogeneous DE with FICs

x(t) = xcr(t)+ y1(t)a1 + y2(t)a2 + ......+ yn(t)an


3.5 Solution of a higher order differential equation with

fuzzy initial condition

3.5.1 Solution of a higher order constant coefficient differential

equation with fuzzy initial condition

Example 3.1 Solve the constant coefficient homogeneous DE with FICs

x′′+4x

′+3x = 0,

x(0) = (−0.5,0,0.5)

x′(0) = (−1,0,0.5)

The problem is homogeneous and initial values are fuzzy numbers with vertices at 0.

Therefore, the solution using the algorithm is obtained as follows:

x1(t) = e−t and x2(t) = e−3t are linearly independent solutions for the equation

x′′+4x

′+3x = 0. Hence s(t) = (e−t ,e−3t) and

W =

∣∣∣∣∣∣1 1

−1 −3

∣∣∣∣∣∣

and

y(t) = (y1(t),y2(t))

=

(32

e−t− 12

e−3t ,12

e−t− 12

e−3t).

xun(t) =(

32

e−t− 12

e−3t)(−0.5,0,0.5)+

(12

e−t− 12

e−3t)(−1,0,0.5),

where arithmetic operations are considered to be fuzzy operations. The fuzzy solution

x(t) form band in the tx-coordinate space (Fig.3.1).

Since the initial values are TFNs, an r-cut of the solution can be determined by similarity

coefficient (1− r), i.e. xr(t) = (1− r)[ x(t), x(t) ].

Solution of a higher order differential equation with fuzzy initial condition 31

FIGURE 3.1: The fuzzy solution obtained by the properties of linear transformations

Example 3.2 Consider the constant coefficient non-homogeneous DE with FICs

x′′+4x

′+3x = e−t ,

x(0) = (0.5,1,1.5)

x′(0) = (1,2,2.5)

We represent the FICs as

A = (0.5,1,1.5) = 1+(−0.5,0,0.5),

B = (1,2,2.5) = 2+(−1,0,0.5).

The crisp solution to the crisp non-homogeneous problem

x′′+4x

′+3x = e−t ,

x(0) = 1

x′(0) = 2

is

xcr(t) =94

e−t− 54

e−3t +t2

e−t

=

(32

e−t− 12

e−3t)+

32

(e−t

2− e−3t

2

)+

t2

e−t


The fuzzy homogeneous problem to find the uncertainty of the solution is as follows:

x′′+4x

′+3x = 0,

x(0) = (−0.5,0,0.5)

x′(0) = (−1,0,0.5)

This problem is the same as Example 3.1 Hence, its solution is

xun(t) =(

32

e−t− 12

e−3t)(−0.5,0,0.5)+

(12

e−t− 12

e−3t)(−1,0,0.5)

Adding this uncertainty to the crisp solution gives the fuzzy solution of the given DE

with FICs

x(t) = xcr(t)+ xun(t)

=

(32

e−t− 12

e−3t)(0.5,1,1.5)+

(12

e−t− 12

e−3t)(1,2,2.5)+

t2

e−t

The fuzzy solution x(t) form band in the tx-coordinate space (Fig.3.2).

FIGURE 3.2: The fuzzy solution obtained by the properties of linear transformationsand the red line represents the crisp solution


Example 3.3 Solve the constant coefficient homogeneous DE with FICs.

x′′′+3x

′′+3x

′+ x = 0,

x(0) = (−0.5,0,1)

x′(0) = (−1,0,1)

x′′(0) = (−1,0,0.5)


Therefore, the solution is by solution algorithm.

x1(t) = e−t , x2(t) = te−t and x3(t) = t2e−t are linearly independent solutions for the

equation x′′′+3x

′′+3x

′+ x = 0. Hence s(t) = (e−t , te−t , t2e−t) and

W =

∣∣∣∣∣∣∣∣∣

1 0 0

−1 1 0

1 −2 2

∣∣∣∣∣∣∣∣∣

and

y(t) = (y1(t),y2(t),y3(t))

=(

e−t + te−t +t2

2e−t , te−t + t2e−t ,

t2

2e−t)

xun(t) =(

e−t +te−t +t2

2e−t)(−0.5,0,1)+(te−t +t2e−t)(−1,0,1)+

t2

2e−t(−1,0,0.5);

where arithmetic operations are considered to be fuzzy operations.


Since the initial values are TFNs, an r-cut of the solution can be determined by similarity

coefficient (1− r), i.e. xr(t) = (1− r)[ x(t), x(t) ].


FIGURE 3.3: The fuzzy solution obtained by the properties of linear transformation

Example 3.4 Consider the constant coefficient non-homogeneous DE with FICs.

x′′′+3x

′′+3x

′+ x = 30e−t ,

x(0) = (2.5,3,4)

x′(0) = (−4,−3,−2)

x′′(0) = (−48,−47,−46.5)

We represent the FICs as

A = (2.5,3,4) = 3+(−0.5,0,1),

B = (−4,−3,−2) =−3+(−1,0,1),

C = (−48,−47,−46.5) =−47+(−1,0,0.5)

We solve crisp non-homogeneous problem

x′′′+3x

′′+3x

′+ x = 30e−t ,

x(0) = 3

x′(0) =−3

x′′(0) =−47


And the crisp solution is

xcr(t) = (3−25t2)e−t +5t3e−t

= 3(

e−t + te−t +t2

2e−t)−3(te−t + t2e−t)− 47

2t2e−t +5t3e−t

The fuzzy homogeneous problem to find the uncertainty of the solution is as follows:

x′′′+3x

′′+3x

′+ x = 0,

x(0) = (−0.5,0,1)

x′(0) = (−1,0,1)

x′′(0) = (−1,0,0.5)

This problem is the same as Example.3.3. Hence, its solution is

xun(t) =(

e−t + te−t +t2

2e−t)(−0.5,0,1)+(te−t + t2e−t)(−1,0,1)+

t2

2e−t(−1,0,0.5)

Adding this uncertainty to the crisp solution gives the fuzzy solution of the given DE.


x(t) =(

e−t + te−t +t2

2e−t)(2.5,3,4)+(te−t + t2e−t)(−4,−3,−2)

+t2

2e−t(−48,−47,−46.5)+5t3e−t



FIGURE 3.4: The fuzzy solution obtained by the properties of linear transformations,and the red line represents the crisp solution

In above example, if we take t → ∞ then the fuzziness in the solution is disappearing.

So, the solution goes nearer to zero if we increase t (Fig.3.5).

FIGURE 3.5: If t increases, fuzziness disappears


3.5.2 Solution of higher order Cauchy-Euler’s variable coefficient

differential equations with fuzzy initial conditions

Example 3.5 Consider the variable coefficient DE with FICs

(t2D2−3tD+4)x = t2

x(1) = (0.5,1,2)

x′(1) = (−0.5,0,0.5)

(t2D2−3tD+4)x = t2

x(1) = 1

x′(1) = 0

Associated non-homogeneous problem has the solution

xcr = [1−2log(t)]t2 +t2

2(log(t))2

Consider the homogeneous DE with FICs.

(t2D2−3tD+4)x = 0

x(1) = (−0.5,0,1)

x′(1) = (−0.5,0,0.5)


Therefore the solution by solution algorithm

x1(t) = t2 and x2(t) = t2 log(t) are linearly independent solutions of

(t2D2−3tD+4)y = 0. Hence s(t) = (t2, t2 log(t)) and

W =

∣∣∣∣∣∣1 0

2 1

∣∣∣∣∣∣

and y(t) = (y1(t),y2(t)) = ([1−2log(t)]t2, t2 log(t))

The fuzzy solution is

xun = ([1−2log(t)]t2)(−0.5,0,1)+(t2 log(t))(−0.5,0,0.5)



FIGURE 3.6: The fuzzy solution obtained by the properties of linear transformations

Adding this uncertainty to crisp solution gives the fuzzy solution of the given DE with

FICs

x(t) = xcr(t)+ xun(t) = [1−2log(t)]t2(0.5,1,2)+(t2 log(t))(−0.5,0,0.5)+t2

2(log(t))2


FIGURE 3.7: The fuzzy solution obtained by the properties of linear transformations,The red line represents the crisp solution


Example 3.6 Consider the variable coefficient DE with FICs

(t3D3−3t2D2 +6tD−6)x =1t

x(1) = (4,5,6)

x′(1) = (12,13,14.5)

x′′(1) = (9.5,10,11)

Consider corresponding crisp problem.

(t3D3−3t2D2 +6tD−6)x =1t

x(1) = 5

x′(1) = 13

x′′(1) = 10

Associated non-homogeneous problem has the solution

xcr(t) =−13824

t +32824

t2− 6924

t3− 124t

Consider the homogeneous DE with FICs.

(t3D3−3t2D2 +6tD−6)x = 0

x(1) = (−1,0,1)

x′(1) = (−1,0,1.5)

x′′(1) = (−0.5,0,1)


Therefore, the solution by solution algorithm

x1(t) = t, x2(t) = t2 and x2(t) = t3 are linearly independent solution of

(t3D3−3t2D2 +6tD−6)x = 0. Hence s(t) = (t, t2, t3) and

W =

∣∣∣∣∣∣∣∣∣

1 1 1

1 2 3

0 2 6

∣∣∣∣∣∣∣∣∣


and

y(t) = (y1(t),y2(t),y3(t))

=

(3t−3t2 + t3,−2t +3t2− t3,

t2− t2 +

t3

2

)

The fuzzy solution is

xun(t) = (3t−3t2 + t3)(−1,0,1)+(−2t +3t2− t3)(−1,0,1.5)+(

t2− t2 +

t3

2

)(−0.5,0,1)


FIGURE 3.8: The fuzzy solution obtained by properties of linear transformation

We add this uncertainty to crisp solution and get the fuzzy solution of the given DE with

FICs


= (3t−3t2 + t3)(4,5,6)+(−2t +3t2− t3)(12,13,14.5)+(

t2− t2 +

t3

2

)(9.5,10,11)− 1

24t

The fuzzy solution x(t) form band in the tx-coordinate space and the red line represents

the crisp solution (Fig.3.9).


FIGURE 3.9: The fuzzy solution obtained by properties of linear transformation, Thered line represents the crisp solution

In above example, if we increase t and as we take t → ∞ then the fuzziness in the

solution increases and goes to infinite (Fig.3.10).

FIGURE 3.10: If we increase t fuzziness increases

Consider the mass-spring system with damping coefficient zero

mx′′(t)+ kx(t) = r(t),

where r(t) is an external force and the resultant motion is a forced motion.

Example 3.7 Consider a unit mass sliding on a frictionless table attached to a spring,

with spring constant k = 16. Suppose the mass is lightly tapped by a hammer every T

seconds. Mass tapped periodically with hammer. Suppose that the first tap occurs at


time t = 0 and before that time the mass is at rest. Describe what happens to the

motion of the mass for the tapping period T = 1.

FIGURE 3.11: Mass tapped periodically with a hammer

The non-homogeneous DE with initial conditions

x′′(t)+16 x(t) = sin t,

x(0) = 0

x′(0) = 1

Consider the homogeneous equation with fuzzy parameters.

x′′(t)+16 x(t) = 0,

x(0) = (−1,0,1)

x′(0) = (−0.5,0,1)

The problem is homogeneous and initial values are fuzzy numbers with vertex 0.

x1(t) = cos4t, x2(t) = sin4t are linearly independent solutions for the equation

x′′(t)+16 x(t) = 0. Hence s(t) = (cos4t, sin4t) and

W =

∣∣∣∣∣∣1 0

0 4

∣∣∣∣∣∣

and y(t) = (y1(t),y2(t)) =(

cos4t,14

sin4t)

The fuzzy solution of the given homogeneous problem is

xun(t) = (cos4t)(−1,0,1)+14

sin4t(−0.5,0,1) (3.4)

Result and Discussion 43

The crisp solution of the given non-homogeneous problem is

xcr(t) =730

sin4t +1

15sin t (3.5)

If we add uncertainty of (3.4) to crisp solution i.e.(3.5) then we get the fuzzy solution

of the given DE with FICs


= cos4t(−1,0,1)+7

30sin4t(0.5,1,2)+

115

sin t

where arithmetic operations are considered to be fuzzy operations.


FIGURE 3.12: The fuzzy solution obtained by the properties of linear transformation,The red line represents the crisp solution

3.6 Result and Discussion

Here, we solved the homogeneous DE with FICs that have both the lower-bounds and

upper bounds of solution (Fig.3.1) and the non-homogeneous DE with crisp initial

conditions that gives us the crisp solution i.e. denoted by the red line in Fig.3.2. If we

add both the solutions we can find the fuzzy solution of the given DE with FICs as in


Examples 3.1 and 3.2. Similarly, we can observe the solution bounds and the crisp

solutions for Examples 3.3 to 3.6. We also solved one application-level problem

related to physics (Example 3.7) for the study of the motion of mass when the mass is

tapped periodically by a hammer.

3.7 Conclusion

The DEs with FICs are investigated as a set of crisp problems. The method based on the

properties of linear transformations is applied for the second and the third-order DEs

with FICs for clarity purpose, but it is also applicable for the nth order DEs with FICs

[66]. By using the properties of linear transformations, we solved homogeneous and

non-homogeneous constant coefficient as well as Cauchy-Euler’s variable coefficient

DEs with FICs.

CHAPTER 4

Solutions of the nth order Differential

Equations with Fuzzy Initial and

Boundary Conditions by using the

Gauss Elimination Method

4.1 Introduction

Differential equations play an important role in modelling the real world problems.

Today, DEs represent a fundamental mathematical tool for studying systems that

change with time, and are used in most of the areas of science and engineering. Hence,

engineering and science students need to be able to model and solve problems using

DEs; thus allowing them to analyse the behaviour of their underlying dynamics.

Generally, the description of a phenomenon is completed with certainty over all its

parameters, including initial and boundary conditions. So, a phenomenon is usually

modelled by DEs and predetermined initial and boundary conditions. Chen et al. [28]

worked on fuzzy boundary value problems. They demonstrated two point boundary

value problems of undamped uncertain dynamical systems[27]. Allahviranloo et al.

[11] offered the method for solving the nth order fuzzy linear differential equations.

Fuzzy approximate solutions of second order fuzzy linear boundary value problems

were obtained by Xiaobin et al. [78]. A boundary value problem for the second order

fuzzy differential equation was studied by Khastan et al. [50]. Solutions of linear

differential equation with fuzzy boundary values were discussed by Gasilov et al. [35].

45

Solutions of the nth order Differential Equations with Fuzzy Initial and BoundaryConditions by using the Gauss Elimination Method 46

O’Regan et al. [62] proposed initial and boundary value problems for fuzzy differential

equations. Here, we consider the DEs with fuzzy parameters and solve them by using

GEM.


Here, we shall convert the DE with FIBCs into the system of linear equations, apply

GEM then use MATLAB programming to solve it. We assume that the approximate

solution to the given DE with FIBCs in the form of a polynomial of power k, where

k = 0,1,2, ....,N. We then solve the nth order linear DE with FIBCs by GEM. We also

compare the approximate and the exact solutions and represent them graphically by

using the Mathematica software.

4.3 Method for solving the nth order differential

equation with FBCs

Consider the second-order DE

y′′+ p(t)y′+q(t)y = g(t), a≤ t ≤ b (4.1)

with fuzzy boundary conditions:

y(a) = α, y(b) = β , (4.2)

where p(t) and q(t) are constants or functions of t. Here, we have to find fuzzy

coefficients for approximating the solution as

yN(t) =N

∑k=0

akφk(t) (4.3)

where φk(t), k = 0,1.....,N, are positive polynomial functions whose all derivatives are

positive. Here, we find the coefficients which are fuzzy in (4.3) by considering errors

Method for solving the nth order differential equation with FBCs 47

to zero as follows:

E = D(y′′+ p(t)y

′+q(t)y, g(t))+D(y(a), α)+D(y(b), β ) (4.4)

Then, substitute (4.3) in (4.4) and the parametric form as under

y′′+ p(t)y

′(t,r)+q(t)y(t,r) = g(t,r)

y(a,r) = α(r)

y(b,r) = β (r)

y′′+ p(t)y′(t,r)+q(t)y(t,r) = g(t,r)

y(a,r) = α(r)

y(b,r) = β (r)

(4.5)

Case:1 p(t)q(t)≥ 0

Suppose that the coefficients p(t),q(t) are non-negative. From (4.5), the equations are

y′′(t,r)+ p(t)y

′(t,r)+q(t)y(t,r) = g(t,r) (4.6)

y′′(t,r)+ p(t)y

′(t,r)+q(t)y(t,r) = g(t,r) (4.7)

y(a,r) = α(r), y(b,r) = β (r), y(a,r) = α(r), y(b,r) = β (r). (4.8)

Equation (4.3) is substituted in (4.6), (4.7) and (4.8) respectively, then

N

∑k=0

ak(r)φ′′k (t)+ p(t)

N

∑k=0

ak(r)φ′k(t)+q(t)

N

∑k=0

ak(r)φk(t) = g(t,r) (4.9)

N

∑k=0


N

∑k=0

ak(r)φ′k(t)+q(t)

N

∑k=0

ak(r)φk(t) = g(t,r) (4.10)

and

N

∑k=0

ak(r)φk(a) = α(r),N

∑k=0

ak(r)φk(b) = β (r) (4.11)

N

∑k=0


∑k=0

ak(r)φk(b) = β (r) (4.12)


By considering

γk = φ′′k (t)+ p(t)φ

′k(t)+q(t)φk(t),k = 0,1, .....,N

σak = φk(a),σbk = φk(b),

the following system is obtained:

N∑

k=0ak(r)γk = g(t,r)

N∑

k=0ak(r)σak = α(r)

N∑

k=0ak(r)σbk = β (r)

N∑

k=0ak(r)γk = g(t,r)

N∑


N∑


(4.13)

(4.13) is a system of linear equations A(t)X(r) = B(r) such that

A =

A1 A2

A2 A1

,

where

A1 =

γ0 γ1 γ2 γ3

σa0 σa1 σa2 σa3

σb0 σb1 σb2 σb3

, A2 =

0 0 0 0

0 0 0 0

0 0 0 0

and

X = (a0,a1,a2,a3,a0,a1,a2,a3)T , B = (g(t,r),α(r),β (r),g(t,r),α(r),β (r))T .

The parameters a0,a1,a2,a3,a0,a1,a2,a3 are obtained by solving (4.13) and taking

t = s,s ∈ [a,b]. These parameters gives us the fuzzy approximate solution

(y(t,r),y(t,r)).


