UNIVERSITI PUTRA MALAYSIA

NUMERICAL METHODS FOR SOLVING OSCILLATORY AND FUZZY DIFFERENTIAL EQUATIONS

ALI KARIMI DIZICHEH

FS 2014 63

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By

ALI KARIMI DIZICHEH

Thesis Submitted to the School of Graduate Studies, Universiti PutraMalaysia, in Fulfilment of the Requirements for the Degree of Doctor

of Philosophy

June 2014

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COPYRIGHT

All material contained within the thesis, including without limitation text, lo-gos, icons, photographs and all other artwork, is copyright material of UniversitiPutra Malaysia unless otherwise stated. Use may be made of any material con-tained within the thesis for non-commercial purposes from the copyright holder.Commercial use of material may only be made with the express, prior, writtenpermission of Universiti Putra Malaysia.

Copyright c© Universiti Putra Malaysia

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DEDICATIONS

To Hazrate FATEMEH ZAHRA (Salamollah alayha),daughter of Prophet MOHAMMAD (Salavatollah alayhe va Aleh)

To Hazrate ABALFAZL ABAS (Alyhesalam),son of Hazrate ALI (Salavatollah alayh)

To My Mother and Father

To My Teachers

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Abstract of thesis presented to the Senate of Universiti Putra Malaysia infulfilment of the requirement for the degree of Doctor of Philosophy

NUMERICAL METHODS FOR SOLVING OSCILLATORY ANDFUZZY DIFFERENTIAL EQUATIONS

By

ALI KARIMI DIZICHEH

June 2014

Chair: Professor Fudziah Ismail, Ph.D.

Faculty: Institute of Mathematical Research

In this thesis we develop five numerical schemes for solving ordinary differentialequations. These include exponentially-fitted Runge-Kutta method, trigonometri-cally fitted hybrid method, Legendre wavelet method on large intervals as well asan iterative spectral collocation method, exponentially-fitted fuzzy Runge-Kuttamethod, exponentially-fitted system of fuzzy Runge-Kutta method. The stabilityanalysis, estimation of local truncation errors and the efficiency of the methods’implementation in computer programs are discussed.

An exponentially-fitted explicit Runge-Kutta method of algebraic order 4 is for-mulated for the first-order ordinary differential equations

y′ = f(x, y), y(x0) = y0.

It integrates exactly the first-order systems where their solutions are expressed aslinear combinations of {exp (wx), exp (−wx)} or {cos (λx), sin(λx)} where w = λi.Stability analysis of our approach as well as a good estimation for the local trunca-tion errors are presented. The efficiency of the exponentially-fitted Runge-Kuttamethod is tested via some numerical experiments and a comparison with otherexisting methods.

A trigonometrically fitted explicit hybrid three-stage method is derived for thesecond-order initial value problems with oscillatory solutions. We compare ourresults with the classical hybrid method and the trigonometrically fitted explicitRunge-Kutta method through several examples. Our results indicate that trigono-

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metrically fitted explicit hybrid method is more efficient than the classical hybridmethod. we analyze the stability, phase- lag (dispersion) and dissipation.

An iterative spectral collocation method are introduced for solving initial valueproblems defined on large intervals. Indeed, the Legendre wavelet method is ex-tended and proved valid for large interval. Then, the Legendre-Guass collocationpoints of the Legendre wavelets are computed. By employing an interpolationbased on Legendre wavelet, we find approximate solution for any order (first-orderand second-order) differential equations. Using this strategy the iterative spectralmethod converts the differential equation to a set of algebraic equations. Solvingthis set of algebraic equations yields an approximate solution.

Using exponentially-fitted Runge-Kutta (EFRK) method, we develop a methodfor numerically solving fuzzy first order linear and nonlinear differential equationsunder generalized differentiability. In addition, this method is applied for the sys-tem of first order fuzzy differential equations with uncertainty. The generalizedHukuhara differentiability are applied to estimate the solutions. For solving thefuzzy problems, the exponentially-fitted Runge-Kutta method is applied.

Finally, some examples are solved to illustrate our proposed approaches. Theresults are compared with those in the literature. We show that our proposedmethods are simple and more accurate than the other existing methods.

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Abstrak tesis yang dikemukakan kepada Senat Universiti Putra Malaysia sebagaimemenuhi keperluan untuk ijazah Doktor Falsafah

KAEDAH BERANGKA UNTUK PENYELESAIAN AYUNAN DANPERSAMAAN PEMBEZAAN SAMAR

Oleh

ALI KARIMI DIZICHEH

Jun 2014

Pengerusi: Professor Fudziah Ismail, Ph.D.Fakulti: Institut Penyelidikan Matematik

Dalam tesis ini kita akan membangunkan lima skim berangka bagi menyelesaikanmasalah pembezaan biasa persamaan. Ini termasuk pesat dipasang kaedah Runge-Kutta, secara trigonometri kaedah hibrid dipasang, Legendre kaedah ombak kecildi selang besar serta kaedah penempatan bersama spektrum lelaran, pesat di-pasang kabur Runge-Kutta kaedah, sistem pesat dipasang kabur kaedah Runge-Kutta.

Analisis kestabilan, anggaran ralat pangkasan setempat dan kecekapan pelak-sanaan kaedah dalam program komputer yang dibincangkan. Satu yang jelaskaedah Runge-Kutta pesat dipasang perintah algebra 4 dirumus untuk pertama-perintah persamaan pembezaan biasa

y′ = f(x, y), y(x0) = y0.

Ia menggabungkan betul-betul sistem tertib pertama di mana penyelesaian merekadinyatakan sebagai gabungan linear bagi {eks(wx), eks(−wx)} atau {kos(λx),sin(λx)} di mana w = iλ.

Analisis kestabilan pendekatan kami dan juga anggaran yang baik bagi tempatanralat pangkasan dibentangkan. Keberkesanan kaedah Runge-Kutta eksponen di-pasang diuji melalui beberapa ujikaji berangka dan perbandingan dengan kaedahlain yang sedia ada. A secara trigonometri dipasang jelas hibrid kaedah tigaperingkat berasal bagi tertib kedua masalah nilai awal dengan penyelesaian ayu-nan. Kami membandingkan keputusan kami dengan kaedah hibrid klasik dan jelaskaedah Runge-Kutta yang secara trigonometri dipasang melalui beberapa contoh.Keputusan kami menunjukkan bahawa secara trigonometri dipasang kaedah hib-

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rid eksplisit adalah lebih cekap daripada kaedah hibrid klasik. Kita menganalisiskestabilan, fasa lag (penyebaran) dan pelepasan.

