Implementation of Chebyshev Polynomials Approximation for

ii

Implementation of Chebyshev Polynomials Approximation for SolvingSecond

Order Ordinary Differential Equations

AmaniAlnumanYagobEbrahim

B.Sc. (Honor) in Geology, University of Khartoum, (2010)

Postgraduate Diploma in Mathematics, University of Gezira (2012)

A Dissertation

Submitted to the University of Gezira in Partial Fulfillment of the Requirements for

the Award of the Degree of Master of Science

in

Mathematics

Department of Mathematics

Faculty of Mathematical and Computer Sciences

December, 2016

iii

Implementation of Chebyshev Polynomials Approximation for Solving Second



Supervision Committee:

Name Position Signature

Dr. Sad Eldin M. Sad Eldain Main Superviser ……………………

Dr.MogtabaAhmed Yousif Co-supervisor ……………………

Date:

iv




Examination Committee:

Name Position Signature

Dr. Sad Eldin M. Sad Eldain ChairPerson ……………………

Dr. Mohsin Hassan Abdalla External Examiner ……………………

Dr. AwadAbdelrahmanAbdallaInternal Examiner ……………………

Date : of Examination: م2018مايو 5

v

Dedication

To my dear father

To my kind mother

To my dear brother and sisters

To my wifely

To my supervisor

To my friends

To who helped me…

vi

Acknowledgements

Thanks to Allah for granting strength, courage and good health throughout the

course.

I would like my gratitude and sincere appreciation to my supervisor proof

SaadEldin Mohamed SaadEldin for his kindness, keen special thanks for

department of mathematics at the faculty of my sincere appreciation also extend to

everyone help me in my study and research.

Finally, thanks are also to my colleagues friends and family for granting me con-

dences. Continuous support, respect, and love.

vii



AmaniAlnumanYagob Ibrahim

Abstract

Differential equations are considered to be the topics of pure and applied mathematics

and are the link between science and engineering. The topics of electrical engineering,

mechanical and structural engineering are not without ordinary differential equations. There are

no general methods to solve differential equations and there are some methods can be

generalized to a whole set on differential equations. Thechebyshev polynomials were discovered

almost a century ago by the Russian mathematicanChebyshev. Thesepolynomials play an

important role in the theory of approximation in the field of applied sciences and

engineering.These polynomials are the solution of the chebyshev differential equations.Their

importance for practical computation was rediscovered fifty years later by C.Ianczos. The aim of

this study is to present chebyshev method to find approximate solutions of the second order

linear differential equations ODEs. We applied chebyshev polynomials of the first and second

kinds to obtain analytical solutions to ordinary differential equations. The achievement of our

results is as follows: We discuss the chebyshev polynomials of the both kinds and theirproperties

and then usethis technique in obtaining approximate solutions of second order ODEs.This

studyrecommend toapply chebyshev polynomials to solve physical models described by partial

differential equations.

viii

من الدرجة الثانيةحدود شيبيشف لحل المعادلات التفاضلية العادية اتكثير تقريبتطبيق

مانى النعمان يعقوب ابراهيم حجرأ

ملخص الدراسة

وهى الرابط بين تعتبر المعادلات التفاضلية من المواضيع المهه فى الرياضيات البحته والتطبيقية

واع المعادلات التفاضلية العلوم والهندسة . فلا تخلو مواضيع الهندسة الكهربائية والميكانيكيه والانشائية من أن

عه الطرق يمكن تعميمها على مجمو. لاتوجد طرق عامه لحل المعادلات التفاضلية وهنالك بعض العادية

. كثيرة حدود ضلية العاديةلحل المعادلات التفا كثيرة حدود شيبيشف. خاصه على المعادلات التفاضلية

هذه الداله تلعب دور . فاكتشفت فى القرون الماضية بواسطة علم الرياضيات الروسى شيبيش شيبيشف

وهذه الدالة حلت كثيرة حدود والهندسة. ها فى مجال العلومأساسى فى نظرية التقريب وتطبيق

تم إعادة الاكتشاف بعد خمسون سنة بواسطة العالم فالتفاضلية. ولاهميتها في الحساب العملي شيبيش

C.lanzos والهدف من هذه الدراسة هو عرض طريقة شيبيشف لايجاد الحل التقريبي للمعادلات التفاضلية

من النوع الاول والثانى للحصول على الحل ف كثيرة حدود شيبيش تم تطبيق. العادية الخطية من الدرجة

كثيرة حدود ناقشنا : كما يلى والانجاز الذي توصلنا اليه من نتائجنا منظم.يلى للمعادلات التفاضلية العاديةالتحل

التقريبة لحل فاستخدمنا كثيرة حدود شيبيشو. خصائها المهه من النوع الاول والثانى وبعض فشيبيش

بعض التوصيات على الخاتمه و البحث هذا . وفى نهاية الدرجة الثانيةمعادلات التفاضلية العادية الخطية من ال

.باستخدام المعادلات التفاضلية الجزيئة وتطبيقها لحل النماذج الفيزيائيةفكثيرة حدود شيبيش

ix

Table of contents

Signatures of the supervision committee members ii

Signatures of the examination committee members iii

Dedication iv

Acknowledgements v

Abstract vi

vii الملخص

Table of contents xi

Chapter One : Introduction

1.1 Introduction 1

1.2 Problem statement 2

1.3 The Objectives 2

1.4 Methodology 2

1.5 The Study Layout 2

Chapter Two: Ordinary Differential Equations

2.1 Introduction 3

2.2 First order differential equations 4

2.2.1 Linear equations

2.2.2 Separable equation 5

2.3 homogenous Second order ODEs 6

2.3.1 Wronskians 6

2.3.2 Characteristic equation 8

2.4 Non homogenousSecond order ODEs 10

Chapter Three: ChebyshevPolynomials Approximation

3.1 Introduction 12

3.2 First kind polynomial Tn (x) 12

3.3 Special values of Tn(x) 14

3.4 Polynomial of second kind 14

3.5 Relation ship between Tn(x) and un(α) 15

3.6 Additional identites of chebyshev polynomials 15

3.7 Derivatives of chebyshev polynomials 14

x

3.8 Integral of chebyshev polynomials 16

3.9 Products of chebyshev polynomials 17

3.10 Basic properties and formula 17

3.11 Orthogonal series of chebyshev polynomials 19

3.12 Chebyshev series of one and two variable

19

3.13 Recurrence Relations 19

Chapter Four: Method of Chebyshev PolynomialsApproximation

4.1 Introduction 25

4.2 Approximation using chebyshev polynomials 27

4.3 An Approximation particular solutions 28

4.3.1 Method of Reduction of order 28

4.3.2 Expansion of Tn (x)and un(x) 29

4.3.3 Examples of Approximation 30

Chapter Five:Conclusion and Recommendation

5.1 Conclusion 41

5.2 Recommendation 41

Reference 42

xi

Chapter One

Introduction

1.1 Introduction

Differential equations are considered to be the topics of pure and applied mathematics

and are the link between science and engineering. The topics of electrical engineering,

mechanical and structural engineering are not without ordinary differential equations. There are

no general methods to solve differential equations and there are some methods can be

generalized to a whole set on differential equations (Philip, 1982).