The system of linear equations A(t)X(r) = B(r) is

γ0 γ1 γ2 γ3 0 0 0 0

σa0 σa1 σa2 σa3 0 0 0 0

σb0 σb1 σb2 σb3 0 0 0 0

0 0 0 0 γ0 γ1 γ2 γ3

0 0 0 0 σa0 σa1 σa2 σa3

0 0 0 0 σb0 σb1 σb2 σb3

a0(r)

a1(r)

a2(r)

a3(r)

a0(r)

a1(r)

a2(r)

a3(r)

=

g(t,r)

α(r)

β (r)

g(t,r)

α(r)

β (r)

Suppose that coefficients p(t),q(t) are negative. Then, from (4.5), the equations are

y′′(t,r)− p(t)y

′(t,r)−q(t)y(t,r) = g(t,r) (4.14)


′(t,r)−q(t)y(t,r) = g(t,r) (4.15)


Equation (4.3) when substituted (4.14), (4.15) and (4.16) gives, respectively,

N

∑k=0

ak(r)φ′′k (t)− p(t)

N

∑k=0

ak(r)φ′k(t)−q(t)

N

∑k=0

ak(r)φk(t) = g(t,r) (4.17)

N

∑k=0


N

∑k=0


N

∑k=0

ak(r)φk(t) = g(t,r) (4.18)

and

N

∑k=0


∑k=0

ak(r)φk(b) = β (r) (4.19)

N

∑k=0


∑k=0

ak(r)φk(b) = β (r) (4.20)


By taking

γk = φ′′k (t)

δk =−p(t)φ′k(t)−q(t)φk(t),k = 0,1, .....,N

σak = φk(a),σbk = φk(b)

the corresponding system of linear equations A(t)X(r) = B(r) is

γ0 γ1 γ2 γ3 δ0 δ1 δ2 δ3

σa0 σa1 σa2 σa3 0 0 0 0

σb0 σb1 σb2 σb3 0 0 0 0

δ0 δ1 δ2 δ3 γ0 γ1 γ2 γ3

0 0 0 0 σa0 σa1 σa2 σa3

0 0 0 0 σb0 σb1 σb2 σb3

a0(r)

a1(r)

a2(r)

a3(r)

a0(r)

a1(r)

a2(r)

a3(r)

=

g(t,r)

α(r)

β (r)

g(t,r)

α(r)

β (r)

Case:2 p(t)q(t)< 0

Suppose that the coefficient p(t) is non-negative and q(t) is negative. From (4.5), the

set of equations are

y′′(t,r)+ p(t)y

′(t,r)−q(t)y(t,r) = g(t,r) (4.21)

y′′(t,r)+ p(t)y

′(t,r)−q(t)y(t,r) = g(t,r) (4.22)


Equation (4.3) when substituted in (4.21), (4.22) and (4.23) gives, respectively,

N

∑k=0


N

∑k=0


N

∑k=0

ak(r)φk(t) = g(t,r) (4.24)

N

∑k=0


N

∑k=0


N

∑k=0

ak(r)φk(t) = g(t,r) (4.25)


and

N

∑k=0


∑k=0

ak(r)φk(b) = β (r) (4.26)

N

∑k=0


∑k=0

ak(r)φk(b) = β (r) (4.27)

By considering

ζk = φ′′k (t)+ p(t)φ

′k(t)

ξk =−q(t)φk(t),k = 0,1, .....,N



N∑

k=0ak(r)ζk−

N∑

k=0ak(r)ξk = g(t,r)

N∑


N∑


N∑

k=0ak(r)ζk−

N∑


N∑


N∑


(4.28)

(4.28) is a system of linear equations A(t)X(r) = B(r) such that

A =

A1 A2

A2 A1

,

where

A1 =

ζ0 ζ1 ζ2 ζ3

σa0 σa1 σa2 σa3

σb0 σb1 σb2 σb3

, A2 =

ξ0 ξ1 ξ2 ξ3

0 0 0 0

0 0 0 0


Similarly, when p(t) is negative and q(t) is non-negative, from (4.5), the set of

equations are:


′(t,r)+q(t)y(t,r) = g(t,r) (4.29)


′(t,r)+q(t)y(t,r) = g(t,r) (4.30)


Equation (4.3) when substituted in (4.29), (4.30) and (4.31) gives, respectively,

N

∑k=0


N

∑k=0

ak(r)φ′k(t)+q(t)

N

∑k=0

ak(r)φk(t) = g(t,r) (4.32)

N

∑k=0


N

∑k=0

ak(r)φ′k(t)+q(t)

N

∑k=0

ak(r)φk(t) = g(t,r) (4.33)

and

N

∑k=0


∑k=0

ak(r)φk(b) = β (r) (4.34)

N

∑k=0


∑k=0

ak(r)φk(b) = β (r) (4.35)

By setting

ζk = φ′′k (t)+q(t)φk(t)

ξk =−p(t)φ′k(t),k = 0,1, .....,N




N∑

k=0ak(r)ζk−

N∑


N∑


N∑


N∑

k=0ak(r)ζk−

N∑


N∑


N∑


(4.36)

where

X = (a0,a1,a2,a3,a0,a1,a2,a3)T , B = (g(t,r),α(r),β (r),g(t,r),α(r),β (r))T .

The parameters a0,a1,a2,a3,a0,a1,a2,a3 are obtained by solving (4.28) and (4.36) by

taking t = s,s ∈ [a,b]. These parameters provide the fuzzy approximate solution

(y(t,r),y(t,r)).

The system of linear equations A(t)X(r) = B(r) is

ζ0 ζ1 ζ2 ζ3 ξ0 ξ1 ξ2 ξ3

σa0 σa1 σa2 σa3 0 0 0 0

σb0 σb1 σb2 σb3 0 0 0 0

ξ0 ξ1 ξ2 ξ3 ζ0 ζ1 ζ2 ζ3

0 0 0 0 σa0 σa1 σa2 σa3

0 0 0 0 σb0 σb1 σb2 σb3

a0(r)

a1(r)

a2(r)

a3(r)

a0(r)

a1(r)

a2(r)

a3(r)

=

g(t,r)

α(r)

β (r)

g(t,r)

α(r)

β (r)

Thus, we obtained the fuzzy approximate solution of the given second order DE with

FBCs as

y(t,r) = a0(r)φ0(t)+a1(r)φ1(t)+a2(r)φ2(t)+a3(r)φ3(t) (4.37)

y(t,r) = a0(r)φ0(t)+a1(r)φ1(t)+a2(r)φ2(t)+a3(r)φ3(t) (4.38)


In general, for the Nth order DE with FIBCs, the system of linear equations

A(t)2N×2(N+1)X(r)2(N+1)×1 = B(r)2N×1, where N = n+1 is:

ζ0 ζ1 · · · ζN ξ0 ξ1 · · · ξN

σa0 σa1 · · · σaN 0 0 · · · 0

σb0 σb1 · · · σbN 0 0 · · · 0

ξ0 ξ1 · · · ξN ζ0 ζ1 · · · ζN

0 0 · · · 0 σa0 σa1 · · · σaN

0 0 · · · 0 σb0 σb1 · · · σbN

a0(r)

a1(r)...

aN(r)

a0(r)

a1(r)...

aN(r)

=

g(t,r)

α1(r)...

αN(r)

g(t,r)

α1(r)...

αN(r)

and

X =(a0,a1, .....,aN ,a0,a1, ....,aN)T , B=(g(t,r),α1(r), ..,αN(r),g(t,r),α1(r), ..,αN(r))T .

The system of linear equations A(t)X(r) = B(r), by taking t = s,s ∈ [a,b]. Here, we

obtained required solution by GEM by the help of MATLAB programming i.e.

X(r) = A−1(s)B(r)

The parameters a0,a1, .....,aN ,a0,a1, ....,aN are obtained by solving the above system.

These parameters provide the fuzzy approximate solution (y(t,r),y(t,r)).

Thus, we obtained the fuzzy approximate solution of the given Nth order DE with

FIBCs as

y(t,r) = a0(r)φ0(t)+a1(r)φ1(t)+ ....+aN(r)φN(t) (4.39)

y(t,r) = a0(r)φ0(t)+a1(r)φ1(t)+ ....+aN(r)φN(t) (4.40)

Solutions of higher order differential equations with fuzzy initial and boundaryconditions 55

4.4 Solutions of higher order differential equations with

fuzzy initial and boundary conditions

4.4.1 Solutions of a second order differential equation with fuzzy

boundary conditions

Example 4.1 Consider the second-order variable coefficient BVP:

(t2 +1)y′′−2ty′+2y = 0, t ∈ [1,2]

y(1) = 0

y(2) = 3

The general solution i.e. exact solution is

y(t) = c1t + c2(t2−1) (4.41)

By applying the boundary conditions, a particular solution is

y(t) = t2−1 (4.42)

Consider the second-order DE with FBCs:

(t2 +1)y′′−2ty

′+2y = 0 = [−0.25+0.25r,0.25−0.25r], t ∈ [1,2]

y(1) = [−0.5+0.5r,0.5−0.5r]

y(2) = [2.1+0.9r,3.5−0.5r]

The lower and upper bounds of the exact solutions are:

Y (t,r) = (−0.5+0.5r)t +13(3.1−0.1r)(t2−1)

Y (t,r) = (0.5−0.5r)t +13(2.5+0.5r)(t2−1)


By using MATLAB calculations

TABLE 4.1: The exact values for lower-bounds.

tr 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

0.0 -0.5000 -0.3330 -0.1453 0.0630 0.2920 0.5417 0.8120 1.1030 1.4147 1.7470 2.10000.1 -0.4500 -0.2787 -0.0868 0.1257 0.3588 0.6125 0.8868 1.1817 1.4972 1.8333 2.19000.2 -0.4000 -0.2244 -0.0283 0.1884 0.4256 0.6833 0.9616 1.2604 1.5797 1.9196 2.28000.3 -0.3500 -0.1701 0.0303 0.2511 0.4924 0.7542 1.0364 1.3391 1.6623 2.0059 2.37000.4 -0.3000 -0.1158 0.0888 0.3138 0.5592 0.8250 1.1112 1.4178 1.7448 2.0922 2.46000.5 -0.2500 -0.0615 0.1473 0.3765 0.6260 0.8958 1.1860 1.4965 1.8273 2.1785 2.55000.6 -0.2000 -0.0072 0.2059 0.4392 0.6928 0.9667 1.2608 1.5752 1.9099 2.2648 2.64000.7 -0.1500 0.0471 0.2644 0.5019 0.7596 1.0375 1.3356 1.6539 1.9924 2.3511 2.73000.8 -0.1000 0.1014 0.3229 0.5646 0.8264 1.1083 1.4104 1.7326 2.0749 2.4374 2.82000.9 -0.0500 0.1557 0.3815 0.6273 0.8932 0.1792 1.4852 1.8113 2.1575 2.5237 2.91001.0 0.0000 0.2100 0.4400 0.6900 0.9600 1.2500 1.5600 1.8900 2.2400 2.6100 3.0000

TABLE 4.2: The exact values for upper-bounds.

tr 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

0.0 0.5000 0.7250 0.9667 1.2250 1.5000 1.7917 2.1000 2.4250 2.7667 3.1250 3.50000.1 0.4500 0.6735 0.9140 1.1715 1.4460 1.7375 2.0460 2.3715 2.7140 3.0735 3.45000.2 0.4000 0.6220 0.8613 1.1180 1.3920 1.6833 1.9920 2.3180 2.6613 3.0220 3.40000.3 0.3500 0.5705 0.8087 1.0645 1.3380 1.6292 1.9380 2.2645 2.6087 2.9705 3.35000.4 0.3000 0.5190 0.7560 1.0110 1.2840 1.5750 1.8840 2.2110 2.5560 2.9190 3.30000.5 0.2500 0.4675 0.7033 0.9575 1.2300 1.5208 1.8300 2.1575 2.5033 2.8675 3.25000.6 0.2000 0.4160 0.6507 0.9040 1.1760 1.4667 1.7760 2.1040 2.4507 2.8160 3.20000.7 0.1500 0.3645 0.5980 0.8505 1.1220 1.4125 1.7220 2.0505 2.3980 2.7645 3.15000.8 0.1000 0.3130 0.5453 0.7970 1.0680 1.3583 1.6680 1.9970 2.3453 2.7130 3.10000.9 0.0500 0.2615 0.4927 0.7435 1.0140 1.3042 1.6140 1.9435 2.2927 2.6615 3.05001.0 0.0000 0.2100 0.4400 0.6900 0.9600 1.2500 1.5600 1.8900 2.2400 2.6100 3.0000

Plotting graph using Mathematica software

0.0

0.5

1.0

r

1.0

1.5

2.0

t

0

1

2

3

YHt,rL

FIGURE 4.1: The lower-bound and upper-bound for exact solution

The approximate solution of the given DE is obtained as under:

If φk(t) = tk; where k = 0,1,2,3


y(t,r) = a0(r)+a1(r)t +a2(r)t2 +a3(r)t

3

y(t,r) = a0(r)+a1(r)t +a2(r)t2 +a3(r)t3

Consider the system of equations

(t2 +1)[2a2(r)+6ta3(r)]−2t[a1(r)+2ta2(r)+3t2a3(r)]

+2[a0(r)+a1(r)t +a2(r)t2 +a3(r)t

3] =−0.25+0.25r

a0(r)+a1(r)+a2(r)+a3(r) =−0.5+0.5r

a0(r)+2a1(r)+4a2(r)+8a3(r) = 2.1+0.9r

(t2 +1)[2a2(r)+6ta3(r)]−2t[a1(r)+2ta2(r)+3t2a3(r)]

+2[a0(r)+a1(r)t +a2(r)t2 +a3(r)t3] = 0.25−0.25r

a0(r)+a1(r)+a2(r)+a3(r) = 0.5−0.5r

a0(r)+2a1(r)+4a2(r)+8a3(r) = 3.5−0.5r

The corresponding linear system is

2 2t 2+4t2 6t +8t3 0 −2t −4t2 −6t3

1 1 1 1 0 0 0 0

1 2 4 8 0 0 0 0

0 −2t −4t2 −6t3 2 2t 2+4t2 6t +8t3

0 0 0 0 1 1 1 1

0 0 0 0 1 2 4 8

a0(r)

a1(r)

a2(r)

a3(r)

a0(r)

a1(r)

a2(r)

a3(r)

=

−0.25+0.25r

−0.5+0.5r

2.1+0.9r

0.25−0.25r

0.5−0.5r

3.5−0.5r

By solving above system we can find approximate solution of the given problem.

Taking t = 1.5, the approximate solution is obtained as follows:

The lower bounds of solution are as:

a0(r) = 7.0833−8.0833r

a1(r) = 17.875r−17.875

a2(r) = 12.8917−11.8917r

a3(r) = 2.6r−2.6


y(t,r)= (7.0833−8.0833r)+(17.875r−17.875)t+(12.8917−11.8917r)t2+(2.6r−2.6)t3

The upper bounds of solution are as:

a0(r) = −0.5r−0.5

a1(r) = 0

a2(r) = 1

a3(r) = 0

y(t,r) = (−0.5r−0.5)+ t2


TABLE 4.3: The approximate values for lower-bounds.

tr 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

0.0 -0.5000 -0.4409 -0.2955 -0.0795 0.1961 0.5021 0.8364 1.1789 1.5141 1.8263 2.10000.1 -0.4500 -0.3758 -0.2219 -0.0025 0.2684 0.5769 0.9088 1.2501 1.5867 1.9047 2.19000.2 -0.4000 -0.3107 -0.1484 0.0744 0.3453 0.6517 0.9811 1.3212 1.6593 1.9831 2.28000.3 -0.3500 -0.2456 -0.0784 0.1514 0.4221 0.7265 1.0535 1.3923 1.7319 2.0614 2.37000.4 -0.3000 -0.1805 -0.0013 0.2283 0.4940 0.8013 1.1258 1.4634 1.8045 2.1398 2.46000.5 -0.2500 -0.1154 0.0723 0.3053 0.5758 0.8760 1.1982 1.5345 1.8771 2.2182 2.55000.6 -0.2000 -0.0503 0.1458 0.3822 0.6526 0.9508 1.2706 1.6056 1.9497 2.2965 2.64000.7 -0.1500 0.0147 0.2194 0.4592 0.7295 1.0256 1.3429 1.6767 2.0222 2.3749 2.73000.8 -0.1000 0.0798 0.2929 0.5361 0.8063 1.1004 1.4153 1.7478 2.0948 2.4533 2.82000.9 -0.0500 0.1449 0.3665 0.6131 0.8832 0.1752 1.4876 1.8189 2.1674 2.5316 2.91001.0 0.0000 0.2100 0.4400 0.6900 0.9600 1.2500 1.5600 1.8900 2.2400 2.6100 3.0000

TABLE 4.4: The approximate values for upper-bounds.

tr 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

0.0 0.5000 0.7100 0.9400 1.1900 1.4600 1.7500 2.0600 2.3900 2.7400 3.1100 3.50000.1 0.4500 0.6600 0.8900 1.1400 1.4100 1.7000 2.0100 2.3400 2.6900 3.0600 3.45000.2 0.4000 0.6100 0.8400 1.0900 1.3600 1.6500 1.9600 2.2900 2.6400 3.0100 3.40000.3 0.3500 0.5600 0.7900 1.0400 1.3100 1.6000 1.9100 2.2400 2.5900 2.9600 3.35000.4 0.3000 0.5100 0.7400 0.9900 1.2600 1.5500 1.8600 2.1900 2.5400 2.9100 3.30000.5 0.2500 0.4600 0.6900 0.9400 1.2100 1.5000 1.8100 2.1400 2.4900 2.8600 3.25000.6 0.2000 0.4100 0.6400 0.8900 1.1600 1.4500 1.7600 2.0900 2.4400 2.8100 3.20000.7 0.1500 0.3600 0.5900 0.8400 1.1100 1.4000 1.7100 2.0400 2.3900 2.7600 3.15000.8 0.1000 0.3100 0.5400 0.7900 1.0600 1.3500 1.6600 1.9900 2.3400 2.7100 3.10000.9 0.0500 0.2600 0.4900 0.7400 1.0100 1.3000 1.6100 1.9400 2.2900 2.6600 3.05001.0 0.0000 0.2100 0.4400 0.6900 0.9600 1.2500 1.5600 1.8900 2.2400 2.6100 3.0000



0.0

0.5

1.0

r

1.0

1.5

2.0

t

0

1

2

3

yHt,rL

FIGURE 4.2: The lower-bound and upper-bound for approximate solution

4.4.2 Solution of fourth order differential equation with fuzzy

initial and boundary conditions

4.4.2.1 Example based on the application of differential equation in civil

engineering

Consider a uniform beam as made up of fibres running lengthwise. We have to find its

deflection under given loadings.

M =EIR

Where R is a radius of curvature of elastic curve.

R =[1+(dy

dx)2]

32

d2ydx2

Thus for small deflection R =1

d2ydx2

Bending moment M = EId2ydx2

Shear force(=

dMdx

)= EI

d3ydx3

Intensity of loading(=

d2Mdx2

)= EI

d4ydx4


Example 4.2 A beam of length L is embedded at both ends. Find the deflection of the

beam if a constant load w0 is uniformly distributed along its length that is,

w(x) = w0,0 < x < L .