Satu kaedah penempatan bersama spektrum lelaran diperkenalkan untuk menye-lesaikan nilai awal masalah ditakrifkan pada selang yang besar. Malah, kaedahwavelet Legendre dilanjutkan dan terbukti sah untuk tempoh yang besar. Kemu-dian, Legendre-Guass titik penempatan bersama daripada riak Legendre dikira.Dengan menggunakan satu interpolasi berdasarkan Legendre ombak kecil, kitamencari penyelesaian anggaran untuk perintah (pertama-perintah dan tertib ke-dua) persamaan pembezaan. Menggunakan strategi ini kaedah spektrum lelaranmenukarkan persamaan pembezaan untuk satu set persamaan algebra. Menyele-saikan set persamaan algebra menghasilkan penyelesaian hampir. Menggunakanpesat dipasang Runge-Kutta (EFRK) kaedah, kami membangunkan satu kaedahyang untuk menyelesaikan secara berangka kabur linear peringkat pertama danpersamaan pembezaan linear bawah kebolehbezaan umum. Di samping itu, kaedahini digunakan untuk sistem perintah pertama persamaan pembezaan kabur denganketidakpastian.

Teritlak Hukuhara kebolehbezaan digunakan bagi menganggarkan penyelesaian.Untuk menyelesaikan masalah kabur, kaedah Runge-Kutta eksponen dipasangdigunakan. Akhir sekali, beberapa contoh diselesaikan untuk menggambarkanpendekatan dicadangkan kami. Keputusan berbanding dengan mereka dalam ke-susasteraan. Kita menunjukkan bahawa kaedah yang dicadangkan kami adalahmudah dan lebih tepat berbanding kaedah lain yang sedia ada.

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ACKNOWLEDGEMENTS

First and foremost, I am grateful to Allah s.w.t for providing an opportunity forme to continue my studies in Malaysia.

My deep gratitude goes to my beloved Hazrat MAHDI (which will emerge (Godwilling)), without his patience, guidance and moral support my survive would nothave been possible.

I am extremely indebted to my supervisor, Prof, Dr Fudziah Ismail for her excel-lent supervision, invaluable guidance, patience and financial support via grant forthe past some years. Her charisma and enthusiasm during these years has madethese challenging life period a very enjoyable and useful experience. She is the onealways leading me to keep faith during some difficult times. She really deservesspecial recognition because without her help, this work would not be accomplished.I have learned immensely from her not only in survival analysis but also in variousother aspects of life. She has helped me so much in so many ways that I do havenot enough words to express them.

Special thanks and appreciation goes to Prof Dato Dr. Malek Abu Hasan as amember of supervisory committee for his helpful comments, suggestions and co-operation. I would like to thank Associate Prof Dr. Norihan Md. Arifin being amember of the supervisory committee for her cooperation.I would like to thank all members of Institute for Mathematical Research (IN-SPEM) and Department of Mathematics for all their friendly support. This par-ticularly goes to Prof Dato Dr. Kamel Arifin M. Atan, for his great effort toprovide a unique academic environment for the graduate students. I am highlygrateful to the Universiti Putra Malaysia (UPM) for all the fruitful years of mystudy that left an enduring positive impression on my life and professional devel-opment.

My special thanks goes to my late grandfathers and my late grandmothers, mybeloved parents, my dear brothers (Majid and Mohammad) and my dear sisters(Akram, Maryam, Zahra, Bahareh) for their love, encouragement, support and alltheir favors.

I am always indebted to Dr Heydar Ghaeid Amini, Mr Mohammad Hoseyn Anaraki,Dr Majid Gazor, Dr Abdollah Hadi, Dr Soheil Salahshour, Dr Majid Tavassoli, DrMohammad Maleki, Dr Dr Sayed Ali Ahmadyan Hosseini, Dr Sarkhosh Seddighiand Dr Saeid Vahdati. I would like to thank all my friends for their encourage-ment.I would like to express my sincere appreciation to all who have helped and sup-ported me and contributed in many ways big and small to the success of thisresearch.

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This thesis was submitted to the Senate of Universiti Putra Malaysia and has beenaccepted as fulfilment of the requirement for the degree of Doctor of Philosophy.

The members of the Supervisory Committee were as follows:

Fudziah Ismail, Ph.D.ProfessorInstitute for Mathematical ResearchUniversiti Putra Malaysia(Chairperson)

Norihan Md. Arifin, Ph.D.Associate ProfessorInstitute for Mathematical ResearchUniversiti Putra Malaysia(Member)

BUJANG KIM HUAT, Ph.D.Professor and DeanSchool of Graduate StudiesUniversiti Putra Malaysia

Date:

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DECLARATION

Declaration by graduate student

I hereby confirm that:• this thesis is my original work;• quotations, illustrations and citations have been duly referenced;• this thesis has not been submitted previously or concurrently for any other

degree at any other institutions;• intellectual property from the thesis and copyright of thesis are fully-owned

by Universiti Putra Malaysia, as according to the Universiti Putra Malaysia(Research) Rules 2012;

• written permission must be obtained from supervisor and the office of DeputyVice-Chancellor (Research and Innovation) before thesis is published (in theform of written, printed or in electronic form) including books, journals, mod-ules, proceedings, popular writings, seminar papers, manuscripts, posters, re-ports, lecture notes, learning modules or any other materials as stated in theUniversiti Putra Malaysia (Research) Rules 2012;

• there is no plagiarism or data falsification/fabrication in the thesis, and schol-arly integrity is upheld as according to the Universiti Putra Malaysia (GraduateStudies) Rules 2003 (Revision 2012-2013) and the Universiti Putra Malaysia(Research) Rules 2012. The thesis has undergone plagiarism detection software.

