Ratio Mathematica Volume 47, 2023

Moduli of continuity of functions in
Hölder’s class Hα,2ωk [0, 1) by first kind

Chebyshev wavelets and its applications in
the solution of Lane-Emden differential

equations

Shyam Lal *
Deepak Kumar Singh †

Abstract

In this paper, two new moduli of continuity W
((
f − S2k−1,0f

)
, 1
2k

)
,

W
((
f − S2k−1,Mf

)
, 1
2k

)
and two estimators E2k−1,0(f) and,E2k−1,M(f)

of a functions f in Hölder’s class Hα,2ωk [0,1) by First kind Cheby-
shev wavelets have been determined. These moduli of continuity and
estimators are new and best possible in wavelet analysis. Applying
this technique, Lane -Emden differential equations have been solved
by the first of kind Chebyshev wavelet method. These solutions ob-
tained by first kind Chebyshev wavelet method approximately coin-
cided with their exact solutions. This is a significant achievement of
this research paper in wavelet analysis.
Keywords: Chebyshev wavelet, Modulus of continuity, Wavelet ap-
proximation, Hölder’s class, Orthonormal basis, Operational matrix
of integration .
2020 AMS subject classifications: 42C40, 65T60, 65L10 , 65L60. 1

*Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi -
221005, India; shyam lal@rediffmail.com.

†Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi -
221005, India; 24deepak97@gmail.com.
1Received on June 12, 2022. Accepted on December 25, 2022. Published online on January 10,
2023. doi: 10.23755/rm.v39i0.794. ISSN: 1592-7415. eISSN: 2282-8214. ©Shyam Lal and
Deepak Kumar Singh. This paper is published under the CC-BY licence agreement.

52



Shyam Lal and Deepak Kumar Singh

1 Introduction
Recently, researchers are making attention on wavelets. Wavelets have con-

nections between several branches of mathematical sciences and play an impor-
tant role in signal processing, engineering and technology.The approximation of
functions of a certain class by trigonometric series is a common places of anal-
ysis.Approximation of functions belonging to some class by wavelet method has
been discussed by many researchers like DeVore[1], Morlet[4] , Meyer[3] and
Debnath[2]. Wavelets are new tools to solve differential equations and to estimate
the moduli of continuity & the approximation of functions.Wavelets help in the
most accurate representation of functions f ∈ Hα,2ωk [0,1) class. Several wavelets
are known like that Haar wavelet , Legendre wavelet , Chebyshev wavelet. Haar
wavelet is one of simplest in wavelet analysis . Due to its simplicity and better ap-
plications, it is used in solution of integral as well as differential equations . Haar
wavelet contains a non-smooth character. This is a difficiency of Haar wavelet
to estimate the moduli of continuity and the approximation of the smooth func-
tion by it. This weak point is almost removed by Chebyshev wavelets and more
accurate the moduli of continuity and approximations of functions are obtained.
Sripathy[14] discussed the chebyshev wavelet based approximation for solving
linear and non-linear differential equations.

Moduli of continuity of functions have been studies by Alexander Babenko[10].
In best of our knowledge, there is no work associated to the modulus of continu-
ity and approximation of a function f in Hölder’s class by first kind Chebyshev
wavelet method. To make an advanced study in this direction, in this paper, the
moduli of continuity and approximation of functions have been determined in
Hölder’s class Hα,2ωk [0,1).

Several linear, as well as non-linear differential equations are solvable by
Galerkin, Collocation, and other known methods. These equations can be solved
by Chebyshev wavelet technique in very efficient and suitable manners. This mo-
tivates us to consider first kind Chebyshev wavelet method for the solution of
differential equations. Also, Babolian and Fattahzadeh[5] suggested a method to
solve the differential equations by using Chebyshev wavelet operational matrix of
integration.