EId4ydx4 = w0,x ∈ [0,L]

y(0) = 0,y′(0) = 0,y(L) = 0,y

′(L) = 0

x

y

Wo

θRθL δR1 R2

L

o

FIGURE 4.3: Elastic or Deflection curve

The general solution i.e exact solution is

y(x) = c1 + c2x+ c3x2 + c4x3 +w0

EIx4 (4.43)

By using the boundary conditions, a particular solution is

y(x) =w0L2

24EIx2− w0L2

12EIx3 +

w0

24EIx4

y(x) =w0

24EIx2(x−L)2

If w0 = 24EI,L = 1 then y(x) = x2(x−1)2

Consider the fourth-order DE with FIBCs:

Here,we take w0 = 24EI,L = 1

d4ydx4 = 24,x ∈ [0,1]

y(0) = y′(0) = y(1) = y

′(1) = [r−1,1− r]


The lower and upper bounds of the exact solutions are:

Y (x,r) = (r−1)+(r−1)x+[(−6r+6)+

23+ r24

]x2 +

[(4r−4)− 2(23+ r)

24

]x3 +

[23+ r

24

]x4

Y (x,r) = (1− r)+(1− r)x+[(6r−6)+

25− r24

]x2 +

[(−4r+4)− 2(25− r)

24

]x3 +

[25− r

24

]x4


TABLE 4.5: The exact values for lower-bounds.

xr 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 -1.0000 -1.0362 -0.9675 -0.8257 -0.6408 -0.4401 -0.2488 -0.0897 0.0165 0.0518 -0.00000.1 -0.9000 -0.9318 -0.8682 -0.7388 -0.5710 -0.3898 -0.2182 -0.0764 0.0174 0.0474 0.00000.2 -0.8000 -0.8274 -0.7689 -0.6518 -0.5011 -0.3396 -0.1857 -0.0630 0.0183 0.0430 -0.00000.3 -0.7000 -0.7729 -0.6695 -0.5648 -0.4313 -0.2893 -0.1569 -0.0496 0.0193 0.0387 -0.00000.4 -0.6000 -0.6185 -0.5702 -0.4778 -0.3614 -0.2391 -0.1262 -0.0362 0.0202 0.0343 -0.00000.5 -0.5000 -0.5141 -0.4792 -0.3908 -0.2916 -0.1888 -0.0956 -0.0228 0.0211 0.0299 0.00000.6 -0.4000 -0.4096 -0.3716 -0.3038 -0.2218 -0.1385 -0.0650 -0.0094 0.0220 0.0256 0.00000.7 -0.3000 -0.3052 -0.2723 -0.2169 -0.1519 -0.0883 -0.0343 0.0039 0.0229 0.0212 0.00000.8 -0.2000 -0.2008 -0.1730 -0.1299 -0.0821 -0.0380 -0.0037 0.0173 0.0238 0.0168 -0.00000.9 -0.1000 -0.0963 -0.0737 -0.0429 -0.0122 0.0122 0.0270 0.0307 0.0247 0.0125 -0.00001.0 0.0000 0.0081 0.0256 0.0441 0.0576 0.0625 0.0576 0.0441 0.0256 0.0081 0.0000

TABLE 4.6: The exact values for upper-bounds.

xr 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 1.0000 1.0524 1.0187 0.9139 0.7560 0.5651 0.3640 0.1779 0.0347 -0.0356 0.00000.1 0.9000 0.9480 0.9194 0.8270 0.6862 0.5148 0.3334 0.1646 0.0338 -0.0312 -0.00000.2 0.8000 0.8436 0.8201 0.7400 0.6163 0.4646 0.3027 0.1512 0.0329 -0.0268 0.00000.3 0.7000 0.7391 0.7207 0.6530 0.5465 0.4143 0.2721 0.1378 0.0319 -0.0225 0.00000.4 0.6000 0.6347 0.6214 0.5660 0.4766 0.3641 0.2414 0.1244 0.0310 -0.0181 0.00000.5 0.5000 0.5303 0.5221 0.4790 0.4068 0.3138 0.2108 0.1110 0.0301 -0.0137 0.00000.6 0.4000 0.4258 0.4228 0.3920 0.3370 0.2635 0.1802 0.0976 0.0292 -0.0094 0.00000.7 0.3000 0.3214 0.3235 0.3051 0.2671 0.2133 0.1495 0.0843 0.0283 -0.0050 -0.00000.8 0.2000 0.2170 0.2242 0.2181 0.1973 0.1630 0.1189 0.0709 0.0274 -0.0006 0.00000.9 0.1000 0.1125 0.1249 0.1311 0.1274 0.1128 0.0882 0.0575 0.0265 0.0037 0.00001.0 0.0000 0.0081 0.0256 0.0441 0.0576 0.0625 0.0576 0.0441 0.0256 0.0081 0.0000


0.0

0.5

1.0

r

0.0

0.5

1.0

x

-1.0

-0.5

0.0

0.5

1.0

YHx,rL

FIGURE 4.4: The lower-bound and upper-bound for exact solution


The approximate solution of the given DE is obtained as under:

If φk(x) = xk where k = 0,1,2,3,4,5

y(x,r) = a0(r)+a1(r)x+a2(r)x2 +a3(r)x

3 +a4(r)x4 +a5(r)x

5

y(x,r) = a0(r)+a1(r)x+a2(r)x2 +a3(r)x3 +a4(r)x4 +a5(r)x5

Consider the system of equations

24a4(r)+120xa5(r) = 23+ r

a0(r) = r−1

a1(r) = r−1

a0(r)+a1(r)+a2(r)+a3(r)+a4(r)+a5(r) = r−1

a1(r)+2a2(r)+3a3(r)+4a4(r)+5a5(r) = r−1

24a4(r)+120xa5(r) = 25− r

a0(r) = 1− r

a1(r) = 1− r

a0(r)+a1(r)+a2(r)+a3(r)+a4(r)+a5(r) = 1− r

a1(r)+2a2(r)+3a3(r)+4a4(r)+5a5(r) = 1− r

The corresponding linear system is

0 0 0 0 24 120x 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 0 0 0 0 0 0

0 1 2 3 4 5 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 24 120x

0 0 0 0 0 0 1 0 0 0 0 0

0 0 0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 1 1 1 1 1 1

0 0 0 0 0 0 0 1 2 3 4 5

a0(r)

a1(r)

a2(r)

a3(r)

a4(r)

a5(r)

a0(r)

a1(r)

a2(r)

a3(r)

a4(r)

a5(r)

=

23+ r

r−1

r−1

r−1

r−1

25− r

1− r

1− r

1− r

1− r

By solving above system we can find approximate solution of the given problem.


Taking x = 0.5, the approximate solution is obtained as follows:

The lower bounds of solution are as:

a0(r) = a1(r) = r−1

a2(r) = 3.9583−2.9583r

a3(r) = 1.9167r−3.9167

a4(r) = 0.04167r+0.9583

a5(r) = 0

y(x,r)= (r−1)+(r−1)x+(3.9583−2.9583r)x2+(1.9167r−3.9167)x3+(0.04167r+0.9583)x4

The upper bounds of solution are as:

a0(r) = a1(r) = 1− r

a2(r) = 2.9583r−1.9583

a3(r) = −1.9167r−0.0833

a4(r) = 1.04167−0.04167r

a5(r) = 0

y(x,r)= (1−r)+(1−r)x+(2.9583r−1.9583)x2+(−1.9167r−0.0833)x3+(1.04167−0.04167r)x4


TABLE 4.7: The approximate values for lower-bounds.

xr 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 -1.0000 -1.0642 -1.0715 -1.0417 -0.9928 -0.9401 -0.8968 -0.8737 -0.8795 -0.9202 -1.00000.1 -0.9000 -0.9570 -0.9618 -0.9332 -0.8878 -0.8398 -0.8014 -0.7820 -0.7890 -0.8274 -0.90000.2 -0.8000 -0.8498 -0.8521 -0.8246 -0.7827 -0.7396 -0.7059 -0.6902 -0.6985 -0.7346 -0.80000.3 -0.7000 -0.7425 -0.7423 -0.7160 -0.6777 -0.6393 -0.6105 -0.5984 -0.6079 -0.6417 -0.70000.4 -0.6000 -0.6353 -0.6326 -0.6074 -0.5726 -0.5391 -0.5150 -0.5066 -0.5174 -0.5489 -0.60000.5 -0.5000 -0.5281 -0.5229 -0.4988 -0.4676 -0.4388 -0.4196 -0.4148 -0.4269 -0.4561 -0.50000.6 -0.4000 -0.4208 -0.4132 -0.3902 -0.3626 -0.3385 -0.3242 -0.3230 -0.3364 -0.3632 -0.40000.7 -0.3000 -0.3136 -0.3035 -0.2817 -0.2575 -0.2383 -0.2287 -0.2313 -0.2459 -0.2704 -0.30000.8 -0.2000 -0.2064 -0.1938 -0.1731 -0.1525 -0.1380 -0.1333 -0.1395 -0.1554 -0.1776 -0.20000.9 -0.1000 -0.0991 -0.0841 -0.0645 -0.0475 -0.0378 -0.0378 -0.0477 -0.0649 -0.0847 -0.10001.0 0.0000 0.0081 0.0256 0.0441 0.0576 0.0625 0.0576 0.0441 0.0256 0.0081 0.0000


TABLE 4.8: The approximate values for upper-bounds.

xr 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 1.0000 1.0804 1.1227 1.1299 1.1080 1.0651 1.0120 0.9619 0.9307 0.9364 1.00000.1 0.9000 0.9732 0.0130 0.0124 1.0030 0.9648 0.9166 0.8702 0.8402 0.8435 0.90000.2 0.8000 0.8660 0.9033 0.9128 0.8979 0.8646 0.8211 0.7784 0.7497 0.7508 0.80000.3 0.7000 0.7587 0.7935 0.8042 0.7929 0.7643 0.7257 0.6866 0.6591 0.6579 0.70000.4 0.6000 0.6515 0.6838 0.6956 0.6878 0.6641 0.6302 0.5948 0.5686 0.5651 0.60000.5 0.5000 0.5443 0.5741 0.5870 0.5828 0.5638 0.5348 0.5030 0.4781 0.4723 0.50000.6 0.4000 0.4370 0.4644 0.4784 0.4778 0.4635 0.4394 0.4112 0.3876 0.3794 0.40000.7 0.3000 0.3298 0.3547 0.3727 0.3727 0.3633 0.3439 0.3195 0.2971 0.2866 0.30000.8 0.2000 0.2226 0.2450 0.2677 0.2677 0.2630 0.2485 0.2277 0.2066 0.1938 0.20000.9 0.1000 0.1153 0.1353 0.1626 0.1626 0.1628 0.1530 0.1359 0.1161 0.1009 0.10001.0 0.0000 0.0081 0.0256 0.0441 0.0576 0.0625 0.0576 0.0441 0.0256 0.0081 0.0000


0.0

0.5

1.0

r

0.0

0.5

1.0

x

-1.0

-0.5

0.0

0.5

1.0

yHx,rL

FIGURE 4.5: The lower-bound and upper-bound for approximate solution


Tables 4.1, 4.2 gives us lower and upper bounds for the exact solution and Tables 4.3,

4.4 gives lower and upper bounds for an approximate solution for Example 4.1 i.e. the

DE with FBCs, respectively. Similarly, the Tables 4.5, 4.6 gives us lower and upper

bounds for the exact solution and the Tables 4.7, 4.8 gives lower and upper bounds for

an approximate solution for Example 4.2 i.e. the non-homogeneous DE which occurred

in civil engineering in that we can estimate the lower and upper bounds of the defection

of the beam under constant load, respectively. In all the above Tables we can observe

that, in between some changes in parameters but for r = 1 endpoints are coincide with

Conclusion 65

given exact conditions of the DEs. Figs. 4.1, 4.2, 4.3, 4.4 shows graphical representation

the lower and upper bounds of the exact and an approximate solution for Examples 4.1,

4.2 respectively. Here, the Figs. 4.1, 4.2 nearly consistent with each other and the Figs.

4.3, 4.4 nearly consistent with each other for Examples 4.1, 4.2 respectively.

4.6 Conclusion

The DEs with FIBCs are investigated as a solution of the matrix. Here the Gauss

elimination method is applied to solve the second-order DE with FBCs and the

fourth-order DE with FIBCs for simplicity purpose but it is also applicable to solve the

nth order DEs with FIBCs. If we assume that, the approximate solution of the problem

in the form of a polynomial of one power more than the order of given DE with FIBCs

then the obtained solution of a given problem is nearer to the exact solution. Here, we

discussed constant-coefficient, variable coefficient, homogeneous and

non-homogeneous higher-order DEs with FIBCs. Approximate and exact solutions of

given examples are nearly consistent with each-other. So, our method is relevant for

solving nth order DEs with FIBCs.

CHAPTER 5

Solutions of Higher Order Differential

Equations with Fuzzy Boundary

Conditions by Finite Difference

Method

5.1 Introduction

Most physical phenomena are modelled by the system of ordinary or partial DEs.

Differential equations with FBCs play an important role in modelling problems in

Science and Engineering because of the way they model dynamical systems under

fuzziness. Opanuga et al. [61] used the finite difference method to solve crisp

boundary value problems. Numerical solutions of FDEs by the predictor-corrector

method were discussed by Bede [16]. An improved predictor-corrector method for

solving fuzzy initial value problems was discussed by Allahviranloo et al. [7]. Ma et

al. [56] used classical Euler’s method to solve FDEs. Friedman et al. [33] proposed a

numerical method for solving fuzzy differential and integral equations. The Euler’s

approximation method was used to solve the FDEs under GHD by Nieto et al. [60].

The Runge-Kutta method was applied by Palligkinis et al. [63] for solving FDEs.

Armand et al. [13] solved two-point FBVP using the variational iteration method.

In general, to find the exact solution of BVPs is difficult, so we have to rely on

numerical methods. Here, we are going to convert DEs with FBCs to the system of

linear equations by using difference equations, and then solve that system of linear

66

Problem formulation 67

equations by FDM and using MATLAB programming. We also solve Airy’s

non-homogeneous DE with FBCs.


In mathematics, in the field of DEs, a BVP is a DE together with a set of additional

constrains, called the boundary conditions. The solution to a BVP is a solution to the

DE which also satisfies the boundary conditions.

On the other hand, we are going to solve DEs with FBCs by using finite

difference method. In that, the DE with FBCs on a particular interval is divided into

small intervals and solved using MATLAB. So, in this way, we can apply MATLAB to

approximate the solution of a particular DE with FBCs and represent it graphically to

determine the lower and upper bounds of the solution.

Theorem: 5.2.1 Lipschitz conditions

Suppose the function f in the DE with FBCs

y′′= f (x,y,y

′), a≤ x≤ b, y(a) = α, y(b) = β

is continuous on the set

D = {(x,y,y′)/a≤ x≤ b,−∞ < y < ∞,−∞ < y′< ∞}

and that the partial derivatives fy and f′y are also continuous in D. If

(1) fy(x,y,y′)> 0, ∀ (x,y,y′) ∈ D

(2) a constant M exists with | fy′ (x,y,y′)| ≤M, ∀ (x,y,y′) ∈ D

then the DE with FBCs has a unique solution in terms of parametric forms of fuzzy

numbers.

The finite difference approximation to the various derivatives are as under:

If y(x) and its derivatives are single valued continuous functions of x then by Taylor’s

expansion, we get

Solutions of Higher Order Differential Equations with Fuzzy Boundary Conditions byFinite Difference Method 68

y(x+h) = y(x)+hy′(x)+

h2

2y′′(x)+

h3

6y′′′(x)+

h4

24yiv(x)(ζ+), (5.1)

for some ζ+ ∈ (x,x+h), and

y(x−h) = y(x)−hy′(x)+

h2

2y′′(x)− h3

6y′′′(x)+

h4

24yiv(x)(ζ−), (5.2)

for some ζ− ∈ (x−h,x).

Equation (5.1) gives

y′(x) =

1h[y(x+h)− y(x)]− h

2y′′(x)− ......

i.e.

y′(x) =

1h[y(x+h)− y(x)]+O(h)

which is the forward difference approximation of y′(x) with an error of the order h.

Similarly, (5.2) gives

y′(x) =

1h[y(x)− y(x−h)]+

h2

y′′(x)+ ......

i.e.

y′(x) =

1h[y(x)− y(x−h)]+O(h)

which is the backward difference approximation of y′(x) with an error of the order h.

Subtracting (5.2) from (5.1), we get

y′(x) =

12h

[y(x+h)− y(x−h)]− h2

6y′′′(η),

for some η ∈ (x−h,x+h)

⇒ y′(x) =

12h

[y(x+h)− y(x−h)]+O(h2)

which is the central-difference approximation of y′(x) with an error of the order h2.


The central difference approximation of y′(x) is better than the forward or backward

difference approximation.

Adding (5.1) and (5.2)

y′′(x) =

1h2 [y(x+h)−2y(x)+ y(x−h)]− h2

24[yiv(x)(ζ+)+ yiv(x)(ζ−)]

The Intermediate Value Theorem can be used to simplify this even further.

y′′(x) =

1h2 [y(x+h)−2y(x)+ y(x−h)]− h2

12yiv(x)(ζ )

for some ζ ∈ (x−h,x+h)

⇒ y′′(x) =

1h2 [y(x+h)−2y(x)+ y(x−h)]+O(h2)

which is the central difference approximation of y′′(x) with an error of the order h2 [61].

Similarly we can derive central difference approximations to higher order derivatives.

Thus, the working expressions for the central difference approximations to the first two

derivatives of yi are:

y′i =

12h

(yi+1− yi−1) (5.3)

y′′i =

1h2 (yi+1−2yi + yi−1) (5.4)

Similarly, we can find central difference approximations to higher order derivatives.

y′′′i =

12h3 (yi+2−2yi+1 +2yi−1− yi−2) (5.5)

yivi =

1h4 (yi+2−4yi+1 +6yi−4yi−1 + yi−2) (5.6)


5.3 Method for solving higher order differential

equation with FBCs

Consider the second order differential equation

y′′(x)+ p(x)y′(x)+q(x)y(x) = r(x), x ∈ [a,b] (5.7)

together with the FBCs

y(a) = α, y(b) = β (5.8)

where p(x),q(x) and r(x) are constants or functions of x on [a,b].

Here, we divide the interval I = [a,b] into a chosen n numbers of subintervals of equal

width. Thus, the step size h of each of the n subintervals is given by h =b−a

n.

yi denotes the value of the function at ith node of the computational grid.

y0 = α, yn = β .

At any mesh point or node point x = xi the finite difference representation of the

differential equation with fuzzy boundaries can be written for as (i = 1,2, ...,n−1):

yi+1−2yi + yi−1

h2 + p(xi)yi+1− yi−1

2h+q(xi)yi = r(xi)+

h2

12[2p(xi)y

′′′(ηi)+ yiv(ζi)].

(5.9)

A Finite difference method with truncation error of the order O(h2) results into the

equation(5.10) after ignoring truncation error [61].

2(yi+1−2yi + yi−1)+hp(xi)(yi+1− yi−1)+2h2q(xi)yi = 2h2r(xi). (5.10)

The boundary conditions provides the solution at the two ends of the grid

i.e. at y0 = α and yn = β .

Method for solving higher order differential equation with FBCs 71

The system of linear equations A(N+1)×(N+1)X(N+1)×1 = B(N+1)×1 is obtained.

A =

1 0 0 · · · · · · 0

(2h2q1−4) (2+hp1) 0 · · · · · · 0

(2−hp2) (2h2q2−4) (2+hp2) 0 · · · 0

0 (2−hp3) (2h2q3−4) (2+hp3) · · · 0...

... . . . . . . . . . ...