Signature:—————————– Date:————————————‘

Name and Matric No.:——————————————————————

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Declaration by Members of Supervisory Committee

This is to confirm that:• the research conducted and the writing of this thesis was under our supervision;• supervision responsibilities as stated in the Universiti Putra Malaysia (Gradu-

ate Studies) Rules 2003 (Revision 2012-2013) are adhered to.

Signature:——————————- Signature:———————Name of Name ofChairman of Member ofSupervisory SupervisoryCommittee:————————— Committee:———————-

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CONTENTS

Page

DEDICATIONS i

ABSTRACT ii

ABSTRAK iv

ACKNOWLEDGEMENTS vi

APPROVAL vii

DECLARATION ix

LIST OF TABLES xiv

LIST OF FIGURES xvi

LIST OF ABBREVIATIONS xvi

CHAPTER

1 INTRODUCTION 11.1 Background 1

1.1.1 Initial value problem 21.2 Runge-Kutta method 2

1.2.1 Local truncation error and order conditions 31.2.2 Stability 5

1.3 Hybrid method 61.3.1 Local truncation error and order conditions 71.3.2 Phase-lag and stability analysis 11

1.4 Legendre polynomial 131.5 Fuzzy differential equations 131.6 Objectives of the thesis 151.7 Outline of the thesis 16

2 LITERATURE REVIEW 182.1 Exponentially-fitted Runge-Kutta method 182.2 Trigonometrically fitted hybrid method 182.3 Legendre Wavelets Spectral approach 202.4 Fuzzy first order differential equations 202.5 System of first order differential equations 21

3 EXPONENTIALLY-FITTED EXPLICIT RUNGE-KUTTA METHOD 233.1 Introduction 233.2 Derivation of exponentially-fitted Runge-Kutta method 23

3.2.1 Simos’s Technique 243.2.2 Vanden Berghe’s technique 25

3.3 Stability 27

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3.4 Derivation of local truncation error 273.5 Numerical experiments 313.6 Discussion 33

4 A TRIGONOMETRICALLY FITTED EXPLICIT HYBRID METHODFOR SECOND-ORDER INITIAL VALUE PROBLEMS WITHOSCILLATING SOLUTIONS 344.1 Introduction 344.2 Trigonometric fitted hybrid method 344.3 Stability analysis 37

4.3.1 Basic definition and properties 374.3.2 Stability and phase-lag analysis 38

4.4 Numerical experiments 404.5 Discussion 41

5 LEGENDRE WAVELETS SPECTRAL METHOD FOR SOLV-ING SECOND ORDER IVPS 425.1 Introduction 425.2 The Legendre wavelets spectral method 42

5.2.1 Legendre wavelets 425.2.2 Interpolation by Legendre wavelets 43

5.3 Solving IVPs on a large domain 455.4 Numerical experiments 465.5 Discussion 49

6 EXPONENTIALLY-FITTED RUNGE-KUTTA METHOD FORSOLVING FUZZY FIRST ORDER DIFFERENTIAL EQUATIONS 506.1 Introduction 506.2 Exponentially-fitted Runge-Kutta method 506.3 First order Fuzzy differential equations 526.4 Exponentially-fitted fuzzy Runge-Kutta method 536.5 Numerical experiments 556.6 Discussion 65

7 SOLUTIONS OF LINEAR SYSTEM OF FIRST-ORDER FUZZYDIFFERENTIAL EQUATIONS WITH FUZZY COEFFICIENT 667.1 Introduction 667.2 Exponentially-fitted Runge-Kutta method 667.3 Descriptions of Method 677.4 Numerical experiments 687.5 Discussion 75

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8 CONCLUSION 768.1 Summary 768.2 Future works 77

REFERENCES/BIBLIOGRAPHY 78

APPENDICES 81A.1 Algorithm for exponentially-fitted fuzzy Runge-Kutta method 82B.1 Algorithm for exponentially-fitted system of first order fuzzy Runge-

Kutta method 96

BIODATA OF STUDENT 108

LIST OF PUBLICATIONS 109

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LIST OF TABLES

Table Page

1.1 Butcher tableau fourth order explicit Runge-Kutta method 31.2 Butcher tableau fourth order explicit Runge-Kutta method 7

3.1 Butcher tableau an explicit Runge-Kutta method 233.2 Butcher tableau an explicit four stage four step Runge-Kutta method 243.3 Butcher tableau an explicit Runge-Kutta method 263.4 Comparison of the Euclidean norms of the end-point global errors. 32

4.1 Comparison of the Euclidean norms of the end-point global errors. 40

5.1 Numerical comparison for Example 5.4.1. 47

5.2 Numerical comparison for Example 5.4.2. 48

5.3 Numerical comparison for Example 5.4.3. 48

5.4 Numerical comparison for Example 5.4.4. 48

6.1 Butcher tableau Eq.(6.1) 506.2 Butcher tableau 516.3 Comparison of the Euclidean norms of the end-point global errors

for Example (6.5.1) in v = h = 0.1, t = 1. 586.4 Comparison of the Euclidean norms of the end-point global errors

for Example (6.5.2) in v = h = 0.1, t = 1. 596.5 Comparison of the Euclidean norms of the end-point global errors

for Example (6.5.3) in t = 1. 606.6 Comparison of the Euclidean norms of the end-point global errors

for Example (6.5.4) in t = 1. 616.7 Comparison of the Euclidean norms of the end-point global errors

for Example (6.5.5) in t = 0.1. 626.8 Comparison of the Euclidean norms of the end-point global errors

for Example (6.5.6) in t = 0.2. 636.9 Comparison of the Euclidean norms of the end-point global errors

for Example (6.5.7) in t = 1. 64

7.1 Numerical comparison for Example (7.4.1). 71

7.2 Numerical comparison for Example (7.4.2). 72

7.3 Numerical comparison for Example (7.4.1). 737.4 Numerical comparison for Example (7.4.2). 74

A.1 Butcher tableau 82B.1 Butcher tableau for Example (7.4.1) in step 1 left hand 96

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B.2 Butcher tableau for Example (7.4.1) in step 2 left hand 98B.3 Butcher tableau for Example (7.4.1) in step 1 right hand 99B.4 Butcher tableau for Example (7.4.1) in step 2 right hand 100B.5 Butcher tableau for Example (7.4.2) in step 1 left hand 102B.6 Butcher tableau for Example (7.4.2) in step 2 left hand 104B.7 Butcher tableau for Example (7.4.2) in step 1 right hand 105B.8 Butcher tableau for Example (7.4.2) in step 2 right hand 107