The chebyshev polynomials were discovered almost a century ago by the Russian

mathematican chebyshev. These polynomials play an important role in the theory of

approximation in the fild of pplied sciences and engineering. This polynomials are the solution

of the chebyshev differential equations. Their importance for practical computation was

rediscovered fifty years later by C.Ianczos. we present chebyshev method to find approximate

solution of the second order linear differential equations Chebeshev polynomials of either kind

are a sequence of orthogonal polynomials that can also be defined recursively . the motivation

for Chebeshev interpolation is to improve control of the interpolation error on the interpolation

interval [-1 1] and the authors Chebeshev polynomials and the trigonometric basis functions to

approximate their equations for each time step. In their two-stage approximation scheme, the use

of Chebeshev polynomials in stage one is because of the high accuracy (spectral convergence of

Chebeshev interpolation.We then look for a particular solution that is expressed as a linear

combination of Chebeshev polynomials. our choice of Chebeshev polynomials is because of

their high accuracy . the Chebeshev polynomials is very close to the minimax polynomials which

(among all polynomials of the same degree) has the smallest maximum deviation from the true

function 𝑓(𝑥). The minimax criterion is that 𝑝𝑛(𝑥) is the polynomials of degree 𝑛 for which the

maximum value of the error, which is defined by 𝑒𝑛(𝑥) = 𝑓(𝑥) − 𝑝𝑛(𝑥), is a minimum with in

the specified rang f −1 ≤ 𝑥 ≤ 1 [1]. This is extremely ideal for polynomial approximations

(Fox, 1968).

xii

1.2 Problem Statement

Differential equations are hard and sometime impossible to be solved analytically, in particular

differential equations of higher order to overcme this difficulties, we propose the chebyshev

polynomials approximation techniques to find approximate solution of second order differential

equation .

1.3 The Objective

The main objectives of this study is to discus Chebyshev polynomials and use then in

solving second order ordinary differential equations.

1.4 The methodology

In this thesis,we use analytic and approximation tools, in finding approximation solution of

second order ordinary differential equation . these tools are :

Reduction of order and chebyshev polynomials approximation .

1.5the study layout

- Chapter two gives introduction to ordinary differential equation and solution of second order

ordinary differential equations.

- Chapter three discuss the chebyshev polynomials of the both kind and their properties.

- Chapter four introduces the approximation method using chebyshev polynomials and we

describe our method for finding a particular solution or an approximate particular solution by the

approximation and reduction of order. The last chapter presents the conclusion and

recommendation.

xiii

Chapter Two

Ordinary Differential Equations

2.1 introduction

In this chapter, we present introduction materials on ordinary differential equations (ODs) of first

and second order. A differential equation is an equation which contains derivatives of the

unknown function Edwards and Denney (2008).

0)...,,( nyyyxf (2.1)

This is general form to ODE of order n. Some concepts related to differential equations

(1) order the order of the differential equations is the order of the highest derivative.

The following equations for y(x) are

y' = xy2

y'' + 3xy' + 2y = x2

y''' = y''y

(2) degree the degree of differential equations is the highest power of the highest derivative

in the equation, the following equations are first, second and third degree respectively

y' – 3y2 = sin x

(y'')2 + 2x cos y = 5

(y)3 + y4 = 0

(3) An equation is said to be linear or non linear equation

let y(x) be the unknown then

a0(x)y'' + a1(x)yn-1 + an(x)y =g(x) (2.2)

is a linear equations. If the equation can be written as * the its non-linear.

to things you must know: identify the linearity of an equation.

xiv

(4) A differential equation is homogeneous if it has no terms that are function of the independent

variable alone. Thus an homogeneous equations is one in which there are terms that are

function of the in dependent variables alone.

1) y'' + xy + y = 0 is homogenous equation

2) y' + y + x2 = 0 is an in homogenous equation

2.2 First order differential equations

We consider the equation

),( yxfdx

dy (2.3)

We shall consider two spatial types of first order ODEs, linear and separable equations.

2.2.1 Linear equations

method of integrating factors Edwars and Denney (2008)

When function f(x, y) in (1.3) is a linear in y we can write

y' = p(x)y + g(x) (2.4)

we give the method of integrating factors: we multiply equation (1.4) by a function µ (α) on both

sides

µ (α) y' + µ(x) P(x)y = µ(x) g(x)

the function µ is chosen such that the equation is integrable, meaning the LHS (left hand side) is

the derivative of something. In particular we require

µ(x)y' +µ (x) p(x) y = (µ (x) y)'

µ(x)y' + µ (x) p(x)y = µ(x)y' + µ'(x)y

which requires

dxxpd

xpxdx

dx

)(

)()()(

Integrating both sides

µ (x) = p(x) dx

which gives a formula to compute µ

µ (x) = exp ( p(x) dx)

therefore, this µ is called the integrating factor. Putting back into equation (1.1) weget

xv

cdxxgxyx

xgxyxdx

d

)()()(

)()())(

which give the formula for the solution

cdxxgxx

xy )()()(

1)(

(2.5)

where

µ (x) = exp ( p(x) dx) (2.6)

As illustration, we consider the following problem

y' + y = e3x

it is clear that p(x) = 1 , g(x) = e3x

µ (x) = exp ( 1 dx) = ex

and

y(x) = e-xex e3x dx = e-xe4x dx

xxxx ceecee 34

4

1)

4

1(

2.2.2 separable equations

We study first order equations that can be written as:

)(

)(),(

yQ

xp

dx

dyyxf (2.7)

p(x) and Q(y) are suitable function of x and y only Hartman, (1982).

Then we have

p(x) dx = Q(y) dy

p(x) dx = Q(y) dy

and we get implicitly defined solution of y(x)

consider y' = xy2

we separate the dependent and independent variables and integrate to find the solution

xvi

cx

y

cx

y

cdxxdyy

dxxdyy

xydx

dy

2

21

2

2

2

2

1

2

2.3 Homogeneous Second Order ODEs

We consider some theoretical aspects of the solutions to a general 2nd order linear equations.

Theorem: (Existence and Uniqueness Theorem)

Consider the initial value problem Hartman (1982).

y'' + p(x)y' + q(x)y = g(x)

y(x0) = y0 , y'(xo) = y'0

if p(x) , q(x) and g(x) are continuous and bounded on an open interval I containing x0, then there

exists exactly one solution y(x) of this equation, valid on I.

2.3.1 Wronskians

Definition Given two functions f(x) , g(x), Edwards and Denny (2008).

the wronskian is defined as

w (f, g)(x) = fg' – f'g

Remark: one way to remember this definition could be using the determinant

gf

gfxgfw

))(,( (2.8)

main property of the wronskian:

if w(f, g) = o, then f and g are linearly dependent

if w(f, g) ≠ o they are linearly independent

example:

check if thr given pair of function are linearly or not

01sincos

cossin

sincos)sin,(cos

22

xx

xx

xxxxw

xvii

so they are linearly independent

04444)44(1)1(4

41

441),(

44)(,1)(

xxxx

xxgfw

xxgxxf

So they are linearly dependent.

Theorem Edward and Denney (2008).

Let Y1 and Y2 be two linearly independent solution of homogeneous equation

X'' + p(x) Y' + q(x)Y = 0 (2.9)

with p and q continuous on the open interval I. if Y is any solution what so ever of eq(q) on

I.Then there exist numbers c1 and c2 such that

Y(x) = c1Y1(x) + c2 X2(x) (2.10)

for all x in I.

proof

choose appoint a of I and consider the simultaneous equation

)()()(

)()()(

2222

2211

aYaYcaYc

axaXcaYc

(2.11)

The determinant of the coefficients in this system of linear equations the unknowns c1 and c2 is

simply the wronskian is linearly independent so bx elementary algebra it follows that the

equation in (2.11) can be solved of c1 and c2 we define the solution G(x) = c1y1(x) + c2x2(x) of

equ (2.9).