0 · · · 0 (2−hpn−2)(2h2qn−2−4) (2+hpn−2)

0 · · · · · · 0 (2−hpn−1) (2h2qn−1−4)

0 0 · · · · · · 0 1

X =

y0

y1

y2

y3...

yn−2

yn−1

yn

B =

α

2h2r1− (2−hp1)α

2h2r2

2h2r3...

2h2rn−2

2h2rn−1− (2+hpn−1)β

β

The above system is a non-singular tridiagonal system and it has a unique solution in

form of fuzzy parameters.


5.4 Solution of the higher order differential equation

with fuzzy boundary conditions

5.4.1 Solution of the second order differential equation with fuzzy

boundary conditions

Example 5.1 Consider the DE with crisp boundary conditions

y′′= x+ y, x ∈ [0,1]

y(0) = 0

y(1) = 0

Exact solution is

Y (x) =sinhxsinh1

− x. (5.11)

Consider the DE with FBCs

y′′= x+ y, x ∈ [0,1]

y(0) = y(1) = [−0.5+0.5r,0.5−0.5r]

Put xn = x0 +nh, x0 = 0, h = 0.1 and let yn to be calculated as yn = y(xn)

Here, we divide the interval [0,1] into 10 sub-intervals.

By using the central difference approximation:

yi+1−2yi + yi−1

h2 = xi + yi, i = 1,2, ....,10,

for i = 1:

100y0−201y1 +100y2 = 0.1

for i = 2:

100y1−201y2 +100y3 = 0.2

Solution of the higher order differential equation with fuzzy boundary conditions 73

for i = 3:

100y2−201y3 +100y4 = 0.3

and so on

for i = 9:

100y8−201y9 +100y10 = 0.9

We set y0 = y10 = −0.5+ 0.5r,r ∈ [0,1], to obtain the system of linear equations with

unknowns y0,y1,y2, ....,y10 as:

1 0 0 0 0 0 0 0 0 0 0

0−201 100 0 0 0 0 0 0 0 0

0 100 −201 100 0 0 0 0 0 0 0

0 0 100 −201 100 0 0 0 0 0 0

0 0 0 100 −201 100 0 0 0 0 0

0 0 0 0 100 −201 100 0 0 0 0

0 0 0 0 0 100 −201 100 0 0 0

0 0 0 0 0 0 100 −201 100 0 0

0 0 0 0 0 0 0 100 −201 100 0

0 0 0 0 0 0 0 0 100 −201 0

0 0 0 0 0 0 0 0 0 0 1

y0

y1

y2

y3

y4

y5

y6

y7

y8

y9

y10

=

0.5r−0.5

0.6−0.5r

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1.4−0.5r

0.5r−0.5


The lower-bounds for the solution are:

y0(0) = 0.5r−0.5

y1(0.1) = 0.00479373r−0.019549

y2(0.2) = 0.00463541r−0.0332936

y3(0.3) = 0.00452343r−0.0453711

y4(0.4) = 0.00445669r−0.0549022

y5(0.5) = 0.00443452r−0.0609824

y6(0.6) = 0.00445669r−0.0626725

y7(0.7) = 0.00452343r−0.0589892

y8(0.8) = 0.00463541r−0.0488959

y9(0.9) = 0.00479373r−0.0312915

y10(1) = 0.5r−0.5

Next, by setting y0 = y10 = 0.5− 0.5r,r ∈ [0,1], we get the system of linear equations

with unknowns y0,y1,y2, ....,y10 as:

1 0 0 0 0 0 0 0 0 0 0

0−201 100 0 0 0 0 0 0 0 0

0 100 −201 100 0 0 0 0 0 0 0

0 0 100 −201 100 0 0 0 0 0 0

0 0 0 100 −201 100 0 0 0 0 0

0 0 0 0 100 −201 100 0 0 0 0

0 0 0 0 0 100 −201 100 0 0 0

0 0 0 0 0 0 100 −201 100 0 0

0 0 0 0 0 0 0 100 −201 100 0

0 0 0 0 0 0 0 0 100 −201 0

0 0 0 0 0 0 0 0 0 0 1

y0

y1

y2

y3

y4

y5

y6

y7

y8

y9

y10

=

0.5−0.5r

−0.4+0.5r

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.4+0.5r

0.5−0.5r


which gives the upper-bounds for the solution:

y0(0) = 0.5−0.5r

y1(0.1) = −0.00479373r−0.00996158

y2(0.2) = −0.00463541r−0.0240228

y3(0.3) = −0.00452343r−0.0363242

y4(0.4) = −0.00445669r−0.0459889

y5(0.5) = −0.00443452r−0.0521134

y6(0.6) = −0.00445669r−0.0537591

y7(0.7) = −0.00452343r−0.0499424

y8(0.8) = −0.00463541r−0.0396251

y9(0.9) = −0.00479373r−0.021704

y10(1) = 0.5−0.5r

By using MATLAB, the calculations obtained are as in Tables 5.1 and 5.2.

TABLE 5.1: Approximate values for the lower-bounds.

xr 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 -0.5000 -0.0195 -0.0333 -0.0454 -0.0549 -0.0610 -0.0627 -0.0590 -0.0489 -0.0313 -0.50000.1 -0.4500 -0.0191 -0.0328 -0.0449 -0.0545 -0.0605 -0.0622 -0.0585 -0.0484 -0.0308 -0.45000.2 -0.4000 -0.0186 -0.0324 -0.0445 -0.0540 -0.0601 -0.0618 -0.0581 -0.0480 -0.0303 -0.40000.3 -0.3500 -0.0181 -0.0319 -0.0440 -0.0536 -0.0597 -0.0613 -0.0576 -0.0475 -0.0299 -0.35000.4 -0.3000 -0.0176 -0.0314 -0.0436 -0.0531 -0.0592 -0.0609 -0.0572 -0.0470 -0.0294 -0.30000.5 -0.2500 -0.0172 -0.0310 -0.0431 -0.0527 -0.0588 -0.0604 -0.0567 -0.0466 -0.0289 -0.25000.6 -0.2000 -0.0167 -0.0305 -0.0427 -0.0522 -0.0583 -0.0600 -0.0563 -0.0461 -0.0284 -0.20000.7 -0.1500 -0.0162 -0.0300 -0.0422 -0.0518 -0.0579 -0.0596 -0.0558 -0.0457 -0.0279 -0.15000.8 -0.1000 -0.0157 -0.0296 -0.0418 -0.0513 -0.0574 -0.0591 -0.0554 -0.0452 -0.0275 -0.10000.9 -0.0500 -0.0152 -0.0291 -0.0413 -0.0509 -0.0570 -0.0587 -0.0549 -0.0447 -0.0270 -0.05001.0 -0.0000 -0.0148 -0.0287 -0.0408 -0.0504 -0.0562 -0.0582 -0.0545 -0.0443 -0.0265 0.0000


TABLE 5.2: Approximate values for the upper-bounds.

xr 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.5000 -0.0100 -0.0240 -0.0363 -0.0460 -0.0521 -0.0538 -0.0499 -0.0396 -0.0217 0.50000.1 0.4500 -0.0104 -0.0245 -0.0368 -0.0464 -0.0526 -0.0542 -0.0504 -0.0401 -0.0222 0.45000.2 0.4000 -0.0109 -0.0249 -0.0372 -0.0469 -0.0530 -0.0547 -0.0508 -0.0406 -0.0227 0.40000.3 0.3500 -0.0114 -0.0254 -0.0377 -0.0473 -0.0534 -0.0551 -0.0513 -0.0410 -0.0231 0.35000.4 0.3000 -0.0119 -0.0259 -0.0381 -0.0478 -0.0539 -0.0555 -0.0518 -0.0415 -0.0236 0.30000.5 0.2500 -0.0124 -0.0263 -0.0386 -0.0482 -0.0543 -0.0560 -0.0522 -0.0419 -0.0241 0.25000.6 0.2000 -0.0128 -0.0268 -0.0390 -0.0487 -0.0548 -0.0564 -0.0527 -0.0424 -0.0246 0.20000.7 0.1500 -0.0133 -0.0273 -0.0395 -0.0491 -0.0552 -0.0569 -0.0531 -0.0429 -0.0251 0.15000.8 0.1000 -0.0138 -0.0277 -0.0399 -0.0496 -0.0557 -0.0573 -0.0536 -0.0433 -0.0255 0.10000.9 0.0500 -0.0143 -0.0282 -0.0404 -0.0500 -0.0561 -0.0578 -0.0540 -0.0438 -0.0260 0.05001.0 0.0000 -0.0148 -0.0287 -0.0408 -0.0504 -0.0565 -0.0582 -0.0545 -0.0443 -0.0265 0.0000

These bounds are shown graphically as in Fig.5.1.

0-0.5

0.11

0

0.20.9

Y v

alue

s

0.5

0.8 0.30.7 0.40.6 0.5

r values0.5

X values0.60.4 0.70.3 0.80.2 0.90.1 01

FIGURE 5.1: The lower-bounds and the upper-bounds for an approximate solution


5.4.2 Airy’s functions

In physical sciences, the Airy’s function (or Airy’s function of the first kind) Ai(x) is a

special function named after the British astronomer George Biddell Airy (1801–92).

They commonly appear in physics, especially in optics, quantum mechanics,

electromagnetic, and radioactive transfer. The Airy’s stress function is employed in

solid mechanics. The function Ai(x) and the related function Bi(x), are linearly

independent solutions to the differential equation

d2ydx2 − xy = 0 (5.12)

known as Airy’s equation or Stokes equation. This is the simplest second-order linear

differential equation with a turning point (a point where the character of the solutions

changes from oscillatory to exponential). Here, we are going to solve Airy’s

non-homogeneous differential equation with FBCs.

5.4.2.1 Solution of Airy’s non-homogeneous differential equation with fuzzy

boundary conditions

Example 5.2 Consider Airy’s non-homogeneous DE with crisp boundary conditions

y′′−(

1− x5

)y = x, x ∈ [1,3]

y(1) = 2

y(3) =−1

The exact solution is difficult to find and we can use Mathematica/MATLAB to find

the solution, however, it is too long to write and interpret. So, we will first apply FDM

then the obtained system is solved by the help of MATLAB.


Consider Airy’s non-homogeneous DE with FBCs

y′′−(

1− x5

)y = x, x ∈ [1,3]

y(1) = [1.9+0.1r,2.01−0.01r]

y(3) = [−0.5−0.5r,−1.01+0.01r]

Let xn = x0 +nh, x0 = 1, h = 0.2, and let yn be calculated by yn = y(xn)

Here, we divide the interval [1,3] into 10 sub-intervals.

By using the central difference approximation:

yi+1−2yi + yi−1

h2 −(

1− xi

5

)yi = xi, i = 1,2, ...,10.

This gives,

yi−1−[

2+(

1− xi

5

)h2]

yi + yi+1 = xih2, i = 1,2, ...,10.

For i = 1:

y0−[

2+(

1− 1.25

)(0.2)2

]y1 + y2 = 1.2(0.2)2

for i = 2:

y1−[

2+(

1− 1.45

)(0.2)2

]y2 + y3 = 1.4(0.2)2

for i = 3:

y2−[

2+(

1− 1.65

)(0.2)2

]y3 + y4 = 1.6(0.2)2

and so on

for i = 9:

y8−[

2+(

1− 2.85

)(0.2)2

]y9 + y10 = 2.8(0.2)2


We set y0 = 1.9 + 0.1r, y10 = −0.5− 0.5r, r ∈ [0,1], to obtain the system of linear

equations with unknowns y0,y1,y2, ....y10 as:

1 0 0 0 0 0 0 0 0 0 0

0−2.0304 1 0 0 0 0 0 0 0 0

0 1 −2.0288 1 0 0 0 0 0 0 0

0 0 1 −2.0272 1 0 0 0 0 0 0

0 0 0 1 −2.0256 1 0 0 0 0 0

0 0 0 0 1 −2.0240 1 0 0 0 0

0 0 0 0 0 1 −2.0224 1 0 0 0

0 0 0 0 0 0 1 −2.0208 1 0 0

0 0 0 0 0 0 0 1 −2.0192 1 0

0 0 0 0 0 0 0 0 1 −2.0176 0

0 0 0 0 0 0 0 0 0 0 1

y0

y1

y2

y3

y4

y5

y6

y7

y8

y9

y10

=

1.9+0.1r

−1.852−0.1r

0.056

0.064

0.072

0.08

0.088

0.096

0.1040

0.612+0.5r

−0.5−0.5r


The lower-bounds for the solution are:

y0(1) = 1.9+0.1r

y1(1.2) = 0.048771r+1.30194

y2(1.4) = 0.791465−0.00097538r

y3(1.6) = 0.359782−0.0507498r

y4(1.8) = 0.00188402−0.101905r

y5(2.0) = −0.213168r−0.49663

y6(2.2) = −0.275443r−0.624419

y7(2.4) = −0.343446r−0.677196

y8(2.6) = 0.359782−0.0507498r

y9(2.8) = −0.418044r−0.638975

y10(3) = −0.5r−0.5

By setting y0 = 2.01− 0.01r, y10 = −1.01+ 0.01r, r ∈ [0,1], we obtained the system

of linear equations with unknowns y0,y1,y2, ...,y10 as:

1 0 0 0 0 0 0 0 0 0 0

0−2.0304 1 0 0 0 0 0 0 0 0

0 1 −2.0288 1 0 0 0 0 0 0 0

0 0 1 −2.0272 1 0 0 0 0 0 0

0 0 0 1 −2.0256 1 0 0 0 0 0

0 0 0 0 1 −2.0240 1 0 0 0 0

0 0 0 0 0 1 −2.0224 1 0 0 0

0 0 0 0 0 0 1 −2.0208 1 0 0

0 0 0 0 0 0 0 1 −2.0192 1 0

0 0 0 0 0 0 0 0 1 −2.0176 0

0 0 0 0 0 0 0 0 0 0 1

y0

y1

y2

y3

y4

y5

y6

y7

y8

y9

y10


=

2.01−0.01r

−1.962+0.01r

0.056

0.064

0.072

0.08

0.088

0.096

0.1040

1.122−0.01r

−1.01+0.01r

The upper-bounds for the solution are:

y0(1) = 2.01−0.01r

y1(1.2) = 1.358362−0.00764766r

y2(1.4) = 0.796018−0.00552781r

y3(1.6) = 0.312599−0.00356715r

y4(1.8) = −0.00170353r−0.0983172

y5(2.0) = 0.000116485r−0.43975

y6(2.2) = 0.0019393r−0.711737

y7(2.4) = 0.00380555r−0.903667

y8(2.6) = 0.00575095r−1.02639

y9(2.8) = 0.00780678r−1.06483

y10(3) = 0.01r−1.01


Calculations obtained using MATLAB are shown in Tables 5.3 and 5.4.

TABLE 5.3: Approximate values for the lower-bounds.

xr 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

0.0 1.9000 1.3019 0.7915 0.3598 0.0019 -0.2840 -0.4966 -0.6244 -0.6772 -0.6390 -0.50000.1 1.9100 1.3068 0.7914 0.3547 -0.0083 -0.2995 -0.5179 -0.6520 -0.7115 -0.6808 -0.55000.2 1.9200 1.3117 0.7913 0.3496 -0.0185 -0.3151 -0.5393 -0.6795 -0.7459 -0.7226 -0.60000.3 1.9300 1.3166 0.7912 0.3446 -0.0287 -0.3307 -0.5606 -0.7071 -0.7802 -0.7644 -0.65000.4 1.9400 1.3215 0.7911 0.3395 -0.0389 -0.3462 -0.5819 -0.7346 -0.8146 -0.8062 -0.70000.5 1.9500 1.3263 0.7910 0.3344 -0.0491 -0.3618 -0.6032 -0.7621 -0.8489 -0.8480 -0.75000.6 1.9600 1.3312 0.7909 0.3243 -0.0694 -0.3774 -0.6245 -0.7897 -0.8833 -0.8898 -0.80000.7 1.9700 1.3361 0.7908 0.3293 -0.0518 -0.3929 -0.6458 -0.8172 -0.9176 -0.9316 -0.85000.8 1.9800 1.3410 0.7907 0.3192 -0.0796 -0.4085 -0.6672 -0.8448 -0.9520 -0.9734 -0.90000.9 1.9900 1.3458 0.7906 0.3141 -0.0898 -0.4241 -0.6885 -0.8723 -0.9863 -1.0152 -0.95001.0 2.0000 1.3507 0.7905 0.3090 -0.1000 -0.4396 -0.7098 -0.8999 -1.0206 -1.0570 -1.0000

TABLE 5.4: Approximate values for the upper-bounds.

xr 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

0.0 2.0100 1.3584 0.7960 0.3126 -0.0983 -04398 -0.7117 -0.9037 -1.0264 -1.0648 -1.01000.1 2.0090 1.3576 0.7955 0.3122 -0.0985 -04397 -0.7115 -0.9033 -1.0264 -1.0640 -1.00900.2 2.0080 1.3568 0.7949 0.3119 -0.0987 -04397 -0.7113 -0.9029 -1.0252 -1.0633 -1.00800.3 2.0070 1.3561 0.7944 0.3115 -0.0988 -04397 -0.7112 -0.9025 -1.0247 -1.0625 -1.00700.4 2.0060 1.3553 0.7938 0.3112 -0.0990 -04397 -0.7110 -0.0921 -1.0241 -1.0617 -1.00600.5 2.0050 1.3545 0.7933 0.3108 -0.0992 -04397 -0.7108 -0.9018 -1.0235 -1.0609 -1.00500.6 2.0040 1.3538 0.7927 0.3105 -0.0993 -04397 -0.7106 -0.9014 -1.0229 -1.0601 -1.00400.7 2.0030 1.3530 0.7921 0.3101 -0.0995 -04397 -0.7104 -0.9010 -1.0224 -1.0594 -1.00300.8 2.0020 1.3522 0.7916 0.3097 -0.0997 -04397 -0.7102 -0.9006 -1.0218 -1.0586 -1.00200.9 2.0010 1.3515 0.7910 0.3094 -0.0999 -04396 -0.7100 -0.9002 -1.0212 -1.0578 -1.00101.0 2.0000 1.3507 0.7905 0.3090 -1.0000 -04396 -0.7098 -0.8999 -1.0206 -1.0570 -1.0000


These bounds are shown graphically as in Fig.5.2.

-23

-1

2.8

0

2.6

Y v

alue

s 1

2.4 00.1

2

2.2 0.2

X values

2

3

0.30.41.8

r values

0.51.6 0.61.4 0.70.81.2 0.91 1

FIGURE 5.2: The lower-bounds and the upper-bounds for an approximate solution



Tables 5.1 and 5.2 shows the values of lower and upper bounds for an approximate

solution of Example 5.1 respectively, Fig. 5.1 shows the graphical representation of

these bounds. Table 5.3 and 5.4 shows the values of lower and upper bounds for an

approximate solution of Example 5.2 respectively, Fig. 5.2 shows the graphical

representation of these bounds of Example 5.2 in which Airy’s function is involved. As

we can observe from the above tables, as r increases from 0 to 1 the fuzziness of the

solution decreases, that is, for r = 1, the lower and upper bounds are the same, and that

gives us the required solution.