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LIST OF FIGURES

Figure Page

3.1 λ = a + ib in Eq.(3.8) 293.2 λ = a + ib in Eq.(3.8) 29

4.1 H-V plane of the new method with 0 < V < 1.5. 39

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LIST OF ABBREVIATIONS

IVP Initial Value ProblemLTE Local Truncation ErrorEFRKMB Exponentially-Fitted Runge-Kutta MethodRKMS Exponentially-Fitted using Simos’s techniqueODEs Ordinary Differential EquationsPDEs Partial Differential EquationsFDE Fuzzy Differential EquationHMC Hybrid Classical MethodHMB3 Hybrid Method three stage using Vanden Berghe TechniqueHMB2 Hybrid Method two stage using Vanden Berghee TechniqueRKM Fourth-order Runge-Kutta MethodRK44M Runge-Kutta Fourth-order Four-stage MethodLWSM Legndre Wavelets Spectral MethodFCC Fuzzy Constant CoefficientsVIM Variational Iteration Method

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CHAPTER 1

INTRODUCTION

1.1 Background

Many research results have been reported in the area of the numerical solution ofinitial value problems in the last century. Particularly, problems exhibit a pro-nounced oscillatory property have taken a special attention. Such problems oftenarise in the filed of celestial mechanics, astrophysics, electronics and moleculardynamic, etc. There are many different approaches to deal with such problems.However, it is of central importance for an approach to specifically address thestructure of the problem and its physical solutions.

Numerous numerical methods have been developed to approximate solutions ofdifferential systems while a challenging important task is to preserve reasonablebounds for errors. Recent developments in computer sciences and technology havegiven the theory of numerical analysis a given momentum.

There exist ways for finding analytical solutions for certain simple ordinary dif-ferential equations. However, most models of real life problems are modeled bynonlinear differential equations which there does not exist any approach for find-ing analytic solutions. Furthermore, any analytic solution must be implementedin a computer for most real applications. Therefore, the most and final practicalsolutions are the numeric solutions.

Generally, there are two types of numerical methods, single-step methods and mul-tistep methods. The single-step method assumes an initial point and approximatesa new solution in a one step process. However, the multistep methods compriseof consecutive phases such that in each phase, the previous solutions and theirassociated derivatives are used. Therefore, the multistep methods require moreinitial points in which they are usually determined via a one-step approach.

Recently fuzzy systems and their associated solutions have been introduced inorder to study problems with real conditions. The systems of first order differentialequations under fuzzy conditions, have been treated in this thesis under generalizedHukuhara differentiability.

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1.1.1 Initial value problem

We deal with the initial value problem of second order ordinary differential equa-tions

y′′ = f(x, y), y(x0) = y0, y′(x0) = y′0, (1.1)where y stands for the vector valued function with y = [y1, y2, . . . , yn] and y′ =[y′1, y

′2, . . . , y

′n].

Theorem 1.1.1 (Lambert 1991) Assume that the function f : R× R −→ R is con-tinuous for all (x, y) in the region D; where D is given by a ≤ x ≤ b, −∞ < y < ∞.Further, assume that f satisfies the Lipschitz condition corresponding with the Lip-schitz constant L, i.e.,

‖f(x, y)− f(x, ŷ)‖ ≤ L‖y − ŷ‖ (1.2)holds for every (x, y) ∈ D, (x, ŷ) ∈ D. Then, there exists a unique continuous and

differentiable solution y(x) for all (x, y) ∈ D associated with the initial problem (1.1).

For the proof of Theorem (1.1.1) see Henrici (1962) and Lambert (1991).

1.2 Runge-Kutta method

The general s-stage Runge-Kutta method is defined by

yn+1 = yn + hs∑

i=1

bif(xn + cih, Yi) (1.3)

where

Y1 = yn;

Yi = yn + hs∑

j=1

aijf(xn + cjh, Yj), i = 2, . . . , s

ci =i−1∑

j=1

aij

In this research, the explicit Runge-Kutta method of concern have the form

yn+1 = yn + h4∑

i=1

bif(xn + cih, Yi) (1.4)

where

Y1 = yn;Y2 = yn + ha21f(xn + c1h, Y1)Y3 = yn + h(a31f(xn + c1h, Y1) + a32f(xn + c2h, Y2))Y4 = yn + h(a41f(xn + c1h, Y1) + a42f(xn + c2h, Y2) + +a43f(xn + c3h, Y3))

2

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0

25

25

35 − 320 341 1944 − 1544 1011

1172

2572

2572

1172

(1.5)

The method chosen in this case is the fourth order explicit Runge-Kutta methodby Dormand (1996) which is given in Table 1.1.

1.2.1 Local truncation error and order conditions

Using Table 1.1, we have c1 = 0, c2 = 25 , c3 =35 , c4 = 1 ;

a21 = 25 ; a31 = − 320 ; a32 = 34 ; a41 = 1944 ; a42 = − 1544 ; a43 = 1011b4 = 1172 ; b1 = b4 , b3 = b2 ; b2 =

12 − b4;

Consider in Taylor series around xn, i.e.

y(xn+1) = y(xn) + hy′(xn) +h2

2y′′(xn) +

h3

6y′′′(xn) +

h4

24y(4) + . . .

where

y′(xn) = fy′′(xn) = fx + f.fyy′′′(xn) = fxx + 2f.fxy + f2.fyy + fy.(fx + f.fy)

y(4)(xn) = fxxx + 3f.fxxy + 3fx.fxy + 5f.fy.fxy + 3f2.fxyy + 3f.fx.fyy+4f2.fy.fyy + f3.fyyy + fy.fxx + fx.f2y + f

3y .f

Also we used Taylor expansion for a function of two variables around the point(xn, yn). At first we put

T (s, r) =s3

6fxxx +

s2r

2fxxy +

sr2

2fxyy +

r3

6fyyy

then

k1 = f

k2 = f + hc2fx + (Y2 − yn)fy + 12c22h

2fxx + c2h(Y2 − yn)fxy + 12(Y2 − yn)2fyy

+T (c2h, Y2 − yn)= f + c2hfx + ha21f.fy +

12c22h

2fxx + (ha21f)hc2fxy +12(ha21f)2fyy

+T (c2h, ha21f)

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and

k3 = f + hc3fx + (Y3 − yn)fy + 12c23h

2fxx + c3h(Y3 − yn)fxy

+12(Y3 − yn)2fyy + T (c3h, Y3 − yn)

whereY3 = yn + h(a31k1 + a32k2).

also

k4 = f + hc4fx + (Y4 − yn)fy+

12c24h

2fxx + c4h(Y4 − yn)fxy + 12(Y4 − yn)2fyy

+T (c4h, Y4 − yn)

whereY4 = t4yn + h(a41k1 + a42k2 + a43k3).