In this paper, the Lane-Emden differential equations has been solved by Cheby-
shev wavelet technique. The main characteristic of this techique is that it reduces
the problem to a system of algebraic equations. The approach is based on convert-
ing the given differential equations into integral equations through integration by
approximating various signals involved in the equation through truncated orthog-
onal Chebyshev wavelet series and using the operational matrix P of integration ,
to eliminate the integral operations.

This paper is organized as follows: Section(1) is introductory in which the

53



Moduli of continuity of functions in Hölder’s class....

importance of moduli of continuity and Chebyshev wavelet of first kind related
literature are studied. In section(2), Chebyshev wavelet of first kind ,approxima-
tion of function and moduli of continuity of functions in class Hα,2ωk [0,1) are de-
fined.In section(3), theorem concerning the moduli of continuity of f−S2k−1,M(f)
has been established and also its detail proof is discussed in section(4).In sec-
tion(5) corollaries are deduced from theorem of section (3) . In section(6) first
kind Chebyshev wavelet operational matrix of integration has been constructed
and the product operational matrix has been obtained in section(7). In section(8),
Lane-Emden differential equations of index 0, 1 & 2 are solved by Chebyshev
wavelet method. Finally, the main conclusions are summarized in section (9).

2 Definitions and Preliminaries

2.1 Chebyshev wavelets of first kind
Wavelets constitute a family of functions constructed from dialation and

translation of a single function Ψ ∈ L2(R) called mother wavelet .We write

Ψb,a(t) = |a|
−1
2 Ψ

(
t − b
a

)
, a ̸= 0. ( Daubechies [6])

If we restrict the values of dialation and translation parameter to a = a−k0 ,b =
(2n − 1)b0a0−k,a0 > 1,b0 > 0 respectively,the following family of discrete
wavelets are constructed:

Ψk,n(t) = |a0|
k
2 Ψ
(
ak0t − (2n − 1)b0

)
.

Now, taking a0 = 2,Ψ(t) = T̃m(t) & b0 = 1 the Chebyshev wavelet Ψ(k,n,m,t)
of first kind , generally denoted by Ψ

(c1)

n,m(t) over the interval [0,1), is obtained as
(Babolian [5])

Ψ
(c1)

n,m(t) =

{
2

k
2 T̃m(2

kt − 2n + 1),
n − 1
2k − 1

≤ t <
n

2k − 1
0 , otherwise

(1)

where T̃m(t) =




1
√
π
, m = 0√
2

π
Tm(t) , otherwise.

where n = 1,2, ...,2k−1 , m = 0,1,2, ...M and k is the positive integer. In above
definition, Tm are the first kind Chebyshev polynomials of degree m on the interval
[-1,1] which are defined by

Tm(t) = cos(mθ), θ = arccos(t) (2)

54



Shyam Lal and Deepak Kumar Singh

and also satisfy the following recursive formula:

T0(t) = 1 T1(t) = t, Tm+1(t) = 2tTm(t) − Tm−1(t), m = 1,2,3, ......

The set of {Tm(t) : m = 0,1,2,3, ...} in the Hilbert space L2[−1,1] is a or-
thogonal set with respect to the weight function ω(t) = 1√

1−t2
. Orthogonality of

Chebyshev polynomial of first kind on the interval [-1,1] implies that

⟨Tm(t),Tn(t)⟩ =
∫ 1
−1

Tm(t)Tn(t)√
1 − t2

dt =




π, m = n = 0
0, n ̸= m.
π

2
, n = m ̸= 0.

In dealing with Chebyshev wavelets, the weight function ω(t) for orthogonal
Chebyshev polynomials has to dilated and translated to construct orthonomal
wavelets. So the first kind Chebyshev wavelets are an orthonormal set with weight
function ( S. Dhawan[7])

ωk(t) =



ω1,k(t), 0 ≤ t < 12k−1 ,
ω2,k(t),

1
2k−1

≤ t < 2
2k−1

,
...
ω2k−1,k(t),

2k−1−1
2k−1

≤ t < 1,

(3)

where ωn,k(t) = ω(2kt−2n+1). Furthermore, the set of wavelets ψn,m(t) makes
an orthonormal basis in Hilbert space L2ωk[0,1), i.e.