5.6 Conclusion

Here, we have considered the DEs with FBCs in which after using the difference

equations, we obtain a system which has a mix of fuzzy and crisp parameters on

right-hand side. Upon solving the system using MATLAB, we found all y parameters

to be fuzzy. We also solved higher-order DEs with fuzzy boundaries by using FDM as

they cannot be solved using analytic methods. The DEs with FBCs is investigated as a

solution of a difference equation. Here, the FDM is applied to solve the second-order

DEs with FBCs for simplicity, but it is also applicable for higher-order DEs with

FBCs. So, in this way, our technique is reasonable and relevant for solving

higher-order DEs with FBCs.

CHAPTER 6

Methods for solving Higher Order

Differential Equations with Fuzzy

Initial and Boundary Conditions by

using Laplace Transforms

6.1 Introduction

Laplace transforms (LT) constitutes an important tool in solving linear ordinary and

partial DEs with constant coefficients and variable coefficients under suitable initial

and boundary conditions. The LT is a powerful integral transform used to switch a

function from the time domain to the s - domain, where s is a parameter. The Laplace

transform can be used in some cases to solve linear differential equations with given

initial conditions. Here, we use the Laplace transform technique mainly to solve

variable coefficient (not a Cauchy-Euler’s) DEs with FIBCs as well as forcing function

being a fuzzy number.

Opanuga et al. [61] used the Laplace transform method to solve BVPs. S.Salahshour

[71] focus on "nth order fuzzy linear DEs under GHD". Allahviranloo et al. [9]

propose a FLT under the SGHD concept; they use it in an analytic solution method for

some FDEs. Allahviranloo et al. [8] described "A new method for solving fuzzy linear

differential equations". Salahshour et al. [72] worked on "Application of fuzzy Laplace

Transform". Ahmad et al. [2] solved fuzzy two-point boundary value problem by FLT.

Salahshour et al. [73] solved the fuzzy heat equation using fuzzy Laplace transform.

85

Methods for solving Higher Order Differential Equations with Fuzzy Initial andBoundary Conditions by using Laplace Transforms 86

"Generalizations of the differentiability of fuzzy number valued functions with

applications to FDEs" were advocated by Bede et al. [18]. Hukurhara or its

generalization of Hukuhara derivatives, under this setting, existence and uniqueness of

solution of FDEs are studied. This approach has disadvantages that it gives the

solution which have increasing length of their support. The SGHD is defined for a

large class of fuzzy valued function than the HD, and FDEs can have solutions which

have a decreasing length of their support. But here, we have four possibilities for

selecting appropriate solution by using SGHD, and so, here we use strongly

generalized differentiability concept in this chapter.

6.2 Laplace transform

Let f (t) be a function defined, ∀ t ≥ 0. Then the integral, L [ f (t)] =∫

∞

0 f (t)e−stdt if

exists, is called LT of f (t). s is a parameter, real or complex number. Clearly L [ f (t)]

being a function of s is written as F(s) or L [ f (t)]. Here the symbol L which

transforms f (t) into F(s) is called the Laplace transform Operator.

6.2.1 Conditions for the existence of Laplace transforms

If f (t) is a piecewise continuous function on every finite interval in the range t ≥ 0

which satisfy | f (t)| ≤Meat , ∀ t ≥ 0 for some positive constants a and M, that is, f (t) is

of exponential order a, then the LT of f (t) i.e.∫

∞

0 f (t)e−stdt exists.

6.3 Strongly generalized Hukuhara differentiability

Let f : (a,b)→ E and t0 ∈ (a,b). We say that f is SGHD at t0 (Bede-Gal differential

[18]) if there exist an element f′(t0) ∈ E such that


(i) for all h > 0 sufficiently small, ∃ f (t0 + h) f (t0), f (t0) f (t0− h) and the limits

(in the metric D)

limh→0

f (t0 +h) f (t0)h

= limh→0

f (t0) f (t0−h)h

= f′(t0).

or

(ii) for all h > 0 sufficiently small, ∃ f (t0) f (t0 +h), f (t0−h) f (t0) and the limits

(in the metric D)

limh→0

f (t0) f (t0 +h)−h

= limh→0

f (t0−h) f (t0)−h

= f′(t0).

or (iii) for all h > 0 sufficiently small, ∃ f (t0 + h) f (t0), f (t0− h) f (t0) and the

limits (in the metric D)

limh→0

f (t0 +h) f (t0)h

= limh→0

f (t0−h) f (t0)−h

= f′(t0).

or

(iv) for all h > 0 sufficiently small, ∃ f (t0) f (t0 +h), f (t0−h) f (t0) and the limits

(in the metric D)

limh→0

f (t0) f (t0 +h)−h

= limh→0

f (t0) f (t0−h)h

= f′(t0)

.


Here we are going to discuss the solution of higher-order DEs with FIBCs by using

properties of LT. The LT is a versatile tool to solve higher order DEs because it converts

the DEs in form of algebraic equations and we know that the algebraic equations are

easy to deal with it. The other advantages are that we can solve the non-homogeneous


DE directly without the need to first solving the corresponding homogeneous DE, or

find the particular solution of the DE without first finding the general solution. Here we

are going to solve some constant coefficient homogeneous and non-homogeneous DEs

with FICs. This method is mainly used to solve higher-order variable coefficient DEs

with FIBCs which are not of the Cauchy-Euler’s form of DEs.

6.5 Method for solving higher order differential

equations with fuzzy initial conditions

Fuzzy Laplace Transform(FLT)

Let f (t) be continuous fuzzy-valued function. Suppose that f (t)e−st improper FRI on

[0,∞). Then, the integral∫

∞

0 f (t)e−stdt is called FLT [9] and is defined as

L [ f (t)] =∫

∞

0 f (t)e−stdt (s > 0 and integer)

We have∫

∞

0 f (t)e−stdt = (∫

∞

0 f (t)e−stdt,∫

∞

0 f (t)e−stdt)

also by using definition of classical LT:

l[ f (t,r)] =∫

∞

0 f (t)e−stdt and

l[ f (t,r)] =∫

∞

0 f (t)e−stdt

then we follow

L [ f (t)] = (l[ f (t,r), l[ f (t,r)).

Theorem:6.5.1 Let f (t) and g(t) be continuous fuzzy-valued functions and c1,c2 are

real constants. Suppose that f (t)e−st ,g(t)e−st are improper FRI on [0,∞). Then

L [c1 f (t)+ c2g(t)] = c1L [ f (t)]+ c2L [g(t)].

Theorem:6.5.2 Let f : IR→ E be a function and denote f (t) = ( f (t,r), f (t,r)), for each

r ∈ [0,1]. Then

If f is (i)-differentiable, then f (t,r) and f (t,r) are differentiable function and f′(t) =

( f′(t,r), f

′(t,r)).

If f is (ii)-differentiable, then f (t,r) and f (t,r) are differentiable function and f′(t) =

( f′(t,r), f

′(t,r)).

Method for solving higher order differential equations with fuzzy initial conditions 89

Theorem:6.5.3 Let f′(t) be an integrable fuzzy-valued function [9], and f (t) is the

primitive of f′(t) on [0,∞). Then

L [ f′(t)] = sL [ f (t)] f (0)

where f is (i)-differentiable

or

L [ f′(t)] = (− f (0)) (−sL [ f (t)])

where f is (ii)-differentiable

Theorem:6.5.4 Let f′′(t) be integrable fuzzy-valued function [9], and f (t), f

′(t) are

primitive of f′(t), f

′′(t) on [0,∞). then

L [ f′′(t)] = s2L [ f (t)] s f (0) f

′(0)

where f is (i)-differentiable and f′is (i)-differentiable or

L [ f′′(t)] = s2L [ f (t)] s f (0)− f

′(0)

where f is (ii)-differentiable and f′is (ii)-differentiable or

L [ f′′(t)] =(−s2)L [ f (t)]− s f (0)− f

′(0)

where f is (i)-differentiable and f′is (ii)-differentiable or

L [ f′′(t)] =(−s2)L [ f (t)]− s f (0) f

′(0)

where f is (ii)-differentiable and f′is (i)-differentiable

Theorem:6.5.5 Let f (t) satisfy the condition of the existence theorem of LT and

L [ f (t)] = F(s), then L [t f (t)] =−F′(s)

Hence if f′(t) satisfies the condition of existence theorem of LT then

L [t f′(t)] =− d

dsL [ f

′(t)] =− d

ds

{sF(s)− f (0)

}=−sF

′(s)− F(s)

Similarly, for f′′

L [t f′′(t)]=− d

dsL [ f

′′(t)]=− d

ds

{s2F(s)−s f (0)− f

′(0)}=−s2F

′(s)−2sF(s)+ f (0)


6.6 Solution of higher order differential equations with

fuzzy initial and boundary conditions

6.6.1 Solution of second order constant coefficient differential

equations with fuzzy initial conditions

Application of simple harmonic motion in different fields

• In the simple harmonic motion, the displacement of the object is always in the

opposite direction of the restoring force.

• Simple harmonic motion is always oscillatory.

• Examples are: the motion of a pendulum, motion of a spring, etc.

• Its applications can be found in clock, guitar, violin, bungee jumping, rubber

bands, diving boards, earthquakes, etc.

6.6.1.1 An example based on the application of differential equation in

mechanical engineering

Here, we consider the motion of an object with mass at the end of a spring that is either

vertical or horizontal on a level surface. By Hooke’s Law, which states that if the spring

is stretched (or compressed) units from its natural length, then it exerts a force that is

proportional to y,

restoring force = ky

where k is a positive constant (called the spring constant). Besides, consider the

motion of a spring that is subject to a frictional force (due to air resistance or friction).

The damping force is proportional to the velocity of the mass and acts in the direction

opposite to the motion.

Solution of higher order differential equations with fuzzy initial and boundaryconditions 91

damping force = by′

where b is the damping coefficient.

By using Newton’s Second Law (force equals mass times acceleration), we have

my′′

= restoring force + damping force = −ky−by′

By using above equations, we get the equation of mass-spring system (vibration of

mass m on an spring) and it is modelled by the homogeneous linear ODE

my′′+by

′+ ky = 0 (6.1)

Here y(t), a function of time t, is the displacement of body of mass m from rest. There

are free motions, that is, motions in the absence of external forces.

Consider non-homogeneous linear ODE

my′′+by

′+ ky = r(t) (6.2)

where r(t) is a external force and the resultant motion is the forced motion.

m

y(t)

kr(t)

b

FIGURE 6.1: The mass-spring system

Example:6.1 Consider vibrating mass system. The mass m = 1, the spring constant

k = 4 lbs/ft and there is no or negligible damping. The forcing function is 2cos t.

Here b = 0. So, from (6.2) the DE of the motion with initial conditions is


y′′+4y = 2cos t, t ∈ [0,1]

y(0) = 2

y′(0) = 0

Consider DE with FICs

y′′+4y = 2cos t, t ∈ [0,1]

y(0) = [2r, 4−2r]

y′(0) = [−2+2r, 2−2r]

Apply LT on both the side

L [y′′]+4L [y] = 2L [cos t]

By using Theorems 6.5.1 and 6.5.4

l[y(t,r)] = (2r)s

s2 +4+(2r−2)

1s2 +4

+2s

(s2 +1)(s2 +4)

l[y(t,r)] = (4−2r)s

s2 +4+(2−2r)

1s2 +4

+2s

(s2 +1)(s2 +4)Hence the solution bounds are:

y(t,r) = (2r)cos2t + r sin2t− sin2t +23

cos t− 23

cos2t

y(t,r) = (4−2r)cos2t + sin2t− r sin2t +23

cos t− 23

cos2t

The y(t,r) and y(t,r) values at r = 0,0.5,0.8,0.9 are represented in Figs.6.2 to 6.5.

when r = 1, we obtain exact solution

y(t) = y(t,1) = y(t,1) =43

cos2t +23

cos t,

as shown in Fig.6.6.

0.2 0.4 0.6 0.8 1.0t

1

2

3

4

y

FIGURE 6.2: The lower-bound and upper-bound of solution at r = 0


0.2 0.4 0.6 0.8 1.0t

0.5

1.0

1.5

2.0

2.5

3.0

y

FIGURE 6.3: The lower-bound and upper-bound of solution at r = 0.5

0.2 0.4 0.6 0.8 1.0t

0.5

1.0

1.5

2.0

y


0.2 0.4 0.6 0.8 1.0t

0.5

1.0

1.5

2.0

y


0.2 0.4 0.6 0.8 1.0t

0.5

1.0

1.5

2.0

y



6.6.1.2 Example based on the application of differential equation in physics

An ideal simple pendulum consists of a point mass m1 suspended from a support by a

massless string of length l. The equilibrium position of the mass is a distance l below

the support. If the mass is displaced from its equilibrium position, it exhibits periodic

motion, moving in a vertical plane along a circular arc. When the string makes an

angle θ with the vertical, then the displacement of the mass from its equilibrium

position along the circular arc is s = lθ . The forces acting on the mass are gravity and

the tension in the string. Only gravity provides a restoring force towards the

equilibrium position. The magnitude of this force is m1 gsinθ . The equation of motion,

F = m1a, therefore yields

restoring force = m1d2sdt2 =−m1 gsinθ ,

or

restoring force = ld2θ

dt2 =−gsinθ .

Here, the displacement from equilibrium is small ⇒ sinθ ∼ θ in radians.(Fig.6.7)

Then the equation of motion becomes

ld2θ

dt2 =−gθ .

And, the resistive force is proportional to velocity

The resistive force = − lλm1

dθ

dt.


Therefore,

ld2θ

dt2 = resistive force + restoring force = − lλm1

dθ

dt−gθ

A simple pendulum of length l oscillating through a small angle θ in a medium

resistance is proportional to velocity. The differential equation of motion is

ld2θ

dt2 +lλm1

dθ

dt+gθ = 0 (6.3)

m1

1g

θ

FIGURE 6.7: The simple pendulum

Example:6.2 A Pendulum of length l = 85 f t is subject to resistive force FR = 32

5dθ

dt due

to damping. Determine displacement function.

Equation (6.3) gives the differential equation as below

85

d2θ

dt2 +325

dθ

dt+32θ = 0

θ(0) = 1

θ′(0) = 2


By simplifying and converting them to DE with FICs, The DE become

θ′′+4θ

′+20θ = 0

θ(0) = [r,2− r]

θ′(0) = [1+ r,3− r]


L [θ′′]+4L [θ ]+20L [θ ] = L [0]

By using Theorems 6.5.1 to 6.5.4

l[θ(t,r)] = rs+2

(s+2)2 +16+

(3r+1

4

)1

(s+2)2 +16

l[θ(t,r)] = (2− r)s+2

(s+2)2 +16+

(7−3r

4

)1

(s+2)2 +16Hence the solution bounds are as follows:

θ(t,r) = re−2t cos4t +(

3r+14

)e−2t sin4t

θ(t,r) = (2− r)e−2t cos4t +(

7−3r4

)e−2t sin4t

The θ(t,r) and θ(t,r) values at r = 0,0.5,0.8,0.9 are represented in Figs.6.8 to 6.11.

when r = 1, we obtained exact solution

θ(t) = θ(t,1) = θ(t,1) = e−2t cos4t + e−2t sin4t,

as shown in Fig.6.12.

0.2 0.4 0.6 0.8 1.0t

0.5

1.0

1.5

2.0

Θ



0.2 0.4 0.6 0.8 1.0t

0.5

1.0

1.5

Θ


0.2 0.4 0.6 0.8 1.0t

0.5

1.0

Θ


0.2 0.4 0.6 0.8 1.0t

-0.2

0.2

0.4

0.6

0.8

1.0

1.2

Θ


0.2 0.4 0.6 0.8 1.0t

-0.2

0.2

0.4

0.6

0.8

1.0

Θ



6.6.2 Solution of second order variable coefficient differential

equations with fuzzy initial and boundary conditions

Example:6.3 Consider variable coefficient DE with initial condition

ty′′− y

′=−1, t ∈ [0,1]

y(0) = 0

Consider variable coefficient DE with initial condition and forcing function both as

TFN

ty′′− y

′=−1, t ∈ [0,1]

y(0) = [r−1,1− r]

1 = [r,2− r]


L [ty′′]−L [y

′] =−L [1]


l[y(t,r)] =r−1

s+

rs2 +

As3

l[y(t,r)] =1− r

s+

2− rs2 +

As3 , A ∈ R.

Hence the solution bounds are as follows:

y(t,r) = (r−1)+ rt +Bt2

y(t,r) = (1− r)+(2− r)t +Bt2, B ∈ R.

The y(t,r) and y(t,r), 0≤ r ≤ 1, are represented in Fig.6.13

y(t) = y(t,1) = y(t,1) = t +Bt2

0.00

0.05

0.10

t

0.0

0.5

1.0

r

-1.0

-0.5

0.0

0.5

1.0

yHt,rL

FIGURE 6.13: Solution by considering parameters TFN


Consider initial condition and forcing function as TraFN

ty′′− y

′=−1, t ∈ [0,1]

y(0) = [r−1,2− r]

1 = [r,3− r]


L [ty′′]−L [y

′] =−L [1]


l[y(t,r)] =r−1

s+

rs2 +

As3

l[y(t,r)] =2− r

s+

3− rs2 +

As3 , A ∈ R.


y(t,r) = (r−1)+ rt +Bt2

y(t,r) = (2− r)+(3− r)t +Bt2, B ∈ R.


y(t,1) = t +Bt2 , y(t,1) = 1+2t +Bt2

0.00

0.05

0.10

t

0.0

0.5

1.0

r

-1

0

1

2

yHt,rL

FIGURE 6.14: Solution by considering parameters TraFN

Consider initial condition and forcing function as GFN

ty′′− y

′=−1, t ∈ [0,1]

y(0) = [−√−(2loger),

√−(2loger)]

1 = [1−√(−2loger),1+

√(−2loger)]


L [ty′′]−L [y

′] =−L [1]



l[y(t,r)] =−√−(2loger)

s+

1−√

(−2loger)s2 +

As3

l[y(t,r)] =

√−(2loger)

s+

1+√

(−2loger)s2 +

As3 , A ∈ R.


y(t,r) = [−√−(2loger)]+ [1−

√(−2loger)]t +Bt2

y(t,r) = [√−(2loger)]+ [1+

√(−2loger)]t +Bt2, B ∈ R.


y(t) = y(t,1) = y(t,1) = t +Bt2

0.00

0.05

0.10

t

0.0

0.5

1.0

r

-5

0

5

10

yHt,rL

FIGURE 6.15: Solution by considering parameters GFN

Example:6.4 Consider variable coefficient DE with boundary conditions

ty′′+2y

′+ ty = 0,

y(0) = 1

y(π) = 0

Consider variable coefficient DE with boundary conditions as TFN

ty′′+2y

′+ ty = 0,

y(0) = [r,2− r]

y(π) = [r−1,1− r]


L [ty′′]+2L [y

′]+L [ty] = L [0]



l[y(t,r)] = (r) tan−1(1s )

l[y(t,r)] = (2− r) tan−1(1s )


y(t,r) = (r)sin t

ty(t,r) = (2− r)

sin tt


y(t) = y(t,1) = y(t,1) =sin t

t

0.0

0.5

1.0r

01 2

3t

0.0

0.5

1.0

1.5

2.0

yHt,rL

FIGURE 6.16: Solution by considering parameters TFN

Consider boundary conditions as TraFN

ty′′+2y

′+ ty = 0,

y(0) = [r,3− r]

y(π) = [r−1,2− r]


L [ty′′]+2L [y

′]+L [ty] = L [0]


l[y(t,r)] = (r) tan−1(1s )

l[y(t,r)] = (3− r) tan−1(1s )


y(t,r) = (r)sin t

ty(t,r) = (3− r)

sin tt


y(t,1) =sin t

t, y(t,1) = 2

sin tt


0.0

0.5

1.0r

01 2

3

t

0

1

2

3

yHt,rL

FIGURE 6.17: Solution by considering parameters TraFN

Consider boundary conditions as GFN

ty′′+2y

′+ ty = 0,

y(0) = [1−√−(2loger),1+

√−(2loger)]

y(π) = [−√(−2loger),

√(−2loger)]


L [ty′′]+2L [y

′]+L [ty] = L [0]


l[y(t,r)] = [1−√

(−2loger)] tan−1(1s )

l[y(t,r)] = [1+√

(−2loger)] tan−1(1s )


y(t,r) = [1−√

(−2loger)]sin t

ty(t,r) = [1+

√(−2loger)]

sin tt

The y(t,r) and y(t,r) , 0≤ r ≤ 1, are represented in Fig.6.18

y(t) = y(t,1) = y(t,1) =sin t

t

0.0

0.5

1.0r

01 2

3

t

0

5

yHt,rL

FIGURE 6.18: Solution by considering parameters GFN

Result and Discussion 103


The Laplace transform method is mainly used to solve variable coefficient (not a

Cauchy-Euler’s) DEs with FIBCs. Here, we use different types of fuzzy numbers as

fuzzy conditions to see the change in the solution. The selection of a fuzzy number

depends on situation and the type of the problem. The graphical representation is done

using the Mathematica software. From the figures, we can observe that for distinct

values of r, we get different lower and upper bounds, whereas at r = 1 the lower and

the upper bound are nearly the same. By using the LT technique we have solved the

DEs with FBCs and second-order DE in which a single condition is given.