Finally we haveyn+1 = yn + h(b1k1 + b2k2 + b3k3 + b4k4).

Now

LTE = yn+1 − y(xn+1) (1.6)= (−f + b1f + b2f + b3f + b4f)h

+[−fx2− f.fy

2+ b2(c2fx + a21f.fy)

+b3(c3fx + (a31f + a32f)fy) + b4(c4fx + (a41f + a42f + a43f)fy)]h2

+[−fxx6− ffxy

3− f

2.fyy6

− fx.fy6

− f.f2y

6+ b2(

c22fxx2

+ c2a21f.fxy

+a221f

2.fyy2

) + b3(a32(c2fx + a21f.fy)fy +c23fxx

2+ c3(a31f + a32f)fxy

+12(a31f + a32f)2fyy) + b4((a42(c2fx + a21f.fy)

+a43(c3fx + (a31f + a32f)fy))fy +12c24fxx

+c4(a41f + a42f + a43f)fxy +12(a41f + a42f + a43f)2fyy)]h3 + . . . .

Let

s1 =4∑

i=1

bi − 1 ,

s2 =4∑

i=2

bici − 12 ,

s3 =4∑

i=2

bic2i −

13

,

s4 = b3a32c2 + b4a42c2 + b4a43c3 − 16 ,

s5 =4∑

i=2

bic3i −

14

,

4

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PMs6 = b3a32a21 + b4a42a21 + b4a43a31 + b4a43a32 − 16 ;

s7 = b3a32c3c2 + b4a42c2c4 + b4a43c3c4 − 18 ;

s8 = b3a32c22 + b4a42c22 + b4a43c

23 −

112

;

s9 = b4a43a32c2 − 124 .

Setting si, i = 1, . . . , 9 to zero, we obtain the order conditions:

∑4i=1 bi = 1 ,

∑4i=2 bici =

12

,

∑4i=2 bic

2i =

13

,

∑4i=2 bic

3i =

14

,

b3a32c2 + b4a42c2 + b4a43c3 =16,

b3a32a21 + b4a42a21 + b4a43a31 + b4a43a32 =16;

b3a32c3c2 + b4a42c2c4 + b4a43c3c4 =18;

b3a32c22 + b4a42c

22 + b4a43c

23 =

112

;

b4a43a32c2 =124

.

Here we use the values of Table 1.1 in (1.6). Then, using Maple code, the coeffi-cients of h3, h4 are zero. Now, the coefficient of h5 is obtained as follow:

C5 =130

.fyy.f.fxx +13

1800.fy.fxxx +

115

.f.f2xy +5

176.fyy.f

2x

+130

.fxy.fxx +1631320

fyy.fx.f.fy +91880

.fyy.f2.f2y

+1031800

.fy.f3.fyyy +

43600

.fy.f.fxxy +73600

.fy.f2.fxyy

+110

.fxy.f2.fyy +

130

.f3.f2yy +115

.fy.fx.fxy +112

.f.f2y .fxy

+120

.fxxy.fx +1

120.f2y .fxx +

110

.f.fxyy.fx +120

.f2.fyyy.fx.

1.2.2 Stability

Assume that H = λh andy′ = −λy, λ > 0. (1.7)

Hence,y = c1 exp(−λx),

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for a constant c1. We use the Runge-Kutta method (1.4) for equation (1.7) andobtain

yn+1 = yn −Hs∑

i=1

biYi,

Yi = yn −Hs∑

j=1

ai,jYj , i = 1, . . . , s.

Alternatively, we have

yn+1 = yn −HbtY, (1.8)Y = eyn −HAY, (1.9)

where e = (1, 1, . . . , 1)t and Y = (Y1, Y2, . . . , Ys)t. From (1.8), we have

Y = (I + HA)−1eyn. (1.10)

Next

yn+1 = yn −HbtY = yn −Hbt(I + HA)−1eyn= (1−Hbt(I + HA)−1e)yn = R(H)yn,

forR(H) = (1−Hbt(I + HA)−1e).

The associated sequence is bounded if and only if

|R(H)| ≤ 1. (1.11)

We define the corresponding stability region by

S = {H ∈ C| |R(H)| ≤ 1}. (1.12)

1.3 Hybrid method

Coleman (2003) studied the following class of hybrid method:

yn+1 = 2yn − yn−1 + h2s∑

i=1

bif(xn + cih, Yi), xn = x0 + nh (1.13)

Yi = (1 + ci)yn − ciyn−1 + h2s∑

j=1

ai,jf(xn + cjh, Yj), i = 1, . . . , s.

The Butcher tableau represents the following where bt = [b1, b2, . . . , bs],A = [aij ] and ct = [c1, . . . , cs].

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c Abt

In this research, our hybrid method follow

Y1 = yn, Y2 = yn−1

Yi = (1 + ci)yn − ciyn−1 + h2s∑

j=3

ai,jf(xn + cjh, Yj), i = 3, . . . , s,

yn+1 = 2yn − yn−1 + h2s∑

i=1

bif(xn + cih, Yi). (1.14)

Here, fn−1 and fn stands for f(tn−1, yn−1) and f(tn, yn), respectively. These ap-proaches are explicit and have s − 1 function evaluations or phases in each stagesof integration. Franco (2006) was the first who introduced this class of explicithybrid method. They are a subclass of methods defined in equation (1.13), bytaking c1 = −1, c2 = 0, a21 = 0 and aij = 0 for j > i.