⟨Ψ
(c1)

n,m ,Ψ
(c1)

n′,m′⟩ωk =
∫ 1
0

Ψ
(c1)

n,m(t)Ψ
(c1)

n
′
m

′ (t)ωk(t)dt = δn,n′δm,m′

in which δ denotes Kronecker delta function defined by

δn,n′ =

{
1, n=n’
0, otherwise

2.2 First kind Chebyshev wavelet expansion and approxima-
tion of function

The function f ∈ L2ωk[0,1) is expressed in the Chebyshev wavelet series as

f(t) =
∞∑
n=1

∞∑
m=0

cn,mΨ
(c1)

n,m(t), (4)

55



Moduli of continuity of functions in Hölder’s class....

where cn,m = ⟨f,Ψ
(c1)

n,m⟩ωk. The (2
k−1,M + 1)th partial sums of above

series (4) is given by

S2k−1,M(f)(t) =
2k−1∑
n=1

M∑
m=0

cn,mΨ
(c1)

n,m(t) = C
T Ψ

(c1)

(t) (5)

in which C and Ψ
(c1)(t) are 2k−1(M + 1) vectors of the form

CT = [c1,0,c1,1, ...c1,M,c2,0,c2,1...,c2,M, ......,c2k−1,0, ...,c2k−1,M] and

Ψ
(c1)

= [Ψ
(c1)

1,0 ,Ψ
(c1)

1,1 , ...,Ψ
(c1)

1,M,Ψ
(c1)

2,0 ,Ψ
(c1)

2,1 , ...,Ψ
(c1)

2,z , ...,Ψ
(c1)

2k−1,0, ...,Ψ
(c1)

2k−1,M−1]
T

The Chebyshev wavelet approximation E2k−1,M(f) of a function f ∈ L2ωk[0,1)
by (2k−1,(M + 1))th partial sums S2k−1,M(f) of its Chebyshev wavelet series is
given by

E2k−1,M(f) = min
S
2k−1,M (f)

∥f − S2k−1,M(f)∥2

where, ∥f∥2 =
(∫ 1

0

|f(t)|2 ωk(t)dt
)1

2

If E2k−1,M(f) → 0 as k,M → ∞ then E2k−1,M(f) is called the best approxima-
tion of f of order (2k−1,M + 1) ( Zygmund[8]).

2.3 Modulus of continuity
The Modulus of continuity of a function f ∈ L2ωk[0,1) is defined as

W (f,δ) = sup
0<h≤δ

||f(· + h) − f(·)||2

= sup
0<h≤δ

(∫ 1
0

|f(t + h) − f(t)|2ωk(t)dt
)1

2

.

It is remarkable to note that W (f,δ) is a non-decresing function of δ and W (f,δ)
→ 0 as δ → 0+ (Chui[9]).

2.4 Function of Hölder’s class
A function f is said to be in Hölder’s class H

α

ωk
[0,1) of order α, 0 < α ≤ 1,

( G.Das[11] ) if f satisfies(
f(x + t) − f(x)

)
ωk(x) = O (|t|α) , ∀x,t,x + t ∈ [0,1).

If

(∫ 1
0

|f(x + t) − f(x)|2 ωk(x)dx
)1

2

= O (|t|α) , ∀x,t,x + t ∈ [0,1).

56



Shyam Lal and Deepak Kumar Singh

then f is said to be a function of Hölder’s class H
α,2

ωk
[0,1) of order α, 0 < α ≤ 1.

The class H
α

ωk
[0,1) is a proper subclass of H

α,2

ωk
[0,1).

