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
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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
Order Ordinary Differential Equations
AmaniAlnumanYagobEbrahim
Supervision Committee:
Name Position Signature
Dr. Sad Eldin M. Sad Eldain Main Superviser ……………………
Dr.MogtabaAhmed Yousif Co-supervisor ……………………
Date:
iv
Implementation of Chebyshev Polynomials Approximation for Solving Second
Order Ordinary Differential Equations
AmaniAlnumanYagobEbrahim
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
Implementation of Chebyshev Polynomials Approximation for Solving Second
Order Ordinary Differential Equations
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).