1.3.1 Local truncation error and order conditions

Consider the autonomous scalar differential equation

y′′ = f(y). (1.15)

Taking a few consecutive differentiations gives rise to

y′′′ = y′f ′(y),

y(4) = y′′f ′(y) + y′2f ′′(y),

y(5) = y′′′f ′(y) + 3y′f(y)f ′′(y) + y′3f ′′′(y),y(6) = f ′(y)f ′(y)f(y) + 5f ′′(y)f ′(y)(y′)2 + 3y′f(y)f ′′(y) + 5f ′′(y)f(y)(y′)2 (1.16)

+f (4)(y)(y′)4.

The class of hybrid method (1.14) is applied to solve (1.15), and thereby,

yn+1 = 2yn − yn−1 + h2Φ.

Here,

Φ = b1f(yn−1) + b2f(yn) +s∑

i=3

bif(Yi).

Let the exact solution of (1.15)

y(xn+1) = 2y(xn)− y(xn−1) + h2∆.

Next, the local truncation error (LTE) follows

LTE = h2(Φ−∆). (1.17)

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Order conditions are defined as relationships between coefficients of a methodwhich cause successive terms in a Taylor expansion of the local truncation errorto vanish (Coleman 2003).We demonstrate on how to get the order conditions by means of an example.Suppose that we want to obtain the order conditions of the problem (1.15) forhybrid method defined in (1.14) with two-stage. We expand f(yn−1) and f(Y3) as aTaylor series about h = 0 giving

Φ = (b1 + b2 + b3 + b4)f(yn) + [b4f ′(yn)c4y′n + b3f′(yn)c3y′n

−b1f ′(yn)y′n]h + {b3[−12f ′(yn)c3y′′ + f ′(yn)a31f(yn)

+f ′(yn)a32f(yn) +12f ′′(yn)c23(y

′n)

2] + b4[−12f′(yn)c4y′′

+f ′(yn)a41f(yn) + f ′(yn)a42f(yn)

+f ′(yn)a43f(yn) +12f ′′(yn)c24(y

′n)

2]

+b1[12f ′(yn)y′′n +

12f ′′(yn)y′n

2]}h2 + . . . (1.18)

f(yn−1) = f(yn)− f ′(yn)y′nh + [12f ′(yn)y′′n +

12f ′′(yn)y′n

2]h2

−16(y(3)f ′(y) + 3y′f(y)f ′′(y) + y′3f ′′′(y))h3 + . . .

f(Y4) = f(yn) + f ′(yn)c4y′nh + [−12f ′(yn)c4y′′

+f ′(yn)a41f(yn) + f ′(yn)a42f(yn) + f ′(yn)a43f(yn)

+12f ′′(yn)c24(y

′n)

2]h2 + . . .

f(Y3) = f(yn) + f ′(yn)c3y′nh + [−12f ′(yn)c3y′′ + f ′(yn)a31f(yn)

+f ′(yn)a32f(yn) +12f ′′(yn)c23(y

′n)

2]h2 + . . .

Taylor series expansions of y(xn−1) and y(xn+1) about h = 0 are given by

y(xn+1) = y(xn) + hy′(xn) +h2

2y′′(xn) +

h3

6y′′′(xn) +

h4

24y(4)(xn) + . . .

y(xn−1) = y(xn)− hy′(xn) + h2

2y′′(xn)− h

3

6y′′′(xn) +

h4

24y(4)(xn) + . . .

and thus, the corresponding expansion for y(xn+1)− 2y(xn) + y(xn−1) is

y(xn+1)− 2y(xn) + y(xn−1) = h2y′′(xn) + h4

12y(4)(xn) +

h6

360y(6)(xn) . . .

and hence,

∆ = y′′(xn) +h2

12y(4)(xn) +

h4

360y(6)(xn) . . . (1.19)

Assume that y(xn) = yn . Then (1.18) and (1.19) are substituted into the right sideof equation (1.17). Next, (1.15) and derivative terms such as those given (1.16)

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are used to obtain

LTE = h2(b1 + b2 + b3 + b4 − 1)f(yn) + h3(b4c4 + b3c3 − b1)f ′(yn)y′n (1.20)+h4[(−b4c4

2− b3c3

2+ b4a41 + b4a42 + b4a43 + b3a31 + b3a32

+b12− 1

12)f ′(yn)f(yn) + (

b4c24

2+

b3c23

2+

b12− 1

12)f ′′(yn)(y′n)

2]

Let

t1 = b1 + b2 + b3 + b4 − 1,t2 = b4c4 + b3c3 − b1,t3 = −b4c42 −

b3c32

+ b4a41 + b4a42 + b4a43 + b3a31 + b3a32

+b12− 1

12,

t4 =b4c

24

2+

b3c23

2+

b12− 1

12.

Setting t1 and t2 to zero, we obtain the order conditions:

b1 + b2 + b3 + b4 = 1,b4c4 + b3c3 − b1 = 0

If the above conditions are fulfilled, then t3 reduces to

t3 = b4a41 + b4a42 + b4a43 + b3a31 + b3a32 − 112Setting t3 to zero, we get the order condition:

b4a41 + b4a42 + b4a43 + b3a31 + b3a32 − 112 = 0.

Finally, by assuming that

b1 + b2 + b3 + b4 = 1,b4c4 + b3c3 − b1 = 0,

b4a41 + b4a42 + b4a43 + b3a31 + b3a32 − 112 = 0.

The following order conditions of order 2 to 9 are listed as given by Coleman(2003).