)()()()(

)()()()(

2211

2211

aXaYcaYcaG

aYaYcaYcaG

thus the two solutions Y and G have the same initial value at a: like wise, so do Y' and G'. by the

uniqueness of solutiondetermined by such initial values, it flows that Y and G agree on I. thus we

sec that

Y(x) = G(x) = c1Y1(x) + c2x2(x) (2.12)

xviii

2.3.2 Characteristic Equation

Constant coefficients

We shall now consider second-order homogeneous linear (ODEs) whoess coefficients a and b

are constant

Y'' + aY' + bY = 0 (2.13)

these equations have important applications, especially in connection with mechanical and

electrical vibrations Shaker (2007).

The solution of the first order linear (DPE) with a constant coefficient r

Y' + rY = 0 (2.13.1)

is an axpontial function Y = Ce-rxthis gives us the idea to try as solution of (1.13.1) the function

Y = erx (2.13.2)

substituting (2.13.2) and its derivatives

y' = rerx , y'' = r2erx

into our equation (2.13), we obtain

(r2 + ar + b) erx = 0

Hence if r is a solution of the important characteristic equation

r2 + ar + b = 0 (2.13.3)

then the exponential function (2.13.2) is a solution of the (ODE) now from elementary algebra r

we recall that roots of this quadratic equation (2.13.3)

baar

baar

4

4

2

21

2

2

21

1

(2.13.4)

(2.13.3) and (2.13.4) will be basic because our derivation shows that the functions xr

ey 1

1 andxr

ey 12

2

are solution of (2.13)

from algebra we further know that the quadratic equation (2.13.3) may have three kinds of roots,

depending on the sign of the discriminant a2 – 4bnamely,

case (1) Two real roots if a2 – 4b > 0

case (2) Areal double roots if a2 – 4b = 0

case (3) Complex conjugate roots if a2 – 4b < 0

xix

case (1) Two Distinct Real Roots

r1 and r2

in this case, abasis of solution of (1) on any interval is xr

ey 1

1 andxr

ey 12

2

because y1 and y2 are defined (and real) for all x and their equation is not constant. The

corresponding general solution is

xrxrececy 21

21 (2.14)

Example

We can now solve y'' – y = 0 the characteristic equation is r2 – 1 = 0. its roots are r1 = 1 and r2

= -1.

Heanc a basic solutions is ex and e-x and gives the same general solution as before

y = c1ex + c2e

-x

case (2) Real Double Root 2

ar

is the discriminant a2 – 4b is zero, we sec directly from (4)

that we get only one root, 2

21

arrr

, hence only are solution,

xa

ey 2

1

the general solution

221 )(

ax

exccy

(2.15)

Example

solve y'' + 6y' + 9x = 0

r2 + 6r + 9 = 0

(r + 3)2 = 0 it has the double root r = -3

Hence a basis e-5x and xe-3x the corresponding solution is

y = (c1 + c2x) e-3x

case (3) complex Roots iqa 2

1 and iqa

2

1:-

this case occure if the discriminant a2 – 4b of the characteristic equation (3) is negative. In this

case, the roots of (3) and thus the solution of (ODE) (1) come at first out complex.

However, we show that from then we can obtain a basis of real solution

xx

)0(sin,cos 22

21

qqxeyqxey

axax

Hence a real general solution in case (3) is

).),()sincos(2 arbitaryBAgxBqxAey

ax

(2.16)

Example

Find general solution for equation

y'' + 2y' + 6y = 0

r2 + 2r + 6 = 0

it has the roots ir 512,1

and a general solution

)5sin5cos( 21 xcxcey x

2.4 Non-homogeneous Second Order ODEs

We now consider the non-homogeneous equation of 2-order

y'' + p(x) y' + q(x)y = g(x) (1)

first solve the homogeneous equation

y'' + p(x) y' + q(x)y (2)

suppose that a single fixed particular solution Yp of the non homogeneous equation (1) and to

find the general solution Yc of (2)

y = Yc + Yp= c1Y1 + c2Y2 + Yp

where Y1 , Y2 are linearly independent solution of the homogeneous equation we call Yc a

complementary function of the non-homogeneous equations Shaker (2007).

Example

Find the general solution of

y'' – 3y' – 4 = 3e2x

findyc =

r2 -3r -4 = (r + 1)(r – 4) = 0 r1= -1 , r2 = 4

yc = c1e-x + c2 e

4x

findyp we guess solution of the same form as the source term

yp = Ae2x , y'p = 2Ae2x , y''p = 4e2x

plug these into the equation

xxi

4Ae2x – 3x 2Ae2x - 4Ae2x = 3e2x

(4A – 6A – 4A)e2x -6A = 3 2

1A

so x

p ey 2

2

1

the general solution of to the non homogenousequation is

xxx

pc eececyyxy 24

212

1)(

xxii

Chapter Three

Method of Chebyshev Polynomials Approximation

3.1 Introduction

The chebyshev polynomials were discovered almost a century ago by the Russian

mathematician chebyshev. Their importance for practical computation was rediscovered fifty

years ago by c.lanczos Mason and Handscomb (2003).

We present here chebyshev method to find the approximate solution of the second – order linear

differential equation chebyshev polynomid from a series of orthogonal polynomials , which play

an important role in the theory of the approximation .

Y'' + Q(x)Y' + p(x)Y = g(x) (3.1)

Chebyshev polynomial are defined to be the solution of the following Chebyshev DE.

(1 – x2)Y''(x) – x'Y(x) + n2Y(x) = 0 (3.2)

There are two kind of chebyshev polynomials denoted by Tn(x) and Un(x).Tn(x) is called the

chebyshev polynomial of first kind and is defined by

Tn(x) = cos(n cos-1x), (3.3)

Un(x) is called the chebyshev polynomial of second kind and is defined by

Un(x) = sin(n cos-1x) , (3.4)

where n is a non-negative integer.

Chebyshev polynomials are also known by the name Tchebicheff polynomials.

In this section we give an introduction to the Chebyshev polynomials and their basic

properties.

3.2 The first kind polynomial Tn(x)

The Chebyshev polynomials Tn(x) of the first kind is a polynomial in X of degree n, defined

by the relation:

Tn(x) = cos(n) when x = cos

If the rang of the variable x is the interval [-1, 1], then the range of the corresponding variable

can taken [0, ] this ranges are traversed in opposite directions, since x = -1 corresponds to =

and x = 1 corresponds to = 0 it is well known (as consequence of de moivre’s theorem) that cos

xxiii

(n) is polynomial of degrre n in cos, and indeed we are familiar with the elementary

formulae:-

cos (0 ) = 1 cos(1.) = cos (),

cos (2.) = 2 cos (2.) – 1

cos (3.) = 4 cos (3.) – 3 cos

cos (4.) = 8 cos (4.) – 8 cos(2.)

we introduce the notation = cos-1x and get

Tn((x))= Tn() = cos(n) where [0,]

we can find arecurrence reaction, using these observation

Tn+1() = cos ((n + 1) )

= cos(n) cos () – sin (n ) sin ()

Tn-1() = cos (n -1)) = cos (n) cos() + sin(n)

Tn+1() + Tn-1() = 2 cos (n) cos () (3.6)

Chebyshev polynomial of the first kind are denote by Tn(x) and the 6th several polynomials are

listed be

T0 (x) = 1;

𝑇1(x) = x ;

𝑇2(x) = 2𝑋2–1;

𝑇3(x) = 4𝑋3-3x;

𝑇4(x) = 8𝑋4-8𝑋2+ 1;

𝑇5(x) = 16𝑋5-20𝑋3+ 5x;

𝑇6(x) = 32𝑋6-48𝑋4+ 18𝑋2-1;

Thefirst six Chebyshev polynomials Tn; n= 0,1,2….6

(3.7)

xxiv

Figure (3.1) Chebyshev polynomials Tn, n = 1,2,…6

chebyshev polynomial can be found using the previous two polynomial s by the recursive

formula:

Tn+1 = 2x Tn(x) – Tn-1(x) for n 1 (3.8)

3.3 special values of Tn(x): Arfken (1985).