Let f ∈ Hαωk[0,1),α = 1. Then(
f(x + t) − f(x)

)
ωk(x) = O(|t|)(∫ 1

0

|f(x + t) − f(x)|2ωk(x)dx
)

=

∫ 1
0

O (|t|)2

ωk(x)
dx

= O
(
|t|2
)2k−1∑

n=1

∫ n
2k−1

n−1
2k−1

√
1 − (2kx − 2n + 1)2dx

= O
(
|t|2
)2k−1∑

n=1

∫ π
0

sin2 θ
dθ

2k

= O
(
|t|2
) π
4(∫ 1

0

|f(x + t) − f(x)|2ωk(x)dx
)1

2

=
(
O (|t|)2

)1
2

√
π

2
= O (|t|)

i.e f ∈ H
α,2

ωk
[0,1), α = 1.

Let, f(t) =

{
t2, 0 ≤ t < 1

2

1, 1
2
≤ t < 1

f

(
1

2
+
ϵ

2

)
− f

(
1

2
−
ϵ

2

)
= 1 − (

1

2
−
ϵ

2
)2 =

3

4
+
ϵ

2
−
ϵ2

4
.(

f
(
1
2
+ ϵ

2

)
− f

(
1
2
− ϵ

2

)
ϵ

)
ωk(x) =

(
3

4ϵ
+

1

2
−
ϵ

4

)
ωk

(
1

2
−
ϵ

2

)
→ +∞ as ϵ → 0+

Therefore, f /∈ H
α

ωk
[0,1),α = 1.∫ 1

0

∣∣f(x + t) − f(x)∣∣2ωk(x)dx = ∫ 12
0

∣∣f(x + t) − f(x)∣∣2ωk(x)dx
+

∫ 1
1
2

∣∣f(x + t) − f(x)∣∣2ωk(x)dx
=

∫ 1
2

0

|(x + t)2 − x2|2ωk(x)dx + 0

≤
∫ 1

2

0

∣∣2xt + t2∣∣ 2√
3
dx, 1 ≤ ω(x) ≤

2
√
3

=
2
√
3

∫ 1
2

0

(
4x2t2 + t4 + 4xt3

)
dx

57



Moduli of continuity of functions in Hölder’s class....

∫ 1
0

∣∣f(x + t) − f(x)∣∣2ωk(x)dx = 2√
3

[
4

3
t2 ×

1

8
+ t4 ×

1

2
+ 2t3 ×

1

4

]
=

2
√
3

[
1

6
× t2 +

1

2
× t2 +

1

2
× t2

]
=

7

3
√
3
t2

(∫ 1
0

∣∣f(x + t) − f(x)∣∣2ωk(x)dx)
1
2

≤

√
7

3
√
3

∣∣t∣∣
(∫ 1

0

|f(x + t) − f(x)|2ωk(x)dx
)1

2

= O (|t|)

f ∈ H
α,2

ωk
[0,1),α = 1

Thus,H
α

ωk
[0,1) ⊊ H

α,2

ωk
[0,1), α = 1.

3 Theorems
In this paper, following theorems have been proved.

Theorem 3.1. If f ∈ Hα,2ωk [0,1) and its first kind Chebyshev wavelet expansion be

f(t) =
∞∑
n=1

∞∑
m=0

cn,mΨ
(c1)

n,m(t)

having (2k−1,M + 1)th partial sums

(
S2k−1,M(f)

)
(t) =

2k−1∑
n=1

M∑
m=0

cn,mΨ
(c1)

n,m(t)

Then the Moduli of continuity of f −
(
S2k−1,M(f)

)
satisfies

i.W((f − S2k−1,0f),
1

2k
) = sup

0<h≤ 1
2k

∥(f − S2k−1,0f)(· + h) − (f − S2k−1,0f)(·)∥

= O

(
1

2(k−1)α

)
ii.W((f − S2k−1,Mf),

1

2k
) = sup

0<h≤ 1
2k

∥(f − S2k−1,Mf)(· + h) − (f − S2k−1,Mf)(·)∥

= O

(
1

2k(α+1/2)
√
M + 1

)
, M ≥ 1.