Order 2:s∑

i=1

bi = 1 (1.21)

Order 3:s∑

i=1

bici = 0 (1.22)

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Order 4:s∑

i=1

bic2i =

16

(1.23)

s∑

j=1

s∑

i=1

biaij =112

(1.24)

Order 5:s∑

i=1

bic3i = 0 (1.25)

s∑

j=1

s∑

i=1

biciaij =112

(1.26)

s∑

j=1

s∑

i=1

biaijcj = 0 (1.27)

Order 6:s∑

i=1

bic4i =

115

(1.28)

s∑

j=1

s∑

i=1

bic2i aij =

130

(1.29)

s∑

j=1

s∑

i=1

biciaijcj =−160

(1.30)

s∑

i=1

s∑

j=1

s∑

k=1

biaijaik =7

120(1.31)

s∑

j=1

s∑

i=1

biaijc2j =

1180

(1.32)

s∑

i=1

s∑

j=1

s∑

k=1

biaijajk =1

360(1.33)

Order 7:s∑

i=1

bic5i = 0 (1.34)

s∑

j=1

s∑

i=1

bic3i aij =

130

(1.35)

s∑

j=1

s∑

i=1

bic2i aijcj = 0 (1.36)

s∑

i=1

s∑

j=1

s∑

k=1

biciaijaik =130

(1.37)

s∑

j=1

s∑

i=1

bic2i aijcj = 0 (1.38)

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i=1

s∑

j=1

s∑

k=1

biciaijajk =−1720

(1.39)

s∑

j=1

s∑

i=1

biciaijc2j =

172

(1.40)

s∑

i=1

s∑

j=1

s∑

k=1

biaijaikck =−1120

(1.41)

s∑

j=1

s∑

i=1

biaijc3j = 0 (1.42)

s∑

i=1

s∑

j=1

s∑

k=1

biaijcjajk =1

360(1.43)

s∑

i=1

s∑

j=1

s∑

k=1

biaijajkck = 0. (1.44)

1.3.2 Phase-lag and stability analysis

Let H = λh and consider the test problem

y′′ = −λ2y, λ > 0, (1.45)

where the exact solution is given by

y = c1 exp(λx) + c2 exp(−λx)

with c1 and c2 are constants. Applying the hybrid method (1.13) to equation(1.45) yields

yn+1 = 2yn − yn−1 −H2s∑

i=1

biYi,

Yi = (1 + ci)yn − ciyn−1 −H2s∑

j=1

ai,jYj , i = 1, . . . , s,

which can be expressed in vector form as

yn+1 = 2yn − yn−1 −H2btY (1.46)Y = (e + c)yn − cyn−1 −H2AY, (1.47)

where e = (1, 1, . . . , 1)t and Y = (Y1, Y2, . . . , Ys)t. From (1.46), we get

Y = (I + H2A)−1(e + c)yn − (I + H2A)−1cyn−1 (1.48)

Then, substituting (1.48) into (1.47) give us

yn+1 − S(H2)yn + P (H2)yn−1 = 0 (1.49)

where S(H2) = 2−H2bt(I + H2A)−1(e + c) and P (H2) = 1−H2bt(I + H2A)−1c.

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The phase-lag and stability properties of the hybrid method (1.13) are determinedby the characteristic equation

ξ2 − S(H2)ξ + P (H2) = 0 (1.50)

which is associated with (1.49). The study of phase-lag has been initiated byBrusa and Nigro (1980) in which the phase-lag was introduced as the truncationerror on exponentials.

According to Van der Houwen and Sommeijer (1987), in the phase analysis, onecompares the phase(s) (or argument(s)) of exp(±iH) with the principal root(s) ofthe characteristic polynomial. Thus, phase-lag is defined as the difference

t = H − θ(H)

where H is the phase of exp(±iH) and is the phase of the principal root of (1.50). Inorder to determine , assume that the characteristic polynomial (1.50) has complexroots. Thus, the discriminant should be negative which means that:

(S(H2))2 − 4P (H2) < 0.

It is noted that √(S(H2))2 − 4P (H2) = i

√4P (H2)− (S(H2))2

and therefore the principal root may be written as

r1 =√

S(H2)2 + i√

4P (H2)− (S(H2))2.

Thus,

tan(θ(H)) =

√4P (H2)− (S(H2))2

S(H2)

and

cos(θ(H)) =S(H2)

2√

P (H2).

Hence the phase of r1 is

θ(H) = arccos

(S(H2)

2√

P (H2)

).

The following is the definition according to the formula and concept given byFranco (2006).

Definition 1.3.1 For the hybrid method corresponding to the characteristic poly-nomial (1.50), the quantity

φ(H) = H − arccos(

S(H2)2√

P (H2)

)

is called phase-lag (or dispersion error) while the quantity

d(H) = 1−√

P (H2)

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is called dissipation (or amplification) error. A hybrid method is said to have thephase-lag of order n if φ(H) = O(Hn+1). If P (H2) = 1 then d(H) = 0 and the methodhaving this property is said to be zero dissipative or dissipative of order infinity.Conversely, if P (H2) 6= 1 then we should have d(H) = O(Hm+1). The method withthis property is said to be dissipative of order m.

Definition 1.3.2 For the hybrid method corresponding to the characteristic poly-nomial (1.50), the interval (0,Hp) is called the interval of periodicity if

P (H2) = 1 and |S(H2)| < 2, ∀ H ∈ (0,Hp).

Definition 1.3.3 For the hybrid method corresponding to the characteristic poly-nomial (1.50), the interval (0,Ha) is called the interval of absolute stability if

|P (H2)| < 1 and |S(H2)| < 1 + P (H2), ∀ H ∈ (0,Ha)

For the hybrid method defined in (1.14), S(H2) and P (H2) are polynomials in H2.

1.4 Legendre polynomial

In Kajani and Vencheh (2004), The Legendre polynomials are obtained by therecursive formulas:

L0(t) = 1,L1(t) = t, (1.51)

Lm+1(t) =2m + 1m + 1

tLm(t)− mm + 1

Lm−1(t), m = 1, 2, . . . .

In Chapter V, the Legendre polynomials are applied for solving differential equa-tions with initial values on large intervals.

1.5 Fuzzy differential equations

Here, we recall some basic definitions and results by Georgiou et al. (2005) whichwill be used later. The set of all real numbers is indicated by R. A fuzzy numberis a mapping u : R→ [0, 1] with the following properties:

1. u is upper semi-continuous,

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

3. u is normal, i.e., ∃x0 ∈ R for which u(x0) = 1,

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4. supp u = {x ∈ R|u(x) > 0} is the support of the u, and its closure cl(supp u) iscompact.

Let E be the set of all fuzzy number on R. The α-level set of a fuzzy numberu ∈ E, 0 ≤ α ≤ 1, denoted by [u]α, is defined as

[u]α ={ {x ∈ R|u(x) ≥ α} if 0 < α ≤ 1

cl(supp u) if α = 0.