Other special values of Tn(x) which are easily derived from the above relations are

)1()1(0)0(

1)1()1()0(

)1()()1()(

12

2

nn

n

n

n

n

n

n

n

TT

TT

xTxT

(3.9)

3.4 polynomials of the second kind:Mason and Handscomb (2003)

We will define the chebyshev polynomials of the second kind as solution to the following

recurrence equation.

Un+1(x) = 2x Un(x) + Un-1 (x) = 0 (3.10)

xxv

u0(x) = 1

u1(x) = 2x

The formula for the product of two polynomials is:

Un(x) Um(x) = Un-m(x) – Un+m+2(x)/2(1 – x2)

The Un(x) can also be defined by the following generating function

0

2)(

21

1

n

n

n txUtx

(3.11)

3.5 Relationships betweenTn(x) and Un(x) Rivlin (1990).

It is easy to derive hundreds of relationships between the Tn(x) and Un(x) by using their

trigonometric forms or the formulas in terms of the

)(

)(1))((

)()()(

)()()()1(

)()(

1

1

1

11

2

1

xU

xUxTU

xUxUxT

xTxTxUx

xnUxTdx

d

m

nm

nn

nnn

nnn

nn

(3.12)

other special values of Un(x)

0)0(

)1()0(

)1()1()1(

1)1(

)()1()(

12

2

n

n

n

n

n

n

n

n

n

U

U

nU

nU

xUxU

(3.13)

3.6 Additional identities of Chebyshev polynomials:

The chebyshev polynomials are both orthogonals and the trigometic cos(cos 𝑛𝜃) fuctions in

disguise, therefor they satisfy alarge number of useful relationships.

We begin with:

𝑇𝑛+1 (𝑥)=cos([𝑛+1) cos−1(𝑥)]

and

𝑇𝑛−1(𝑥) = cos[(𝑛 − 1) cos−1(𝑥)]

The differentiation and integration properties are very important in analytical work

xxvi

3.7 Derivatives of chebyshev Polynomials

The following expression for the derivatives chebxshev

polynomials,Press,Flannery,Teutolskt and Vetterling (1990).

oddnnTTTTnwhenTTTnT nnnnn 0231131 ]...[2...[2

where the notation (-) indicates a derivative with resped to x, can be proved by mathematical

induction. Indeed, they are verified for the lowest polynomials.

etcTTxTTTxT

xTTmTT

)....(42,32

,2

134023

1201

then remains to prove that if it is correct for n it is still correct for n + 1.

taking a dervative of the basic recurrence for chebxshev poly.

equa (2.8 ) leads to

11 2 nnn TxTT (3.15)

Therefor:

𝑇′2 (𝑥) = 4𝑇1 (𝑥)

𝑇′1 (𝑥) = 𝑇0(𝑋)

𝑇′0 (x) = 0

3.8 integral of chebxshev polynomials Abramowitz and Stegun (1984).

The following recurrence relationship is easy prove

1

)(

1

)()(2 11

n

xT

n

xTxT nn

n (3.16)

with the help of equ ( ) it then follows that

1

1 01

2

1

2

)(2

oddn

evennnndxxTn (3.17)

these two equations are easily combined to yield

∫𝑇1 (𝑋)𝑑𝑥=

1

4𝑇2 (𝑥)+𝑐

∫𝑇0 (𝑋)𝑑𝑥= 𝑇1 (𝑋)+𝐶

xxvii

3.9 products of chbyshev polynomials

The products of two chbyshev polynomials satisfies the following relationship

2Tn(x) Tn(x) = Tn+m(x) + Tn-m(x), n m (3.18)

3.10 basic Properties and formula:

We observe that the chbyshev polynomials from an orthogonal set on the interval -1 x 1

with weighting function (1 – x2)-1/2 Abramowitz and Stegun (1964).

Property (1) Recurrence Relationchbyshev polynomials

generated in the following way set T0 = 1 , T1 = x and the use the recurrence relation

Tn(x) = 2xTn – Tn – 1 for n = 1, 2, 3…. (3.19)

is often used as the definition for higher-order chbyshev polynomials.

Let us show that T3(x) = 2x T2(x) – T1(x)

Using the expression for T1(x) and T2(x) in (2.7) we obtain

2x T2(x) – T1(x) = 2x(2x2 -1) – x = 4x3 = T3(x) (3.20)

Property (2) leading Coefficient

is provide by observing that recurrence relation doules the leading coefficient of Tn-1(x) the

coefficient in

Tn(x) is 2-1 when N 1

Property (3) Symmetry

is established by showing that T2n(x) involves only even power of and T2m+1(x) involves only

odd power of x.

then n = 2m , T2m is an even function that is

T2m(-x) = T2m(x)

when n = 2m + 1, T2m+1 , T2m+1 is an odd function that is,

T2m+1(-x) = -T2m+1 (3.21)

Property (4) Trigonometric Representation on [-1, 1] using the trigonometric

identity

Formula (3) is used in (a) to establish the general case

Tn(x) = 2xTn-1(x) – Tn-2(x)

xxviii

= 2x cos (N -1) cos-1(x) – cos (N – 2) cos-1(x)

= cos (N cos-1(x)) for -1 x 1 (3.22)

Property (5) zero and extremai in [-1, 1]

The chebyshev polynomial of degree n 1 has h simple zero in [-1,1] at

nKTn

kkx ,...2,1,

2

12cos

Chebyshev , Tn assumes its absolution extemat

1,...2,1)1()(2

2cos

nkk

kxnTwithkx

| Tn(x) | 1 for -1 x 1

orthogonality property of Tn(x) we can determine theorthogonality properties for the

chebyshev polynomials of the first kind from our knowledge of the orthogonality of the cosine

functions, namely,

w

onm

onm

nm

dnm0

)(

)(2

)(0

)cos()cos(

(3.23)

to obtain the orthogonality properties of the chebyshev polynomials:

1

12

)(

)(2

)(0

1

)()(

onm

onm

nm

x

dxxTxT nm

(3.24)

we observe that the chebyshev polynomials from an orthogonal set on the interval -1 x 1 with

the weighting function 2

121

x

Orthogonal Series of Chebyshev polynomials

an arbitrary function f(x) which is continuous and single valued defined over the interval -1 x

1, can be expanded as aseries of chebyshev polynomials:

0

221100

)(

...)()()()(

n

nn xTA

xTAxTAxTAxf

(3.25)

xxix

where the coefficients An are given by

1

12

0 01

)(1n

x

dxxfA

(3.26)

w

onm

onm

nm

dnm0

)(

)(2

)(0

)cos()cos(

(3.27)

to obtain the orthogonality properties of the chebyshev polynomials:

1

12

)(

)(2

)(0

1

)()(

onm

onm

nm

x

dxxTxT nm

(2.28)we observe that the chebyshev polynomials from on

orthogonal set on the interval -1 x 1 with the weighting function 2

121

x

3.11Orthogonal Series of Chebyshev polynomials

an arbitrary function f(x) which is continuous and single valued defined over the interval -1 x

1, can be expanded as a series of chebyshev polynomials:

0

221100

)(

...)()()()(

n

nn xTA

xTAxTAxTAxf

(3.30)

where the coefficients An are given by

1

12

0 01

)(1n

x

dxxfA

(3.31)

1

12

,...2,11

)()(2n

x

dxxTxfA n

n

(3.32)

3.12 Chebyshev Series of one and two variables

the function f(x) may be approximated by chebyshevseries form (Bjorck, Dahalquist 1987) .

m

n

nnm TCcxf1

02

1)( (3.33)

in the discrete case coefficient of the series are calculated from the formula

xxx

m

k

knkn xTxfm

c0

)()(1

2 (3.34)

the function f(x,y) function may be means of chebyshevseries in form (leng 1997)

i

n

j

m

jmnmn yTxTCyxf0 0

)()(),( (3.35)

Figure 3: 2 Plot of f(x) with T1

xxxi

Figure 3:3 the plot of f(x) with T4

Fi

xxxii

Chapter Four

4.1 Introduction

In this thesis , we use Chebeshev polynomials for approximating equations and their

particular solutions. Chebeshev polynomials of the first kind are solutions to the Chebeshev

differential equations Fox and Parker (1964). .

.

(1 − 𝑥2)𝑑2𝑦

𝑑𝑥2− 𝑥

𝑑𝑦

𝑑𝑥+ 𝑛2𝑦 = 0 for |𝑥| < 1 (4.1)

And Chebeshev polynomials of either kind are a sequence of orthogonal polynomials that can

also be defined recursively . the motivation for Chebeshev interpolation is to improve control of

the interpolation error on the interpolation interval [-1 1] and the authors Chebeshev polynomials

and the trigonometric basis functions to approximate their equations for each time step. In their

two-stage approximation scheme, the use of Chebeshev polynomials in stage one is because of

the high accuracy (spectral convergence of Chebeshev interpolation.

If the right and side function 𝑓(𝑥) in Eq. (4.1) is a polynomial of degree 𝑛, i.e.,

𝑓(𝑥) = 𝑝𝑛(𝑥) = 𝑑𝑛𝑥𝑛 + 𝑑𝑛−1𝑥

𝑛−1 + 𝑑1𝑥 + 𝑑0 (4.2)

Where 𝑑𝑛 ≠ 0, we can find a particular solution of Eq. (4.1) that is polynomial.

If 𝑓(𝑥) is not a polynomial, we approximate it using Chebeshev polynomials .

We then look for a particular solution that is expressed as a linear combination of Chebeshev

polynomials . our choice of Chebeshev polynomials is because of their high accuracy . the

Chebeshev polynomials is very close to the minimax polynomials which (among all polynomials

of the same degree) has the smallest maximum deviation from the true function 𝑓(𝑥). The

minimax criterion is that 𝑝𝑛(𝑥) is the polynomials of degree 𝑛 for which the maximum value of

the error, which is defined by 𝑒𝑛(𝑥) = 𝑓(𝑥) − 𝑝𝑛(𝑥), is a minimum with in the specified rang f

−1 ≤ 𝑥 ≤ 1 [1]. This is extremely ideal for polynomial approximations.

xxxiii

In chapter 4 we describe our method for finding a particular solution or an approximate

particular solution by the approximation and reduction of order. these two methods are compared

through examples .

4.2 Approximation using Chebeshev polynomials

The Chebeshev polynomials of the first and second kind are denoted by 𝑇𝑛(𝑥) and 𝑈𝑛(𝑥)

respectively . the subscript 𝑛 is the degree of these polynomials . theChebeshev polynomials of

the first and second kind are closely related, for example , a Chebeshev polynomials of first kind

can be represented as a linear combination of two Chebeshev polynomials of second kind Fox

and Parker (1964),

𝑇𝑛(𝑥)𝑑𝑥 =1

2(𝑈𝑛(𝑥) − 𝑈𝑛−2(𝑥)) (4.3)

And the derivative of a Chebeshev polynomials of first kind can be written in terms of

Chebeshev polynomial of second kind,

𝑇′𝑛(𝑥) = 𝑛𝑈𝑛−1(𝑥), 𝑛 = 1,2, …. (4.4)

In this paper . we direct our attention to the Chebeshev polynomials of first kind and we use

them for approximating a function and a particular solution for second order ODEs.

A function𝑓(𝑥) can be approximated by an n-th degree polynomials 𝑝𝑛(𝑥) expressed in terms of

𝑇0, … , 𝑇𝑛,

𝑝𝑛 = 𝐶0𝑇0(𝑥) + 𝐶1𝑇1(𝑥) + ⋯+ 𝐶2𝑇𝑛(𝑥) −1

2𝐶0 (4.5)

Where

𝐶𝑗 =2

𝑛∑ 𝑓(𝑥𝑘)𝑛+1𝑘=1 𝑇𝑗(𝑥𝑘), 𝑗 = 0,1, … , 𝑛. (4.6)

And 𝑥𝑘 , 𝑘 = 1, … , 𝑛 + 1 are zeroes of 𝑇𝑛+1.

Since

𝑇𝑗(𝑥) = cos(𝑗 arccos 𝑥), (4.7)

xxxiv

4.3 An Approximate Particular Solution

4.3.1 Method of Reduction of Order

we recall that we can reduce an initial value differential equation problem to an algebraic Here

we propose Rivlin (1990).

we first approximate 𝑓(𝑥) in Eq. (4.1) by 𝑝𝑛(𝑥) using Chebeshev polynomials .then we find a

particular solution 𝑦𝑝 of the equation

𝑎𝑦′′(𝑥) + 𝑏𝑦′(𝑥) + 𝑐𝑦(𝑥) = 𝑃𝑛(𝑥). (4.8)

We let the particular solution be in the form of

𝑦𝑝 = ∑ 𝑞𝑗𝑇𝑗(𝑥).𝑚𝑗=0 (4.9)

The coefficients 𝑞𝑗 for (4.9) are to be determined. We let 𝑚 = 𝑛 if 𝑐 ≠ 0 in Equ. (4.8); 𝑚 = 𝑛 +

1 if 𝑐 = 0, 𝑏 ≠ 0;𝑚 = 𝑛 + 2 if 𝑐 = 0, 𝑏 = 0.

Substituting (4.9) into (4.8),

𝑎∑𝑞𝑗𝑇′′𝑗(𝑥) + 𝑏∑𝑞𝑗𝑇

′𝑗(𝑥) + 𝑐∑𝑞𝑗𝑇𝑗(𝑥) = 𝑃𝑛(𝑥).

𝑚

𝑗=0

𝑚

𝑗=0

𝑚

𝑗=0

(4.10)

Our next goal is to use linear combinations of Chebeshev polynomials to represent 𝑇′𝑗(𝑥) and

𝑇′′𝑗(𝑥) for each . that first and second order derivatives of Chebeshev polynomials 𝑇′𝑗(𝑥) and

𝑇′′𝑗(𝑥) are represented in terms of 𝑇𝑘(𝑥), 𝑘 = 0,… , 𝑗. Then we arrive at a system of algebraic

equations of 𝑞𝑗 by eqating coefficients to find the solution 𝑦𝑝 given by (4.9).