6.8 Conclusion

The LT technique offers us a solution of higher-order DEs with FICs but here for the

simplicity purpose we have considered second order constant coefficient and variable

coefficient DEs with FIBCs by using the SGHD notion. The usefulness of the method

was illustrated by solving some examples, including application-level examples. The

uniqueness of the solution is lost in this new procedure because we have four

possibilities and the solution should be appropriately selected among the four cases of

SGHD, but such situation in a fuzzy context is assumed. Here we solved DEs with

FIBCs by using the fuzzy numbers like TFN, TraFN and GFN.

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[65] K R Patel and N B Desai, Solution of fuzzy initial value problem by fuzzy Laplace transform, vol. 2,Kalpa Publications in Computing, ICRISET2017, Selected Papers in Computing (2017), 25–37.

[66] K R Patel and N B Desai, Solution of nth order fuzzy initial value problem, PRAJNA- Journal ofPure and Applied Sciences, (24-25), (2017), 30–37.

[67] K R Patel and N B Desai, Solution of variable co-efficient fuzzy differential equations byfuzzy Lapalce transform, International journal recdent and innovative trends in computing andcommunication, 5(6), (2017), 927–942.

[68] K R Patel and N B Desai, Solution of higher order differential equations with fuzzy boundaryconditions by Finite difference method, Journal of Emerging Technologies and InnovativeResearch, 6(5), (2019), 30–49.

[69] M L Puri and D A Ralescu, Differentials of fuzzy functions, Journal of Mathematical Analysis andApplications, 91(2), (1983), 552–558.

[70] T J Ross, Fuzzy Logic with Engineering Applications, Wiley (2013).

[71] S Salahshour, Nth-order fuzzy differential equations under generalized differentiability, Journal ofFuzzy Set Valued Analysis, 2011, (2011), 1–14.

[72] S Salahshour and T Allahviranloo, Applications of fuzzy Laplace transforms, Soft computing, 17(1),(2013), 145–158.

[73] S Salahshour and E Haghi, Solving fuzzy heat equation by fuzzy Laplace transforms,in: International Conference on Information Processing and Management of Uncertainty inKnowledge-Based Systems, Springer (2010), 512–521.

[74] S Seikkala, On the fuzzy initial value problems, Fuzzy Sets and Systems, 24(3), (1987), 319–330.

[75] S Song and C Wu, Existence and uniqueness of solutions to Cauchy problem of fuzzy differentialequations, Fuzzy sets and Systems, 110(1), (2000), 55–67.

[76] S Tapaswini and S Chakraverty, Numerical solution of n-th order fuzzy linear differential equationsby homotopy perturbation method, International Journal of Computer Applications, 64(6).

[77] H C Wu, The fuzzy Riemann integral and its numerical integration, Fuzzy Sets and Systems, 110(1),(2000), 1–25.

[78] G Xiaobin, S Dequan and L Xiaoquan, Fuzzy approximate solutions of second-order fuzzy linearboundary value problems, Boundary Value Problems, 2013(1), (2013), 212.

References 108

[79] B Yuan et al., Fuzzy Sets, Fuzzy Logic, and Fuzzy Systems: Selected Papers by Lotfi A Zadeh,World Scientific (1996).

[80] L A Zadeh, Fuzzy sets, Information and control, 8(3), (1965), 338–353.

[81] L A Zadeh, The concept of a linguistic variable and its application to approximate reasoning—I,Information sciences, 8(3), (1975), 199–249.

[82] H J Zimmermann, Fuzzy set theory and its applications, Springer Science & Business Media (2011).

List of Publications

List of Publications Arising From the Thesis

1. Solution of fuzzy initial value problems, ADIT Journal of Engineering, ISSN Number : 0973-

3663,(UGC approved journal), 12(1), (2015), 53-58.

2. Solution of variable coefficient fuzzy differential equations by fuzzy Laplace transform,

International Journal on Recent and Innovation Trends in Computing and Communication,

ISSN Number : 2321-8169,(UGC approved journal), 5(6), (2017), 927-942.

3. Solution of nth order fuzzy initial value problem, PRAJNA- Journal of Pure and Applied

Sciences,ISSN Number : 0975-2595,(UGC approved journal) 24-25,(2017), 30-37.

4. Solution of higher order differential equations with fuzzy boundary conditions by finite difference

method, Journal of Emerging Technologies and Innovative Research, ISSN Number : 2349-

5162,(UGC approved journal), 6(5), (2019), 30-49.

Details of the Work Presented in Conference From the Thesis

1. The paper entitled as "Solution of fuzzy initial value problems by fuzzy Laplace transform"

presented in ICRISET2017 at B. V. M. Engineering College, V. V. Nagar, Anand (Gujarat) on

18th February, 2017 and published in Kalpa Publications in Computing, ICRISET2017, Selected

Papers in Computing, 2, (2017), 25-37.

Details of the Work Accepted From the Thesis

1. The paper entitled as "Solution of nth Order Differential Equation with Fuzzy Conditions by

Gauss Elimination Method" is accepted by the "Continuity, Consistency and Innovation in

Applied Sciences and Humanities" on August 13, 2020.

109

Solution of Variable Coefficient Fuzzy

Differential Equations by Fuzzy Laplace

Transform

Komal R.PatelNarendrasinh B.Desai

Assistant ProfessorInstitute of Technology and Management Universe

Vadodara-390510,Gujarat(INDIA)

Associate ProfessorHead of Department of Applied Sciences and Humanities

A. D. Patel Institute of TechnologyV. V. Nagar-388121,Gujarat(INDIA)

Email:[email protected]@yahoo.co.in

Abstract

In this paper we propose a fuzzy Laplace transform to solve variablecoefficient fuzzy differential equations under strongly generalized differ-entiability concept.The fuzzy Laplace transform of derivative was usedto solve second order variable coefficient fuzzy initial value problems andfuzzy boundary value problems if t is multiplied with first or second deriva-tive term.To illustrate applicability of proposed method we solve fuzzydifferential equations using different types of fuzzy numbers i.e. triangu-lar,trapezoidal,Gaussian etc and compare the solutions.We plot 3D plotsfor different values of r-level sets by mathematica software.

Keywords:Fuzzy Number.Fuzzy valued function.triangular,trapezoidal and Gaus-sian fuzzy numbers.Fuzzy Laplace Transform.Strongly generalized differential.Fuzzyinitial value problem.Fuzzy boundary value problem..

1 Introduction

The fuzzy differential equation is very much important topic in field of scienceand engineering to solve dynamic problem.The concept of a fuzzy derivativewas first introduced by Chang and Zadeh [56],followed up by Dubois and Prade

1

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[17] who used the extension principle in their approach.Other fuzzy derivativeconcepts were proposed by Puri and Ralescu [45], and Goetschel and Vaxman[26]as an extension of the Hukuhara derivative of multivalued functions.Kandeland Byatt [33] applied the concept of fuzzy differential equation to the analysisof fuzzy dynamical problems.The fuzzy differential equations and fuzzy initialvalue problems are studied by Kaleva[31, 32]and Seikkala [51]

Two analytical methods for solving an nth-order fuzzy linear differentialequation with fuzzy initial conditions presented by Buckley and Feuring [12,13].Mondal and Roy [42] described the solution procedure for first order linearnon-homogeneous ordinary differential equation in fuzzy environment.Existenceand uniqueness of fuzzy boundary value problem has been proved by Esfahani etal.[18].Lakshikantham et al. [38] investigated the solution of two point boundaryvalue problems associated with non-linear fuzzy differential equation by usingthe extension principle.Generalized differentiability concept is used by Bade etal.[11] to investigated first order linear fuzzy differential equations.Based on theidea of collocation method Allahviranloo et al. [5] solved nth order fuzzy lin-ear differential equations.Far and Ghal-Eh [19] proposed an iterative methodto solve fuzzy differential equations for the linear system of first order fuzzydifferential equation with fuzzy constant coefficient.Variation of constant for-mula has been handle by Khastan et al.[37] to solve first order fuzzy differentialequations.Akin et al.[2] developed an algorithm based on α-cut of fuzzy set forsolution of second order fuzzy initial value problems.A new approach has beendeveloped by Gasilov et al.[22] to get the solution of fuzzy initial value problem.

The concept of generalized H-differentiability is studied by Chalco-Cano andRoman Flores [14] to solve fuzzy differential equation.Hasheni et al.[29, 28] stud-ied homotopy analysis method for solution of system of fuzzy differential equa-tion s and obtained analytic solution of fuzzy Wave like equations with variablecoefficients.As regards, methods to solve nth order fuzzy differential equationare discussed in [5, 25, 30, 35, 48, 55].the Variational iteration method (VIM)was successfully applied by Jafari et al.[30] for solving nth order fuzzy differen-tial equation.A new result on multiple solutions for nth order fuzzy differentialequations under generalized differentiability has been proposed by Khastan etal.[35].Based on idea of collocation method allahviranloo et al.[5] solved nth or-der fuzzy linear differential equations.Integral form of nth order fuzzy differentialequations has been developed by Salahshour [48]under generalized differentiabil-ity.Mansuri and Ahmady [41] implemented characterization theorem for solvingnth order fuzzy differential equations.Also Tapaswini and Chakraverty[53] im-plemented homotopy perturbation method for the solution of nth order fuzzylinear differential equations.Bade[10] found Solutions of fuzzy differential equa-tions based on generalized differentiability.

Paper is organized as In section 2 preliminaries,In section 3 Examples byusing fuzzy Laplace transform ,In section 4 Result and Discussion,In section 5conclusion.

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2 Preliminaries

Definition 2.1 Fuzzy NumberA fuzzy number is a fuzzy set like µ : R→ I = [0, 1] which satisfies:

(a) µ is upper semi-continuous,

(b) µ is fuzzy convex i.e µ(λx+(1−λ)y) ≥ min{µ(x), µ(y)}∀x, y ∈ R, λ ∈ [0, 1],

(c) µ is normal i.e ∃x0 ∈ R for which µ(x0) = 1,

(d) supp µ = {x ∈ R | µ(x) > 0} is support of u, and its closure cl(supp µ) iscompact.

Definition 2.2 r-cutIt is crisp set derived from its parent fuzzy set A where r-cut is defined asAr = {x ∈ R | µ(x) ≥ r}Definition 2.3 Triangular Fuzzy NumberConsider triangular fuzzy number A = (a, b, c) is depicted in Fig.1 The mem-bership function µ(x) of A will be defined as follows.

µ(x) =

0 , x < ax−ab−a , a ≤ x ≤ bc−xc−b , b ≤ x ≤ c0 , x > c

The triangular fuzzy number A = (a, b, c) can be represented with an order pairof function of r-cut approach i.e.[µ(r), µ(r)] = [a+ (b− a)r, c− (c− b)r] , wherer ∈ [0, 1]

Fig.1 Triangle membership function

Definition 2.4 Trapezoidal Fuzzy NumberConsider trapezoidal fuzzy number A = (a, b, c, d) is depicted in Fig.2 The mem-bership function µ of A will be defined as follows.

µ(x) =

0 , x < ax−ab−a , a ≤ x ≤ b1 , b ≤ x ≤ cd−xd−c , c ≤ x ≤ d0 , x ≥ d

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The trapezoidal fuzzy number A = (a, b, c, d) can be represented with an orderpair of function of r-cut approach i.e.[µ(r), µ(r)] = [a + (b− a)r, d− (d− c)r] ,where r ∈ [0, 1]

Fig.2 Trapezoidal fuzzy number

Definition 2.5 Gaussian Fuzzy NumberThe asymmetric Gaussian fuzzy number A = (α, σl, σr).The membership func-tion µ(x) of A will be defined as follows.

µ(x) =

e− (x−α)2

2σ2l , x ≤ α

e− (x−α)2

2σ2r , x ≥ αwhere ,the modal value (center) denote as α and σl, σr denote left and right handspreads(fuzziness i.e.width) corresponding to the Gaussian Distribution.For sym-metric Gaussian fuzzy number the left and right-hand spreads are equal i.e.σl =σr = σ.So symmetric Gaussian fuzzy number may be written as A = (α, σ, σ)

and corresponding function may be defined as µ(x) = e−β(x−α)2

,∀x ∈ R whereβ = 1

2σ2 . The symmetric Gaussian fuzzy number in parametric form can berepresented as

A = [µ(r), µ(r)] =[α−

√− (loger)

β , α+√− (loger)

β

]where r ∈ [0, 1]

Fig.3 Gaussian fuzzy number

For all the above type of fuzzy numbers the left and right bound of fuzzy numberssatisfy the following requirements

1. µ(r) is a bounded monotonic increasing left continuous function over [0, 1],

2. µ(r) is a bounded monotonic decreasing left continuous function [0, 1],

3. µ(r) ≤ µ(r), 0 ≤ r ≤ 1.

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Definition 2.6 Fuzzy arithmeticFor any arbitrary two fuzzy numbers u =(u(r), u(r)), v = (v(r), v(r)), 0 ≤ r ≤ 1and arbitrary k ∈ R.we define addition,subtraction,multiplication,scalar multi-plication by k.(see in [21])

u+ v = (u(r) + u(r), v(r) + v(r)),u− v = (u(r)− v(r), u(r)− v(r)),

u · v =(min{u(r)v(r), u(r)v(r), u(r)v(r), u(r)v(r)},max{u(r)v(r), u(r)v(r), u(r)v(r), u(r)v(r)})

ku =

{(ku(r), ku(r)), k ≥ 0(ku(r), ku(r)), k < 0

Definition 2.7 Hukuhara-differenceLet x, y ∈ E.If there exists z ∈ E such that x = y + z,then z is called theHakuhara-difference of fuzzy numbers x and y,and it is denoted by z = x y.The sign stands for Hukuhara-difference,and x y 6= x+ (−1)y.Definition 2.8 Hukuhara-differentiabilityLet f : (a, b)→ E and t0 ∈ (a, b).We say that f is Hukuhara-differential at t0,ifthere exists an element f

′(t0) ∈ E such that for all h > 0 sufficiently small,

∃f(t0 + h) f(t0), f(t0) f(t0 − h) and the limits holds(in the metric D)

limh→0

f(t0 + h) f(t0)

h= limh→0

f(t0) f(t0 − h)

h= f

′(t0).

Definition 2.9 Generalized Hukuhara differentiabilityLet f : (a, b) → E and t0 ∈ (a, b).We say that f is (1)-differential at t0,if thereexists an element f

′(t0) ∈ E such that for all h > 0 sufficiently small,

∃f(t0 + h) f(t0), f(t0) f(t0 − h) and the limits holds(in the metric D)

limh→0

f(t0 + h) f(t0)

h= limh→0

f(t0) f(t0 − h)

h= f

′(t0).

and f is (2)-differentiable if for all h > 0 sufficiently small,∃f(t0) f(t0 +h),∃f(t0 − h) f(t0)and the limits(in the metric D)

limh→0

f(t0) f(t0 + h)

−h = limh→0

f(t0 − h) f(t0)

−h = f′(t0).

If f′(t0) exist in above cases then i.e called Generalized fuzzy derivative of f(t).

Definition 2.10 Strongly generalized differentiabilityLet f : (a, b)→ E and t0 ∈ (a, b).We say that f is strongly generalized differen-tial at t0(Bede-Gal differential)if there exist an element f

′(t0) ∈ E such that

(i)for all h > 0 sufficiently small,∃f(t0 + h) f(t0),∃f(t0) f(t0 − h)and the

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limits(in the metric D)

limh→0

f(t0 + h) f(t0)

h= limh→0

f(t0) f(t0 − h)

h= f

′(t0).

or(ii)for all h > 0 sufficiently small,∃f(t0) f(t0 + h),∃f(t0 − h) f(t0)and thelimits(in the metric D)

limh→0

f(t0) f(t0 + h)

−h = limh→0

f(t0 − h) f(t0)

−h = f′(t0).

or(iii)for all h > 0 sufficiently small,∃f(t0 + h) f(t0),∃f(t0 − h) f(t0)and thelimits(in the metric D)

limh→0

f(t0 + h) f(t0)

h= limh→0

f(t0 − h) f(t0)

−h = f′(t0).

or(iv)for all h > 0 sufficiently small,∃f(t0) f(t0 + h),∃f(t0) f(t0 − h)and thelimits(in the metric D)

limh→0

f(t0) f(t0 + h)

−h = limh→0

f(t0) f(t0 − h)

h= f

′(t0).

(h and −h at denominators mean 1h and 1

−h ,respectively)

Theorem 2.1 [14].Let f : R→ E be a function and denotef(t) = (f(t, r), f(t, r)),foreach r ∈ [0, 1].Then

1. If f is (i)-differentiable,then f(t, r) and f(t, r) are differentiable functionand

f′(t) = (f

′(t, r), f

′(t, r))

2. If f is (ii)-differentiable,then f(t, r) and f(t, r) are differentiable functionand

f′(t) = (f

′(t, r), f

′(t, r))

Definition 2.11 Piecewise Continuous Functionf(t) is piecewise continuous function in a ≤ t ≤ b if there exist a finite numbersof points t1, t2, ......, tN such that f(t) is continuous on each open subintervala < t < t1,t1 < t2,.......,tN < t < b, and has a finite limit as t approaches eachendpoint from the interior of that subinterval.Definition 2.12 Exponential Orderf(t) is of exponential order as t→∞ if there exist real constants K, c, T 3|f(t)| ≤ e−ct, t ≥ T.