It is clear that the α-level set of a fuzzy number is a closed and bounded interval[u(α), u(α)],where u(α) denotes the left-hand endpoint of [u]α and u(α) denotes theright-hand endpoint of [u]α. Since each y ∈ R can be regarded as a fuzzy number ỹdefined by

ỹ(t) ={

1 if t = y,0 if t 6= y,

R can be embedded in E.

Indeed, for Hausdorff distance we have the following metric properties: D : E×E −→R+

⋃0,

D(u, v) = supα∈[0,1]

max{|u(α)− v(α)|, |ū(α)− v̄(α)|},

It is easy to see that D is a metric in E and has the following properties(i) D(u⊕ w, v ⊕ w) = D(u, v), ∀u, v, w ∈ E,(ii) D(k ¯ u, k ¯ v) = |k|D(u, v), ∀k ∈ R, u, v ∈ E,(iii) D(u⊕ v, w ⊕ e) ≤ D(u,w) + D(v, e), ∀u, v, w ∈ E,(iv) (D,E) is a complete metric space.

Definition 1.5.1 Let x, y ∈ E. If there exists z ∈ E such that x = y ⊕ z, then z iscalled the H-difference of x and y, and it is denoted by xª y.

In this thesis we consider the following definition of differentiability for fuzzy-valued functions which was introduced by Bede and Gal (2005).

Definition 1.5.2 Let f : (a, b) → E and x0 ∈ (a, b). We say that f is strongly gener-alized differential at x0. If there exists an element f ′(x) ∈ E , such that(i) for all h > 0 sufficiently small, ∃f(x0 + h)ª f(x0), ∃f(x0)ª f(x0 − h) and the lim-its(in the metric d)

limh→0

f(x0 + h)ª f(x0)h

= limh→0

f(x0)ª f(x0 − h)h

= f ′(x0),

(ii) for all h > 0 sufficiently small, ∃f(x0) ª f(x0 + h), ∃f(x0 − h) ª f(x0) and thelimits(in the metric d)

limh→0

f(x0)ª f(x0 + h)−h = limh→0

f(x0 − h)ª f(x0)−h = f

′(x0),

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(iii) for all h > 0 sufficiently small, ∃f(x0 + h) ª f(x0), ∃f(x0 − h) ª f(x0) and thelimits(in the metric d)

limh→0

f(x0 + h)ª f(x0)h

= limh→0

f(x0 − h)ª f(x0)−h = f

′(x0),

(iv) for all h > 0 sufficiently small, ∃f(x0) ª f(x0 + h), ∃f(x0) ª f(x0 − h) and thelimits(in the metric d)

limh→0

f(x0)ª f(x0 + h)−h = limh→0

f(x0)ª f(x0 − h)h

= f ′(x0).

Remark 1.5.1 In this thesis, only cases (i)- and (ii)-differentiability in Definition1.5.2 will be used.

Theorem 1.5.1 (see Bede (2006)). Let f : (a, b) → E be a function and denote[F (t)]α = [fα(t), gα(t)], for each α ∈ [0, 1]. Then

(1) If f is (i)-differentiable, then fα(t) and gα(t) are differentiable functions and

[F ′(t)]α = [f ′α(t), g′α(t)],

(2) If f is (ii)-differentiable, then fα(t) and gα(t) are differentiable functions and

[F ′(t)]α = [g′α(t), f′α(t)].

1.6 Objectives of the thesis

In this thesis, we propose some new efficient methods for numerically solving linearand nonlinear first order differential equations and the system of first order differen-tial equations with deterministic and fuzzy conditions based on the exponentially-fitted Runge-Kutta method. In particular, the objectives of the thesis are:

1. Applying exponentially-fitted Runge-Kutta method to solve second orderdifferential equations that has been reduced to first order ODE and obtainingthe local truncation error and the stability regions.

2. To derive an explicit trigonometrically fitted hybrid method for solving os-cillatory second order ordinary differential equations.

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3. To propose a reliable algorithm based on Legendre wavelets-spectral methodfor solving first order and second order nonlinear oscillatory differential equa-tions.

4. To solve first order ordinary differential equations with uncertainty, repre-sented by fuzzy numbers and fuzzy-valued functions, involving character-ization theorem and some other new results under generalized Hukuharadifferentiability.

5. Solving system of first order fuzzy differential equations under generalizedHukuhara differentiability both exactly and numerically.

1.7 Outline of the thesis

In Chapter I, some preliminaries and basic concepts of Runge-Kutta method aregiven. Hybrid method and its Phase-lag and stability analysis is also discussed.Chapter I also covers some basic concepts on Legendre wavelets and fuzzy differ-ential equations.

Chapter II is devoted to some discussion on earlier research on Runge-Kuttamethod, exponentially-fitted Runge-Kutta method, hybrid-type methods, exponentially-fitted hybrid-type methods and Legendre wavelet spectral method and fuzzy dif-ferential equations, system of first order fuzzy differential equations.

Chapter III focuses on the derivation of the exponentially Runge-Kutta methodusing the techniques introduced by Simos (1998) and Berghe et al. (1999) for solv-ing oscillatory first order ordinary differential equations. Stability analysis and thelocal truncation error of the methods are also given.

In Chapter IV, we develop the trigonometrically fitted hybrid method based on thehybrid method of order five given in Franco (2006) for solving oscillatory secondorder ordinary differential equations. The stability region of the method whenapplied to linear second order ODEs is also depicted.

In Chapter VI, we solve first order fuzzy differential equations, firstly the fuzzy dif-ferential equation are transformed to ordinary differential equations using charac-terization theorem. The equations are first order then solved using exponentially-fitted Runge-Kutta method and numerical comparisons with other existing meth-ods are made.

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Chapter VII, we solve the fuzzy linear system of first order differential equationsanalytically under the generalized Hukuhara differentiability. Then the same fuzzydifferential equations are solved using exponentially-fitted Runge-Kutta methodand comparisons are made between the exact values and the computed values.

Finally, Chapter VIII summarizes the conclusion of the research and recommen-dation for future research will be suggested.

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Allahviranloo, T., Ahmady, N. and Ahmady, E. 2007. Numerical solution offuzzy differential equations by predictor-corrector method. Information Sciences177 (7): 1633–1647.

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