According to [4.6] and its tables for representation coefficients , we reduce the first and second

order derivatives as follows

𝑇′𝑗(𝑥) = ∑ 𝑏𝑘𝑇𝑘(𝑥).𝑗−1𝑘=0 (4.11)

Where

xxxv

𝑏2𝑙 = 0, for 𝑙 = 0,1, … ,𝑗

2− 1,

𝑏2𝑙ℎ = 2𝑗, for 𝑙 = 0,1, … ,𝑗

2− 1,

For even 𝑗and 𝑏0 = 𝑗, 𝑏2𝑙 = 2𝑗, for 𝑙 = 0,1, … ,𝑗−1

2, 𝑏2𝑙+1 = 0, for 𝑙 = 0,1, … ,

𝑗−3

2,

For odd .Next, we expand the second order derivative𝑇′′𝑗(𝑥) in term of Chebeshev polynomials

𝑇𝑖(𝑥), 𝑖 = 0,… , 𝑗 − 2. With 𝑐𝑖 being the representation coefficients for 𝑇′′𝑗(𝑥) that is,

𝑇′′𝑗(𝑥) = ∑ 𝑐𝑖𝑇𝑖(𝑥),𝑗−2𝑖=0 (4.12)

We use the tables provided to calculate 𝑐𝑖 fori = 0,… , j − 2. By Eq.(4.11),

𝑇′′𝑗(𝑥) = (𝑇′𝑗(𝑥))

′

= (∑ 𝑏𝑘𝑇𝑘(𝑥)𝑗−1𝑘=0 )

′

=∑ 𝑏𝑘𝑇′𝑘(𝑥)

𝑗−1𝑘=0

=∑ 𝑏𝑘(∑ 𝑏𝑖𝑇𝑖(𝑥)𝑘−1𝑖=0 )

𝑗−1𝑘=0

=∑ ∑ 𝑏𝑘𝑏𝑖𝑇𝑖(𝑥)𝑘−1𝑖=0

𝑗−1𝑘=0 (4.13)

4.3.2 Expansions of Tn(x) and Un(x)

Let x = cos , it follows that

2/r2rrrnn

0r

nr

r2rn

n

0r

nr

r2rn

n

0r

nr

n2

22

nnn

x1(i)1(1XC2

1

x1iXCx1iXC2

1

x1ixx1ix2

1

)sini(cos))sini(cos2

1ncos)x(T

xxxvi

now if r is odd, 1 + (-1)r =0

and if r is even 1 + (-1)r =2

there fore,

,)x1(XC2

1)x(T s225n

2/n

0s

n25n

25 = n , n = 25 + 1 , 25 = 0 , n = 25 + 1

These two results can be expressed as:

r22/n

0r

xnr

n )x1()!r2n()!r2(

X!n)1()x(T

again

n2

n2

n

x1ixx1ixi2

1

nsin)x(U

therefore

2/r2rrrnnr2n )x1(i)1(1(XC

i2

1)x(U

when r is even, 1 – (-1)r = 0

and when r is odd, 1 – (-1)r = 2 therefore taking r = 25 + 1

2

1s21251252

2/)1n(

0s

n125n )x1()i(XC

i

1)x(U

2/)1n(

0s

2

1r2

1r2*nr

n )x1.()!1r2n()!1r2(

X!n)1()x(U

xxxvii

where (n/2) and (n – 1)/2 represent the integer not greater than the corresponding quantities for

any non-negative integral value of n.

4.3.3 Recurrence Relations

now we prove the following recurrence relations of the chebyshev polynomials

(1) Tn + (x) – 2 x Tn(x) + Tn-1(x) = 0

Proof

Since Tn(x) = cos n where = cos-1x we have

Tn+1(x) + Tn-1(x) = cos(n+1) + cos (n – 1)

= (cos n cos - sin n sin ) + cos n cos

+ sin n sin )

= 2 cos n cos

= 2x Tn(x)

(2) (1 + x2) T'n(x) + nx Tn(x) – n Tn-1(x) = 0

proof

since Tn(x) = cos (n cos-1x) we have

nsinsin

n)xcosnsin(

x1

n)x(nT 1

2

so

)1ncos()1ncos(n2

1

nsinsinnTsin n2

hence

1n1n2

1n

2 T)x(Tn)x(T)x1( by equation (20.8)

n1n2

1

1nn1n

xT2T2n

TTx2T

this gives

xxxviii

#0)x(nT)x(Tnx)x(T)x1( 1nnn2

(3) Un+1(x) – 2x Un(x) + Un-1(x) = 0

proof

since Un(x) = sin n where = cos-1x

we have

Un+1(x) + Un-1(x) = sin (n + 1) + sin (n – 1)

= sin n cos + cos n sin + sin n cos - cos n sin

= 2 sin n cos = 2x sin n

= 2x un(x). #

(4) 0)x(nU)x(nxU)x(U)x1( 1nnn2

proof

we know that Un(x) = sin n and also that

= sin (n cos-1x)

ncossin

n

)xcosncos(

x1

n)x(U 1

2n

0)()()()1(

)()(

)sincos)1(sin(

sincos)()1(

cossin)(sin

1

2

1

2

2

xnUxnxUxUx

xnXUxnU

nnn

nnxUx

nnxU

nnn

nn

n

n

xxxix

Example

Prove that

xU

x1

n)x(T n

2n

since

)xcosnsin()x(U

)xcosncos()x(T

1n

1n

)x(U

x1

n

)xcosnsin(

x1

n)x(T

n2

1

2n

Example

Prove that

(Tn(x))2 – Tn+1(x) Tn-1(x) = 1 – x2

(Tn(x))2 – Tn+1(x) Tn-1(x)

= cos2n - cos (n + 1) . cos(n – 1)

= cos2n - (cos2n - sin2)

= sin2 = 1 – x2 sinc x = cos

Example

Show that

)x(U

x1

1n

2

xl

Satisfies the (DE) (1 – x2 )Y''(x) – 3xy'(x) + (n2 – 1)y

Solution

Taking

)x(U

)x1(

x)x(U

x1

1)x(y

xU

x1

1)x(y

n

2

32

n2

n2

and

)x(U

)x1(

x3)x(U

)x1(

1)x(U

)x1(

x2)x(U

x1

1)x(y n

2

52

2

n

2

32

n

2

32

n2

Therefore

0)x(Un)x(Ux)x(U)x1(

x1

1

)x(y)1n()x(yx3)x(y)x1(

n2

nn2

2

22

Since Un(x) is the solution of

(1 – x2)y"(x) – x y'(x) + n2y(x) = 0

Example

Show that

)x(xU)x(U)x(Tx1 n1nn2

Since Tn(x) = cos n

Un(x) = sin n

we have

xli

xxU)x(U

nsincos)1nsin(

ncossin)x(T)x1(

nin

n2

Example

show that

nsinsin)x(Ux1

nsin)x(U

cosxcesin

)x(T)x(T)x(xT)x(U)x1(

n2

n

1nnnn2

there for xTn(x) – Tn-1(x) = coscos n - cos(n + 1) = sin sin n

4.3.4 Examples of Approximation

Example

We consider the equation

𝑦′′(𝑥) + 2𝑦(𝑥) = 𝑝4(𝑥)

With 𝑝4(𝑥) = 2𝑇1(𝑥) + 5𝑇4(𝑥). We look for a particular solution 𝑦𝑝 = ∑ 𝑞𝑗𝑇𝑗(𝑥),4𝑗=0 by (4.11)-