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Definition 2.13 Fuzzy Laplace TransformFuzzy Laplace Transform is an example of integral transform relation of theformF (s) =

∫ baK(t, s)f(t)dt,

where t is time and K(t, s) is kernel of transform which transform f(t) to F (s)i.e. which transform time domain to frequency domain.The most well knownintegral transform is Laplace transformwhere a = 0 and b =∞K(t, s) = e−st

F (s) = limb→∞∫ b0e−stf(t)dt,

limb→∞∫ b0e−stf(t)dt = (limb→∞

∫ b0e−stf(t)dt, limb→∞

∫ b0e−stf(t)dt)

also by using definition of classical Laplace transform:

l[f(t, r)] = limb→∞∫ b0e−stf(t)dt and

l[f(t, r)] = limb→∞∫ b0e−stf(t)dt

then we followL[f(t)] = (l[f(t, r), l[f(t, r))

Theorem 2.2 [4].Let f′(t) be an integrable fuzzy-valued function,and f(t) is

the primitive of f′(t) on [0,∞).Then

L[f′(t)] = sL[f(t)] f(0)

where f is (i)-differentiableorL[f

′(t)] = (−f(0)) (−sL[f(t)])

where f is (ii)-differentiableTheorem 2.3 [4].Let f(t) and g(t) be continuous fuzzy-valued functions andc1, c2 are constants.suppose that f(t)e−st, g(t)e−st are improper fuzzy Riemann-integrable on [0,∞],thenL[(c1f(t)) + (c2g(t))] = (c1L[f(t)]) + (c2L[g(t)]).Theorem 2.4 Let f

′′(t) be integrable fuzzy-valued function,and f(t),f

′(t) are

primitive of f′(t),f

′′(t) on [0,∞].Then

L[f′′(t)] = s2L[f(t)] sf(0) f ′(0)

where f is (i)-differentiable and f′

is (i)-differentiable orL[f

′′(t)] = s2L[f(t)] sf(0)− f ′(0)

where f is (ii)-differentiable and f′

is (ii)-differentiable orL[f

′′(t)] = (−s2)L[f(t)]− sf(0)− f ′(0)

where f is (i)-differentiable and f′

is (ii)-differentiable orL[f

′′(t)] = (−s2)L[f(t)]− sf(0) f ′(0)

where f is (ii)-differentiable and f′

is (i)-differentiableTheorem 2.5 Let f(t) satisfies the condition of existence theorem of Laplacetransform and L[f(t)] = F (s) thenL[tf(t)] = −F ′(s)Hence if f

′(t) satisfies the condition of existence theorem of Laplace transform

then

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L[tf′(t)] = − d

dsL[f

′(t)] = − d

ds

{sF (s)− f(0)

}= −sF ′(s)− F (s)

similarly for f′′

L[tf′′(t)] = − d

dsL[f

′′(t)] = − d

ds

{s2F (s)− sf(0)− f ′(0)

}= −s2F ′(s)− 2F (s) + f(0)

3 Examples

Example 3.1 Consider Variable coefficient differential equation ty′′ − y′ = −1

subject to initial condition y(0) = 0Consider Variable coefficient fuzzy differential equation where we are consider-ing initial condition and forcing function both triangular fuzzy number

ty′′ − y′ = −1, t ∈ [0, 1]

y(0) = [r − 1, 1− r]1 = [r, 2− r]

By using fuzzy Laplace transform method,we have:L[ty

′′]− L[y

′] = −L[1]

By using FLT of derivative and Differentiation of FLTl[y(t, r)] = r−1

s + rs2 + A

s3

l[y(t, r)] = 1−rs + 2−r

s2 + As3 ,where A is constant.

Hence solution is as follows:y(t, r) = (r − 1) + rt+Bt2

y(t, r) = (1− r) + (2− r)t+Bt2,where B is constant.The y(t, r) and y(t, r) at r ∈ [0, 1] are presented in Fig.4

y(t) = y(t, 1) = y(t, 1) = t+Bt2

0.00

0.05

0.10

t

0.0

0.5

1.0

r

-1.0

-0.5

0.0

0.5

1.0

yHt,rL

Fig.4 Solution y(t,r) by using Triangular fuzzy number

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Consider Trapezoidal fuzzy number

ty′′ − y′ = −1, t ∈ [0, 1]

y(0) = [r − 1, 2− r]1 = [r, 3− r]


′′]− L[y

′] = −L[1]

By using FLT of derivative and Differentiation of FLTl[y(t, r)] = r−1

s + rs2 + A

s3

l[y(t, r)] = 2−rs + 3−r

s2 + As3 ,where A is constant.

Hence solution is as follows:y(t, r) = (r − 1) + rt+Bt2

y(t, r) = (2− r) + (3− r)t+Bt2,where B is constant.The y(t, r) and y(t, r) at r ∈ [0, 1] are presented in Fig.5

y(t, 1) = t+Bt2 , y(t, 1) = 1 + 2t+Bt2

0.00

0.05

0.10

t

0.0

0.5

1.0

r

-1

0

1

2

yHt,rL

Fig.5 Solution y(t,r) by using Trapezoidal fuzzy number

Consider Gaussian fuzzy number

ty′′ − y′ = −1, t ∈ [0, 1]

y(0) = [−√−(2loger),

√−(2loger)]

1 = [1−√

(−2loger), 1 +√

(−2loger)]


′′]− L[y

′] = −L[1]

By using FLT of derivative and Differentiation of FLT

l[y(t, r)] =−√−(2loger)s +

1−√

(−2loger)s2 + A

s3

l[y(t, r)] =

√−(2loger)

s +1+√

(−2loger)s2 + A

s3 ,where A is constant.Hence solution is as follows:y(t, r) = [−

√−(2loger)] + [1−

√(−2loger)]t+Bt2

y(t, r) = [√−(2loger)] + [1 +

√(−2loger)]t+Bt2,where B is constant.

The y(t, r) and y(t, r) at r ∈ [0, 1] are presented in Fig.6

y(t) = y(t, 1) = y(t, 1) = t+Bt2

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0.00

0.05

0.10

t

0.0

0.5

1.0

r

-5

0

5

10

yHt,rL

Fig.6 Solution y(t,r) by Gaussian fuzzy number

Example 3.2 Consider Variable coefficient differential equation ty′′

+2y′+ty = 0

subject to boundary conditionsy(0) = 1y(π) = 0Consider Variable coefficient fuzzy differential equation where we are consider-ing boundary conditions triangular fuzzy number

ty′′

+ 2y′+ ty = 0,

y(0) = [r, 2− r]y(π) = [r − 1, 1− r]


′′] + 2L[y

′] + L[ty] = −L[0]

By using FLT of derivative and Differentiation of FLT l[y(t, r)] = (r)tan−1( 1s )

l[y(t, r)] = (2− r)tan−1( 1s )

Hence solution is as follows:y(t, r) = (r) sintty(t, r) = (2− r) sinttThe y(t, r) and y(t, r) at r ∈ [0, 1] are presented in Fig.7

y(t) = y(t, 1) = y(t, 1) = sintt

0.0

0.5

1.0r

01 2

3t

0.0

0.5

1.0

1.5

2.0

yHt,rL

Fig.7 Solution y(t,r) by using Triangular fuzzy number

Consider Trapezoidal fuzzy number

ty′′

+ 2y′+ ty = 0,

y(0) = [r, 3− r]y(π) = [r − 1, 2− r]

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′′] + 2L[y

′] + L[ty] = −L[0]

By using FLT of derivative and Differentiation of FLTl[y(t, r)] = (r)tan−1( 1

s )

l[y(t, r)] = (3− r)tan−1( 1s )

Hence solution is as follows:y(t, r) = (r) sintty(t, r) = (3− r) sinttThe y(t, r) and y(t, r) at r ∈ [0, 1] are presented in Fig.8

y(t, 1) = sintt , y(t, 1) = 2 sintt

0.0

0.5

1.0r

01 2

3

t

0

1

2

3

yHt,rL

Fig.8 Solution y(t,r) by using Trapezoidal fuzzy number

Consider Gaussian fuzzy number

ty′′

+ 2y′+ ty = 0,

y(0) = [1−√−(2loger), 1 +

√−(2loger)]

y(π) = [−√

(−2loger),√

(−2loger)]


′′] + 2L[y

′] + L[ty] = 0

By using FLT of derivative and Differentiation of FLTl[y(t, r)] = [1−

√(−2loger)]tan

−1( 1s )

l[y(t, r)] = [1 +√

(−2loger)]tan−1( 1

s )Hence solution is as follows:y(t, r) = [1−

√(−2loger)]

sintt

y(t, r) = [1 +√

(−2loger)]sintt

y(t) = y(t, 1) = y(t, 1) = sintt

0.0

0.5

1.0r

01 2

3

t

0

5

yHt,rL

Fig.9 Solution y(t,r) by using Gaussian fuzzy number

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4 Result and Discussion

From,Examples 1,2 we see that the solution of second order FIVP and FBVPare depends on the derivative i.e.(i)-differentiable or (ii)-differentiable.Thus,asin above examples,the solution can be adequately chosen among four cases ofthe strongly generalize differentiability.On the other hand,In this new procedureunicity of the solution is lost because we have four possibilities,but flexibility isgained in fuzzy context.In above Examples,for Triangular and Gaussian fuzzynumbers at r = 1 upper bound and lower bounds are same and that is same asExact solution of given differential equation but for Trapezoidal fuzzy number,itis some interval that contain Exact solution as lower bound or upper bound.

5 Conclusion

The Fuzzy Laplace transform method provided solutions to variable coefficientsecond order FIVPs and FBVPs by using the strongly generalize differentiabilityconcept.Here we solved FIVPs and FBVPs by using different types of fuzzy num-bers like Triangular,Trapezoidal and Gaussian.In that Triangular fuzzy numberis easy to use in conclusion where as Trapezoidal fuzzy number required longcalculations and Gaussian fuzzy number include logarithmic function i.e. againdifficult to deal with it if asymmetric Gaussian fuzzy number occurs in calcula-tion.The efficiency of method was described by solving numerical examples.

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PRAJÑĀ - Journal of Pure and Applied Sciences Vol. 24 - 25 : 30 - 37 (2017) ISSN 0975 - 2595

SOLUTION OF NTH ORDER FUZZY INITIAL VALUE PROBLEM

Komal R.Patel1, Narendrasinh B.Desai2*

1Assistant Professor, Applied Sciences and Humanities Department, ITM Universe, Vadodara-390510, Gujarat, India

2Associate Professor, Head of Applied Sciences and Humanities Department, ADIT, V.V.Nagar-388121, Gujarat, India

[email protected], [email protected] *Corresponding author

ABSTRACT

In this paper we consider higher order linear differential equations with fuzzy initial values that occurs in almost all engineering branches. Here ,We find solution for constant coefficient and variable coefficient third order linear differential equation by using method based on properties of linear trasformation.We show that fuzzy problem has unique solution if corresponding crisp Problem has unique solution. We will also prove that if the initial values are triangular fuzzy numbers, then the values of the solution at a given time are also triangular fuzzy numbers. We are going to propose a method to find fuzzy solution. We present three examples, One is homogeneous and another is non-homogeneous linear differential equation with constant coefficient and third is non-homogeneous linear differential equation with variable coefficient (Cauchy-Euler equation) to illustrate applicability of proposed method. We also plot graphs to show difference between exact and fuzzy solutions. This shows that our method is practical and applicable to solve nth order fuzzy initial value problems. AMS Subject Classification code: 34A07:Fuzzy differential equations. Keywords: Fuzzy initial value problem, Fuzzy set, Fuzzy number and Linear transformation.

INTRODUCTION

Fuzzy initial value problem occurs in almost all Engineering branches. The term "Fuzzy differential equation" was put forward for the first time by Kandel and Byatt [1],The fuzzy initial value problem was studied by Seikkala [26].The fuzzy initial problem has been investigated by many authors so far Buckley and Feuring [7,8]; Buckley, Feuring, and Hayashi [9]; Lakshmikantham and Nieto [18]; Bede and Gal [5]; Bede, Rudas, and Bencsik [3]; Perfilieva et al.[25]; Chalco-Cano and Román-Flores[10,11]; Khastan, Bahrami, and Ivaz [14]; Gasilov, Amrahov, and Fatullayev [17]; Khastan,Nieto,Rodríguez-López [16]; Patel, Desai [22].Gasilo et al.[13],Gomes and Barros [4,28] proposed concepts of fuzzy calculus, analogically to classical calculus, and studied fuzzy differential equations in terms of this calculus. Under certain conditions, they established the existence of a solution for the first order fuzzy initial value problem and suggested a solution method. Gasilov et al. [17] benefitted from properties of linear transformations and proposed a method to find fuzzy bunch of Solution functions for linear equation. The method is applicable to higher order linear differential equations with constant coefficients. Most of the researchers assume that derivative in the differential equation as a derivative of a fuzzy function in some sense. In earlier researches the derivative was considered as

Hukuhara derivative. A study in this direction was made by Kaleva [19,20,21]. When Hukuhara derivative is used, then uncertainty of the solution may increase infinitely with time. Furthermore, Bede and Gal [5] showed that a simple fuzzy function, generated by multiplication of differentiable crisp function and a fuzzy number, may not have Hukuhara derivative. In order to overcome this difficulty Bede and Gal [5] developed the generalized derivative concept and after that the studies about this subject were accelerated (Bede[2]); Bede et al [6]; Khastan and Nieto[15]; Chalco-Cano,Román-Flores[10,11,29]. But in case of generalized Hukuhara derivatives there are four different cases for second order fuzzy differential equation. Khastan A.,Gasilov.N.A., Fatullayev.A.G, Amrahov.S.E.[27], found solution of constant coefficient FBVP. Patel, Desai [23,24] solve fuzzy initial value problems by fuzzy Laplace transform. In this paper we consider the fuzzy initial value problem as a set of crisp problem using properties of linear transformations. We solve higher order Cauchy-Euler’s equation. For clarity we explain the

proposed method for third order fuzzy linear differential equations, but the results are true for higher-order equations too. The fuzzy solution proposed by our method coincides with extension principle’s results.

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PRELIMINARIES Definition: Membership function:

A fuzzy set 𝐴 can be defined as a pair of the universal set U and the membership function 𝜇: 𝑈 −> [0,1] for each x ∈ 𝑈, the number 𝜇𝐴 is called the membership degree of x in 𝐴 .

Definition: 𝜶 − 𝒄𝒖𝒕 𝒔𝒆𝒕 :

𝐹𝑜𝑟 𝑒𝑎𝑐ℎ 𝛼 ∈ 0,1 the crisp set 𝐴𝛼 = 𝑥 ∈ 𝑈ǀ𝜇𝐴 𝑥 ≥ 𝛼 is called 𝛼 − 𝑐𝑢𝑡 𝑠𝑒𝑡 of 𝐴 .We use the notation 𝑢 = 𝑢𝐿, 𝛼 , 𝑢𝑅(𝛼) 0 ≤ 𝛼 ≤ 1 to indicate a fuzzy

number in parametric form. We denote 𝑢 = 𝑢𝐿 0 and 𝑢 = 𝑢𝑅 0 to indicate the left and the right end-points of 𝑢 respectively. an α-cut of 𝑢 is an interval [𝑢𝐿, 𝛼 , 𝑢𝑅(𝛼)], which we denote as 𝑢𝛼= [u𝛼, 𝑢 𝛼].

Definition:Fuzzy Number A fuzzy number is a fuzzy set like 𝑈: 𝑅 → 𝐼 = [0,1] which satisfies:

(a) 𝑢 is upper semi-continuous on 𝑅 ∀ ∈ > 0 ∃ 𝛿 > 0 ∋ 𝑢 𝑥 − 𝑢 𝑥0 < ∈, 𝑥 − 𝑥0 < 𝛿

(b) 𝑢 is fuzzy convex i.e.(𝜆𝑥 + 1 − 𝜆 𝑦) ≥min 𝑢 𝑥 , 𝑢 𝑦 ∀𝑥, 𝑦 ∈ 𝑅, 𝜆 ∈ [0,1]

(c) 𝑢 is normal i.e ∃𝑥0 ∈ 𝑅 for which 𝑢(𝑥0) = 1

(d)𝐹𝑜𝑟 𝑒𝑎𝑐ℎ 𝛼 ∈ 0,1 , 𝑠𝑢𝑝𝑝 𝑢 = 𝑥 ∈ 𝑅ǀ𝑢𝛼(𝑥) >0 is support of 𝑢,and its 𝑐𝑙(𝑠𝑢𝑝𝑝 𝑢) is compact.

Definition: Triangular Fuzzy Number:

The Triangular fuzzy number as 𝑢 = (a, b, c) for which 𝑢𝐿, 𝛼 = a + 𝛼 (b − a), 𝑢𝑅(𝛼) = b - 𝛼 (c − b)

and u = a, 𝑢 = c. In geometric interpretations, we refer to the point 𝑏 as a vertex. Let us consider a triangular fuzzy number 𝑢 = (p, 0, q) the vertex of which is 0 (Note that p < 0 and q > 0 in this case).Then 𝑢𝐿(𝛼) = (1−α) p and

𝑢𝑅(𝛼) = (1−α) q and consequently, α-cuts are intervals

[(1 − 𝛼) 𝑝, (1 − 𝛼) 𝑞] = (1 − 𝛼)[𝑝, 𝑞] From the last representation one can see that an α-cut is similar to the interval [p, q] (i.e. to the 0-cut) with similarity coefficient (1−α). We often express a fuzzy number 𝑢 as 𝑢 = 𝑢𝑐𝑟 + 𝑢 𝑢𝑛 (crisp part + uncertainty). Here 𝑢𝑐𝑟 is a number with membership degree 1 and represents the crisp part (the vertex) of 𝑢 ,while 𝑢 𝑢𝑛 represents the uncertain part with vertex at the origin.

For a triangular fuzzy number 𝑢 = (a, b, c) we have 𝑢𝑐𝑟 = 𝑏 and 𝑢 𝑢𝑛 = (𝑎 − 𝑏, 0 , 𝑐 − 𝑏). Properties of Fuzzy Valued Number

For arbitrary 𝑢 =(𝑢 𝑟 , 𝑢 (𝑟)) 𝑣 = (𝑣 𝑟 , 𝑣 (𝑟)),

0 ≤ 𝑟 ≤ 1 and arbitrary 𝑘 ∈ 𝑅. We define addition, subtraction, multiplication, scalar multiplication by 𝑘.

𝑢 + 𝑣 = (𝑢 𝑟 + 𝑣 𝑟 , 𝑢 𝑟 + 𝑣 (r)) 𝑢 − 𝑣 = (𝑢 𝑟 − 𝑣 𝑟 , 𝑢 𝑟 − 𝑣(r))

𝑢 ∙ 𝑣 = (min u r v r , 𝑢 𝑟 𝑣 𝑟 , 𝑢 𝑟 𝑣 r , 𝑢 𝑟 𝑣 𝑟 ,

max u r v r , 𝑢 𝑟 𝑣 𝑟 , 𝑢 𝑟 𝑣 r , 𝑢 𝑟 𝑣 𝑟 )

𝑘𝑢 = (𝑘𝑢 𝑟 , 𝑘𝑢 𝑟 ) 𝑖𝑓 𝑘 ≥ 0

(𝑘𝑢 𝑟 , 𝑘𝑢 𝑟 ) 𝑖𝑓 𝑘 < 0

Definition:Hukuhara difference

Let 𝑥, 𝑦 ∈ 𝐸.If there exists 𝑧 ∈ 𝐸 such that 𝑥 = 𝑦 + 𝑧, then 𝑧 is called the Hukuhara difference of fuzzy numbers x and y, and it is denoted by 𝑧 = 𝑥 Ɵ 𝑦. The Ɵ sign stands for Hukuhara-difference and 𝑥 Ɵ 𝑦 ≠ 𝑥 + −1 𝑦.