(4.13), we list the coefficients for 𝑇′𝑗(𝑥), 𝑗 = 0,1, … ,4 in table 1,which is equivalent to the

following,

(𝑇0)′ = 0,

(𝑇1)′ = 𝑇0(𝑥),

(𝑇2)′ = 4𝑇1(𝑥),

(𝑇3)′ = 3𝑇0(𝑥) + 6𝑇2(𝑥),

(𝑇4)′ = 8𝑇1(𝑥) + 8𝑇3(𝑥),

xlii

𝑏3 = 8

𝑏2 = 6 𝑏2 = 0

𝑏1 = 4 𝑏1 = 0 𝑏1 = 8

𝑏0 = 1 𝑏0 = 0 𝑏0 = 3 𝑏0 = 1

j=0

j=1

j=2

j=3

j=4

Table 1:coefficients for 𝑇′𝑗(𝑥), 𝑗 = 0,1, … ,4

𝑗 = 0

𝑗 = 1 𝑏0 = 1,

𝑗 = 2 𝑏0 = 0

𝑏1 = 4, 𝑏1, 𝑏0 = 4.1

𝑗 = 3 𝑏0 = 3,

𝑏1 = 0, 𝑏1, 𝑏0 = 0.1

𝑏2 = 6, 𝑏2, 𝑏0 = 6.0, 𝑏2, 𝑏1 = 6.4

𝑗 = 4 𝑏0 = 0,

𝑏1 = 8, 𝑏1, 𝑏0 = 8.1

𝑏2 = 0, 𝑏2, 𝑏0 = 0.0, 𝑏2, 𝑏1 = 0.4

𝑏3 = 8, 𝑏3, 𝑏0 = 8.3, 𝑏3, 𝑏1 = 8.0, 𝑏3, 𝑏2 = 8.6

Table 2:Coefficients for 𝑇′′𝑗(𝑥), 𝑗 = 0,1, … ,4

Thus,

𝑦′𝑝= (𝑞1 + 3𝑞3)𝑇0(𝑥) + (4𝑞2 + 8𝑞4)𝑇1(𝑥) + 6𝑞3𝑇2(𝑥) + 8𝑞4𝑇3(𝑥) (4.14)

The coefficients for second order derivatives 𝑇′′𝑗(𝑥), 𝑗 = 0,1, … ,4 are listed in Table 2 below,

Now we obtain the second order derivatives for 𝑇𝑗 , 𝑗 = 0,1, … ,4. We list the results as follows:

(𝑇0)′′ = 0,

(𝑇1)′′ = 0,

(𝑇2)′′ = 4𝑇0(𝑥),

(𝑇3)′′ = 24𝑇1(𝑥),

xliii

(𝑇4)′′ = 32𝑇0(𝑥) + 64𝑇2(𝑥),

We notice, for example ,𝑐0 = 32, 𝑐1 = 0, and 𝑐2 = 48 for (12) when 𝑗 = 4 therefore,

𝑦′′𝑝= 4𝑞2𝑇0(𝑥) + 24𝑞3𝑇1(𝑥) + 32𝑞4𝑇0(𝑥) + 48𝑞4𝑇2(𝑥)

= (4𝑞2 + 32𝑞4)𝑇0(𝑥) + 24𝑞3𝑇1(𝑥) + 48𝑞4𝑇2(𝑥).

We obtain the following linear system of equations by comparing coefficients of 𝑇𝑗 , 𝑗 = 0,1, … ,4.

{

2𝑞0 + 4𝑞2 + 32𝑞4 = 0,

2𝑞1 + 24𝑞3 = 2,2𝑞2 + 48𝑞4 = 0,

2𝑞3 = 0,2𝑞4 = 5.

(4.15)

The solution for (4.15) is 𝑞4 =5

2, 𝑞3 = 0, 𝑞2 = −60, 𝑞0 = 80. The plots of the right hand side

function and the particular solution are shown in figures 7-8.

Assume we use 𝑃𝑛 defined by (4.5)-(4.6) with 𝑛 = 4to approximate the function 𝑓(𝑥) in Eq. (1).

Without loss of generality , we assume 𝑐 ≠ 0 in (1). Using our method of reduction of order ,the

coefficients 𝑞𝑗 , 𝑗 = 0,1, … ,4 of the particular solution satisfy to the following system of

equations,

𝑐𝑞0 + 𝑏𝑞1 + 4𝑎𝑞2 + 3𝑏𝑞3 + 32𝑎𝑞4 =1

2𝐶0,

𝑐𝑞1 + 4𝑏𝑞2 + 24𝑎𝑞3 + 8𝑏𝑞4 = 𝐶1, (4.16)

𝑐𝑞2 + 6𝑏𝑞3 + 48𝑎𝑞4 = 𝐶2,

𝑐𝑞3 + 8𝑏𝑞4 = 𝐶3,

𝑐𝑞4 = 𝐶4.

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Figure (4.1) Plot of the right hand side function in Example

Example

in this example, we consider

𝑦′′(𝑥) + 𝑦′(𝑥) + 1.25𝑦(𝑥) = 3𝑥3 + 𝑥2 + 2𝑥 + 7.

We notice here the function 𝑓(𝑥) = 3𝑥3 + 𝑥2 + 2𝑥 + 7. We can still look for a

particularsolution in the form of 𝑦𝑝 = ∑ 𝑞𝑗𝑇𝑗(𝑥),4𝑗=0 but 𝑞4 will be 0 as expected.

The Chebeshev polynomials representation for 𝑓(𝑥) is

𝑓(𝑥) =∑𝑝𝑗𝑇𝑗(𝑥)

3

𝑗=0

With 𝑝0 = 15/2, 𝑝1 = 17/4, 𝑝2 = 1/2, 𝑝3 = 3/4. According to the system of equations (4.16),

we obtain the following system for 𝑞𝑗 , 𝑗 = 0,… ,4,

1.25𝑞0 + 𝑞1 + 4𝑞2 + 3𝑞3 + 32𝑞4 =15

2,

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1.25𝑞1 + 4𝑞2 + 24𝑞3 + 8𝑞4 =17

4,

1.25𝑞2 + 6𝑞3 + 48𝑞4 =1

2,

1.25𝑞3 + 8𝑞4 =3

4,

1.25𝑞4 = 0.

Using backward substitution, we get

𝑞4 = 0, 𝑞3 =3

5, 𝑞2 = −

62

25, 𝑞1 = −

23

125, 𝑞0 =

7902

625.

Example

in this example we consider the Cauchy-Euler equations that are expressible in the form of

𝑎𝑥2𝑦′′(𝑥) + 𝑏𝑥𝑦′(𝑥) + 𝑐𝑦(𝑥) = ℎ(𝑥),

Where 𝑎, 𝑏 and 𝑐 are constants. This important class of variable coefficient differential equations

can be solved by the particular solutions method of reduction of orde illustration , we solve a

specific Cauchy-Euler equation as follows,

𝑥2𝑦′′(𝑥) − 2𝑥𝑦′(𝑥) + 2𝑦(𝑥) = 𝑥3 (4.17)

By a change of variable 𝑥 = 𝑒𝑡 the equation (4.17) can be transformed into the

constantcoefficient equation in the new independent variable 𝑡

𝑦′′(𝑡) − 2𝑦′(𝑡) + 2𝑦(𝑡) = 𝑒3𝑡

Using the method described above, we approximate the right hand side function by Chebyshev

polynomial approximation, then we use the reduction of order method to solve for the

approximate particular solution.

The approximation to 𝑓(𝑡) = 𝑒3𝑡 by 𝑃4 is

𝑓(𝑡) =𝐶02𝑇0(𝑥) + 𝐶1𝑇1(𝑥) + 𝐶2𝑇2(𝑥) + 𝐶3𝑇3(𝑥) + 𝐶4𝑇4(𝑥),

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Where the numerical values of 𝐶0, … , 𝐶4 are given as follows,

𝐶02= 4.88075365707809,

𝐶1 = 7.90647046809621,

𝐶2 = 4.48879683378407,

𝐶3 = 1.9105629481,

𝐶4 = 0.608043176705983.

Solving the resulting system of equations,

2𝑞0 − 3𝑞1 + 4𝑞2 − 9𝑞3 + 32𝑞4 =1

2𝐶0,

2𝑞1 − 12𝑞2 + 24𝑞3 − 24𝑞4 = 𝐶1,

2𝑞2 − 18𝑞3 + 48𝑞4 = 𝐶2,

3𝑞3 − 24𝑞4 = 𝐶3,

2𝑞4 = 𝐶4.

We get

𝑞0 = 344.481628331765,

𝑞1 = 170.637487061598,

𝑞2 = 36.3797465297907,

𝑞3 = 4.60354053428590,

𝑞4 = 0.304021588352991,

Which are coefficients of 𝑇𝑗(𝑥) for an approximate particular solution 𝑦𝑝(𝑡)of the constant

coefficient equation. The plots of the right hand side function and the particular solution are

shown in figures (4.2) . An approximate particular solution of the Cauchy

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Euler equation is 𝑦𝑝(ln 𝑥).

Figure (4.2) Plot of f(x) and P4 of Example.

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\Example

As another example we consider the initial value problem,

𝑦′′ + 3𝑦′ − 4𝑦 = sin(2𝑥),

With initial data,

𝑦(0) = 1,

𝑦′(0) = 2.

First the Chebyshev approximation of sin(2𝑥), by 𝑃4 is,

sin(2𝑥) ≈𝐶02𝑇0(𝑥) + 𝐶1𝑇1(𝑥) + 𝐶2𝑇2(𝑥) + 𝐶3𝑇3(𝑥) + 𝐶4𝑇4(𝑥),

Where the coefficients 𝐶0, … , 𝐶4 are

𝐶02= 5.55111512312878𝑒−17,

𝐶1 = 1.15344467691257,

𝐶2 = 0,

𝐶3 = 0.257536611098051,

𝐶4 = 2.22044604925031𝑒−16.

The resulting system by the reduction of order method with 𝑛 = 4 gives

−4𝑞0 + 3𝑞1 + 4𝑞2 + 9𝑞3 + 32𝑞4 =1

2𝐶0,

−4𝑞1 + 12𝑞2 + 24𝑞3 + 24𝑞4 = 𝐶1,

−4𝑞2 + 18𝑞3 + 48𝑞4 = 𝐶2,

−4𝑞3 + 24𝑞4 = 𝐶3,

−4𝑞4 = 𝐶4.

The coefficients 𝑞0, … , 𝑞4 for the particular solution of the IVP is given by

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𝑞0 = 1.15994038863409,

𝑞1 = 0.9671298098748495,

𝑞2 = 0.289728687485306,

𝑞3 = 0.0643841527745125,

𝑞4 = −5.55111512312578𝑒−17,

So we arrive at the particular solution,

𝑦𝑝(𝑥) = 𝑞0𝑇0(𝑥) + 𝑞1𝑇1(𝑥) + 𝑞2𝑇2(𝑥) + 𝑞3𝑇3(𝑥) + 𝑞4𝑇4(𝑥).

The derivative of 𝑦𝑝 is given by (14) and thus

𝑦′𝑝(0) = (𝑞1 + 3𝑞3)𝑇0(0) + (4𝑞2 + 8𝑞4)𝑇1(0) + 6𝑞3𝑇2(0) + 8𝑞4𝑇3(0)

= (𝑞1 + 3𝑞3) ∗ 1 + (4𝑞2 + 8𝑞4) ∗ 0 + 6𝑞3 ∗ (−1) + 8𝑞4 ∗ 0

=𝑞1 − 3𝑞3.

This together with

𝑦𝑝(0) = 𝑞0𝑇0(0) + 𝑞1𝑇1(0) + 𝑞2𝑇2(0) + 𝑞3𝑇3(0) + 𝑞4𝑇4(0)

= 𝑞0 ∗ 1 + 𝑞1 ∗ 0 + 𝑞2 ∗ (−1) + 𝑞3 ∗ (0) + 𝑞4 ∗ 1

= 𝑞0 − 𝑞2 + 𝑞4.

Gives the initial data for the corresponding homogeneous problem,

𝑦′′ + 3𝑦′ − 4𝑦 = 0

That is a subject to the initial data,

𝑦(0) = 1 − 𝑦𝑝(0) = 1 − 𝑞0 − 𝑞2 + 𝑞4=,

𝑦′(0) = 2 − 𝑦′𝑝(0) = 2 − 𝑞1 − 3𝑞3 =.

We need to find the solution 𝑦ℎ to the above homogeneous problem so that we can obtain The

homogeneous solution is,

𝑦ℎ = 𝑐1𝑒−4𝑥 + 𝑐2𝑒

𝑥

Where 𝑐1 = −0.219246869919495 and 𝑐2 = 0.349035168770711.

l

𝒚(𝑥) = 𝑦𝑝 −−0.219246869919495 𝑒−4𝑥 + 0.349035168770711𝑒𝑥 .The exact solution of the

problem is

𝑦(𝑥) =1

20cos(2𝑥) −

1

20sin(2𝑥) −

9

40𝑒−4𝑥 +

6

5𝑒𝑥.

Figure 11 shows the right hand side function and its approximation.

li

Chapter Five

Conclusions and Recommendation

5.1 Conclusions

In this thesis the Chebyshev polynomials has applied to ordinary differential equations. We first

presented Chebyshev polynomials and their main properties. Second we applied the

approximation techniques to obtain solutions to some types of ordinary differential equations.

5.2 Recommendation

Recommend to study at Chebyshev polynomials and apply Chebyshev polynomials to solve

physical models described by the partial differential equations.

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References

1. Edwards, Henry and Denney, Davide,–Elementary Differential Equations, (2008).

2. Hartman, Philp – Ordinary Differential Equations- Birhauser Boston, (1982).

3. Shaker, Email – Ordinary Differential Equations and Laplace Transforms (2007).

4. Arfken, George – mathematical methods for Physicists – Acardemic Press, third edition,

(1985).

5. Mason, J.C, and Handscomb, D.C- Chebyshev Polynomials, Chapman and Hall /CRC,

(2003).

6. Rivlin, Theodore- Chebyshev Polynomials from Approximation Theory to Algebra and

Number theory. John Wiley Sons (1990).

7. Press, W.H. Flannery, B.P. Teutolskt, S.A and Vetterling, W.T. Numerical Recipes. The

Art of Scientific Computing. Cambridge University Press. Cambridg (1990).

8. Abramo witz. M and Stegun, I.A. Hand book of mathematical functions. Dover

Publications, in c, Newyork, (1964).

9. Fox, L and Parker, I.B. Chebyshev Polynomials in Numerical Analysis, Oxford

University Presses (c) (1968).

Documents

Implementation of Chebyshev Polynomials Approximation for