Definition:Hukuhara differential

Let 𝑓: (𝑎, 𝑏) → 𝐸 and 𝑡0 ∈ (𝑎, 𝑏) if there exists an element 𝑓 ′(𝑡0) ∈ E such that for all ℎ > 0 sufficiently small,exists 𝑓 𝑡0 + ℎ Ɵ 𝑓(𝑡0), 𝑓 𝑡0 Ɵ𝑓(𝑡0 − ℎ) and the limits holds(in the metric D)

limℎ→0f 𝑡0+h Ɵ f(𝑡0)

ℎ = limℎ→0

f 𝑡0 Ɵ f(𝑡0−h)

ℎ =𝑓 ′(𝑡0)

Here derivative is considered as Hukuhara

derivative.

FUZZY INITIAL VALUE PROBLEMS (FIVPS) In this section, We have described a fuzzy initial value problem (FIVP) and concept of solution. We investigate a fuzzy Initial value problem with crisp linear differential equation and fuzzy initial values. FIVP can arise in modeling of a process the dynamics of which is crisp but there are uncertainties in initial values.

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Consider the nth order fuzzy initial value problem where 𝑏𝑛 𝑥 ≠ 0.

𝑏𝑛 𝑥 𝑦𝑛 + 𝑏𝑛−1 𝑥 𝑦𝑛−1 + ⋯ + 𝑏1 𝑥 𝑦′+𝑏0 𝑥 𝑦 = 𝑓 𝑥

𝑦 𝑙 = 𝐴1

𝑦′ 𝑙 = 𝐴2

𝑦′′ 𝑙 = 𝐴3

……

𝑦𝑛−1 𝑙 = 𝐴𝑛

1

Where 𝐴1 , 𝐴2

, 𝐴3 ,…..,𝐴𝑛

are fuzzy numbers and 𝑏0(𝑥), 𝑏1(𝑥), … , 𝑏𝑛 (𝑥) and 𝑓 (𝑥) are continuous crisp functions or constants and 𝑙 is any integer.Let us represent the initial values as 𝐴1

= 𝑎1 + 𝑎1 , 𝐴2 = 𝑎2+ 𝑎2 ,….., 𝐴𝑛

= 𝑎𝑛+ 𝑎𝑛 where 𝑎1 , 𝑎2 , … . . , 𝑎𝑛 are crisp numbers while 𝑎1 𝑎2 , … … , 𝑎𝑛 are fuzzy numbers. We split the FIVP in Eq.(1) to the following problems: Associated crisp problem (which is non-homogeneous)

𝑏𝑛 𝑥 𝑦𝑛 + 𝑏𝑛−1 𝑥 𝑦𝑛−1 + ⋯ + 𝑏1 𝑥 𝑦′+𝑏0 𝑥 𝑦 = 𝑓 𝑥

𝑦 𝑙 = 𝑎1

𝑦′ 𝑙 = 𝑎2

𝑦′′ 𝑙 = 𝑎3

……

𝑦𝑛−1 𝑙 = 𝑎𝑛

(2)

Homogeneous problem with fuzzy initial values

𝑏𝑛 𝑥 𝑦𝑛 + 𝑏𝑛−1 𝑥 𝑦𝑛−1 + ⋯ + 𝑏1 𝑥 𝑦 ′+𝑏0 𝑥 𝑦 = 0

𝑦 𝑙 = 𝑎1

𝑦′ 𝑙 = 𝑎2

𝑦′′ 𝑙 = 𝑎3 ……

𝑦𝑛−1 𝑙 = 𝑎𝑛

(3)

It is easy to see if 𝑦𝑐𝑟 𝑥 and 𝑦 un(x) are solutions of Eq.(2) and Eq.(3) respectively then 𝑦 (𝑥) = 𝑦𝑐𝑟 𝑥 + 𝑦 un(𝑥) is a solution of the given problem in Eq.(1). Hence, Eq.(1) is reduced to solving a non-homogeneous equation with crisp conditions in Eq.(2) and homogeneous equation with fuzzy initial conditions in Eq.(3). Therefore, we will investigate how to solve Eq.(1).To determine 𝑦 (𝑥) we consider Linear transformation 𝑇: 𝑅𝑛 → 𝑅 , 𝑇 𝑢 =𝑣 𝑢, where 𝑣 is fixed 𝑛 𝑥 𝑛 determinant and 𝑢 = [𝑎1 , 𝑎2 , … 𝑎𝑛 ]𝑇.

THE SOLUTION ALGORITHM The solution algorithm consists of four steps: Represent the initial values as 𝐴1

=𝑎1 + 𝑎1 ,

𝐴2 = 𝑎2 + 𝑎2 ,….. , 𝐴𝑛

= 𝑎𝑛+ 𝑎𝑛 Find linear independent solutions 𝑦1 𝑥 ,

𝑦2 𝑥 ,……,𝑦𝑛 𝑥 of the crisp differential equation 𝑏𝑛 (𝑥)𝑦𝑛 + 𝑏𝑛−1 𝑥 𝑦𝑛−1 +

⋯ 𝑏1 𝑥 𝑦 ′+𝑏0 𝑥 𝑦 = 𝑓(𝑥).Constitute the vector-function 𝑠 𝑥 = (𝑦1 𝑥 , 𝑦2 𝑥 , … . , 𝑦𝑛 (𝑥)), the determinant W and calculate the vector-function

𝑡 𝑥 =𝑠 𝑥 𝑊−1 = 𝑡1 𝑥 , 𝑡2 𝑥 , … . , 𝑡𝑛 𝑥

by formula at 𝑥 = 𝑙. The Wronskain

𝑊 = 𝑦1(𝑙) 𝑦2 𝑙 … 𝑦𝑛 𝑙

⋮ ⋮ ⋱ ⋮𝑦1

𝑛−1(𝑙) 𝑦2𝑛−1 𝑙 … 𝑦𝑛

𝑛−1(𝑙)

and det(𝑊) ≠ 0 ∴ 𝑊 −1 𝑒𝑥𝑖𝑠𝑡.

Find the solution 𝑦𝑐𝑟 𝑥 of the non-homogeneous crisp problem.

The solution of the given problems For homogeneous FIVP

y (x) = 𝑡1(𝑥) 𝑎1 + 𝑡2(𝑥) 𝑎2 + … +(𝑥) 𝑎𝑛 . For non-homogeneous FIVP 𝑦 (x) = 𝑦𝑐𝑟 𝑥 + 𝑡1(𝑥) 𝑎1 + 𝑡2(𝑥) 𝑎2 + … +𝑡𝑛 (𝑥) 𝑎𝑛 . EXAMPLES Example:5.1 Solve the 3rd order FIVP with constant coefficient homogeneous equation.

𝑦 ′′′ + 3𝑦 ′′ + 3𝑦 ′ + 𝑦 = 0

𝑦 0 = (−0.5,0,1)

𝑦 ′ 0 = (−1,0,1)

𝑦 ′′ 0 = (−1,0,0.5)

(4)

The problem is homogeneous and initial values are fuzzy numbers with vertices at 0.therefore solution by solution algorithm.𝑦 𝑥 = 𝑒−𝑥 , 𝑦2 𝑥 =𝑥𝑒−𝑥and 𝑦3 𝑥 = 𝑥2𝑒−𝑥are linearly independent solution for the equation 𝑦 ′′′ + 3𝑦 ′′ + 3𝑦 ′ + 𝑦 = 0. Hence 𝑠(𝑥) = (𝑒−𝑥 , 𝑥𝑒−𝑥 , 𝑥2𝑒−𝑥) and

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𝑊 = 1 0 0

−1 1 01 −2 2

and

𝑡(𝑥) = 𝑠 𝑥 𝑊−1 = (𝑡1(𝑥), 𝑡2(𝑥), 𝑡3(𝑥)),

𝑡(𝑥) = (𝑒−𝑥 + 𝑥𝑒−𝑥 +𝑥2

2𝑒−𝑥 , 𝑥𝑒−𝑥

+ 𝑥2𝑒−𝑥 ,𝑥2

2𝑒−𝑥)

𝑦 𝑢𝑛 𝑥 = 𝑒−𝑥 + 𝑥𝑒−𝑥 + 𝑥2𝑒−𝑥 −0.5,0,1

+ 𝑥𝑒−𝑥 + 𝑥2𝑒−𝑥 (−1,0,1)

+𝑥2

2𝑒−𝑥(-1,0,0.5)

Where the arithmetic operations are considered to be fuzzy operations. The fuzzy solution 𝑦 𝑥 form band in the 𝑥𝑦-coordinate space (Fig.1) Since the initial values are triangular fuzzy numbers, an α-cut of the solution can be determined by similarity coefficient (1−α), i.e. 𝑦𝛼 𝑥 = 1 − 𝛼 [y 𝑥 , 𝑦 𝑥 ]

Fig.1 The fuzzy solution, obtained by the proposed method, for Example 5.1 Example:5.2 Consider the 3rd order FIVP with constant coefficient non-homogeneous equation.

𝑦 ′′′ + 3𝑦 ′′ + 3𝑦 ′ + 𝑦 = 30𝑒−𝑥

𝑦 0 = 2.5,3,4

𝑦 ′ 0 = −4, −3, −2

𝑦 ′′ 0 = −48, −47, −46.5

5

We represent the initial values as 𝐴 = (2.5,3,4) = 3 + (-0.5,0,1), 𝐵 = (-4,-3,-2) = -3 + (-1,0,1), 𝐶 =(-48,-47,-46.5) = - 47 + (-1,0,0.5).

we solve crisp non-homogeneous crisp problem

𝑦 ′′′ + 3𝑦 ′′ + 3𝑦 ′ + 𝑦 = 30𝑒−𝑥

𝑦 0 = 3

𝑦 ′ 0 = −3

𝑦 ′′ 0 = −47

(6)

And the crisp solution 𝑦𝑐𝑟 𝑥 = 3 − 25𝑥2 𝑒−𝑥 + 5𝑥3𝑒−𝑥

𝑦𝑐𝑟 𝑥 = 3 𝑒−𝑥 + 𝑥𝑒−𝑥 + 𝑥2𝑒−𝑥 − 3 𝑥𝑒−𝑥 + 𝑥2𝑒−𝑥

−47𝑥2

2𝑒−𝑥 + 5𝑥3𝑒−𝑥

Fuzzy homogeneous problem to find the uncertainty of

the solution is as follows:

𝑦 ′′′ + 3𝑦 ′′ + 3𝑦 ′ + 𝑦 = 0

𝑦 0 = (0.5,0,1)

𝑦 ′ 0 = (−1,0,1)

𝑦 ′′ 0 = (−1,0,0.5)

(7)

This problem is the same as Example 1. Hence, the solution is

𝑦 𝑢𝑛 𝑥 = 𝑒−𝑥 + 𝑥𝑒−𝑥 + 𝑥2𝑒−𝑥 −0.5,0,1 + 𝑥𝑒−𝑥 + 𝑥2𝑒−𝑥 (−1,0,1)

+𝑥2

2𝑒−𝑥(-1,0,0.5)

Fig. 2 The fuzzy solution 𝑦 𝑥 , obtained by the proposed method, for Example 5.2.Red line represents the crisp solution.

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We add this uncertainty to the crisp solution and get the fuzzy solution of the given FIVP

𝑦 𝑥 = 𝑦𝑐𝑟 𝑥 + 𝑦 𝑢𝑛 𝑥

= 𝑒−𝑥 + 𝑥𝑒−𝑥 + 𝑥2𝑒−𝑥 2.5,3,4 + 𝑥𝑒−𝑥 + 𝑥2𝑒−𝑥 (−4, −3, −2)

+𝑥2

2𝑒−𝑥(-48,-47,-46.5) + 5𝑥3𝑒−𝑥

In above example if we take 𝑥 → ∞ then fuzziness in solution is disappeared. So the solution goes nearer to zero if we increase 𝑥.(see Fig.3)

Fig. 3 If x increases fuzziness disappears Example:5.3 Consider 3rd order FIVP with variable coefficient.

(𝑥3𝐷3 − 3𝑥2𝐷2 + 6𝑥𝐷 − 6)𝑦 =

12

𝑥

𝑦 1 = 4,5,6

𝑦′ 1 = 12,13,14.5

𝑦′′ 1 = 9.5,10,11

(8)

(𝑥3𝐷3 − 3𝑥2𝐷2 + 6𝑥𝐷 − 6)𝑦 =

12

𝑥

𝑦 1 = 5

𝑦 ′ 1 = 13

𝑦 ′′ 1 = 10

(9)

Associated non-homogeneous problem has the solution i.e. exact solution of non-homogeneous problem

𝑦𝑐𝑟 𝑥 =−138

24𝑥 +

328

24𝑥2 +

−69

24𝑥3 −

1

24𝑥

Consider homogeneous equation

(𝑥3𝐷3 − 3𝑥2𝐷2 + 6𝑥𝐷 − 6)𝑦 = 0

𝑦 1 = −1,0,1

𝑦 ′ 1 = −1,0,0.5

𝑦 ′′ 1 = −0.5,0,1

(10)

The problem is homogeneous and initial values are fuzzy numbers with vertices at 0.therefore solution by solution algorithm.𝑦1 𝑥 = 𝑥 , 𝑦2 𝑥 = 𝑥2 and 𝑦2 𝑥 = 𝑥3 are linearly independent solution of equation ( 𝑥3𝐷3 − 3𝑥2𝐷2 + 6𝑥𝐷 − 6)𝑦 = 0 Hence 𝑠 𝑥 =(𝑥, 𝑥2 , 𝑥3 ) and

𝑊 = 1 1 11 2 30 2 6

And 𝑡(𝑥)= 𝑠 𝑥 𝑊−1 = (𝑡1(𝑥), 𝑡2(𝑥), 𝑡3(𝑥))

𝑡 𝑥 = 3𝑥 − 3𝑥2 + 𝑥3 , −2𝑥 + 3𝑥2 − 𝑥3 ,𝑥

2− 𝑥2 +

𝑥3

2

Fig.4 The fuzzy solution, obtained by the proposed method, for Example 5.3 The fuzzy solution is

𝑦 𝑢𝑛 𝑥 = 3𝑥 − 3𝑥2 + 𝑥3 −1,0,1 + −2𝑥 + 3𝑥2 − 𝑥3 −1,0,0.5

+ (𝑥

2− 𝑥2 +

𝑥3

2)(−0.5,0,1)

We add this uncertainty to the crisp solution and get the fuzzy solution of the given FIVP

𝑦 𝑥 = 𝑦𝑐𝑟 𝑥 + 𝑦 𝑢𝑛 𝑥 = 3𝑥 − 3𝑥2 + 𝑥3 4,5,6

+ −2𝑥 + 3𝑥2 − 𝑥3 (12, 13, 14.5)

+( 𝑥

2− 𝑥2 +

𝑥3

2)(9.5,10,11) -

1

24𝑥

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𝑦 𝑥 = −6𝑥 + 14𝑥2 − 3𝑥3 −1

24𝑥

Fig.5 The fuzzy solution, obtained by the proposed method, for Example 5.3.Red line represents the crisp solution. In above example if we increase 𝑥 and as we take 𝑥 → ∞ then fuzziness in solution increases and goes to infinite.

Fig. 6 If we increase 𝑥 fuzziness increases Example:5.4 Consider a unit mass sliding on a frictionless table attached to a spring, with spring constant k =16. Suppose the mass is lightly tapped by a hammer every T seconds. Suppose that the first tap occurs at time t = 0 and before that time the mass is at rest. Describe what happens to the motion of the mass for the tapping period T =1.

Fig. 7

𝑥 ′′ 𝑡 + 16 𝑥 𝑡 = 𝑠𝑖𝑛𝑡𝑥 0 = 0

𝑥 ′ 0 = 1

(11)

Consider the homogeneous equation with fuzzy parameters.

𝑥 ′′ 𝑡 + 16 𝑥 𝑡 = 0 𝑥 0 = (−1,0,1)

𝑥 ′ 0 = (−0.5,0,1)

(12)

The problem is homogeneous and initial values are fuzzy numbers with vertex 0 therefore solution by solution algorithm. 𝑥1(𝑡) = 𝑐𝑜𝑠4𝑡 , 𝑥2(𝑡) = 𝑠𝑖𝑛4𝑡 are linearly independent solution for the equation 𝑥 ′′(𝑡) + 16𝑥(𝑡) = 0 Hence 𝑠(𝑡) = (𝑐𝑜𝑠4𝑡, 𝑠𝑖𝑛4𝑡) and

𝑊 = 1 00 4

and 𝑦(𝑡) = (𝑦1(𝑡), 𝑦2(𝑡)) Here 𝑦(𝑡) = (𝑐𝑜𝑠4𝑡, (1/ 4) 𝑠𝑖𝑛4𝑡) 𝑥 𝑢𝑛 𝑡 = (cos4t)(−1,0,1) + (1/ 4)sin4t(−0.5,0,1) Crisp solution of given non-homogeneous problem 𝑥𝑐𝑟 (𝑡) = (7/ 30)𝑠𝑖𝑛4𝑡 + (1/15)𝑠𝑖𝑛𝑡 . 𝑥 (𝑡) = 𝑥𝑐𝑟 (𝑡)+𝑥 𝑢𝑛 𝑡

=𝑐𝑜𝑠4𝑡(−1,0,1)+(7/30)𝑠𝑖𝑛4𝑡(0.5,1,2) + (1/15)𝑠𝑖𝑛𝑡 = (7/ 30)𝑠𝑖𝑛4𝑡 + (1/ 15)𝑠𝑖𝑛𝑡 where arithmetic operations are considered to be fuzzy operations. the fuzzy solution 𝑥 (𝑡)form band in the tx-coordinate space.(Fig.8)

Fig.8:The periodic fuzzy solution by properties of Linear transformation

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If we apply periodic force to block, it will move in given open interval that we denote by triangular fuzzy number, Here the damping force is negligible. The spring mass system occurs in all most all engineering branches as well as in the many real life situations. CONCLUSIONS In this paper we have represented the fuzzy initial value problems as a set of crisp problems. We have proposed a solution method based on the properties of linear transformations. For clarity we have explained the proposed method for third order linear differential equation but it is applicable for nth order also. Here we have solved third order Cauchy-Euler equation i.e. we solved variable coefficient initial value problems based on the properties of linear transformations. We also solved one application level problem. AIM AND SCOPE OF THE WORK The aim of paper is to find solution of any Fuzzy differential equation having uncertain initial conditions. In future we are going to solve Fuzzy differential equations having the applications in different branches of engineering. we will solve higher order FBVP based on the properties of linear transformations. REFERENCES Periodical articles

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