58



Shyam Lal and Deepak Kumar Singh

4 Proof

Proof of the theorem (3.1)
(i) Error between f(x) and its Chebyshev wavelet expansion in interval

[
n−1
2k−1

, n
2k−1

)
is given by

en(t) = cn,0Ψ
(c1)

1,0 (t) − fχ[ n−1
2k−1

, n
2k−1 )

(t)

=

(∫ n
2k−1

n−1
2k−1

f(x)Ψ
(c1)

1,0 (x)ωk(x)dx

)
Ψ

(c1)

1,0 (t) − fχ[ n−1
2k−1

, n
2k−1 )

(t)

=
2

k
2

√
π

(∫ n
2k−1

n−1
2k−1

f(x)ωk(x)dx

)
2

k
2

√
π
− fχ[ n−1

2k−1
, n
2k−1 )

(t)

=
2k

π

(∫ n
2k−1

n−1
2k−1

f(x)ωk(x)dx

)
−

2k

π
fχ[ n−1

2k−1
, n
2k−1 )

(t)

∫ n
2k−1

n−1
2k−1

ωk(x)dx

=
2k

π

(∫ n
2k−1

n−1
2k−1

(f(x) − f(t))ωk(x)dx

)
.

|en(t)| ≤
2k

π

(∫ n
2k−1

n−1
2k−1

|(f(x) − f(t))||ωk(x)|dx

)

=
2k

π

(∫ n
2k−1

n−1
2k−1

|(f(x) − f(t))||
√
ωk(x)||

√
ωk(x)|dx

)

≤
2k

π

(∫ n
2k−1

n−1
2k−1

(|(f(x) − f(t))|)2 |ωk(x)dx

)1
2
(∫ n

2k−1

n−1
2k−1

|ωk(x)|dx

)1
2

∫ n
2k−1

n−1
2k−1

|ωk(x)|dx =
∫ n

2k−1

n−1
2k−1

|ω(2kx − 2n + 1)|dx =
π

2k

|en(t)| ≤
2k

π

(∫ n
2k−1

n−1
2k−1

(|(f(x) − f(t))|)2 |ωk(x)dx

)1
2 ( π

2k

)1
2

=
2k

π

(∫ n
2k−1

n−1
2k−1

(|(f(x) − f(t))|)2 |ω(2kx − 2n + 1)dx

)1
2 ( π

2k

)1
2

=
2k

π

(∫ 1
−1

(
|f
(
v + 2n − 1

2k

)
− f(t)|

)2
|ω(v)|

dv

2k

)1
2 ( π

2k

)1
2

59



Moduli of continuity of functions in Hölder’s class....

|en(t)| ≤
1
√
π

(∫ 1
−1

(
|f
(
v + 2n − 1

2k

)
− f(t)|

)2
|ω(v)|dv

)1
2

=
1
√
π

(∫ 1
0

∣∣∣∣f
(
v + 2n − 1

2k

)
− f(t)

∣∣∣∣2 |ω(v)|dv
+

∫ 0
−1

∣∣∣∣f
(
v + 2n − 1

2k

)
− f(t)

∣∣∣∣2 |ω(v)|dv
)1

2

=
1
√
π

(∫ 1
0

∣∣∣∣f
(
v + 2n − 1

2k

)
− f(t)

∣∣∣∣2 |ω(v)|dv
+

∫ 1
0

∣∣∣∣f
(
−v + 2n − 1

2k

)
− f(t)

∣∣∣∣2 |ω(−v)|dv
)1

2

=
1
√
π

(∫ 1
0

∣∣∣∣f
(
v + 2n − 1

2k

)
− f(t)

∣∣∣∣2 |ω(v)|dv
)1

2

+
1
√
π

(∫ 1
0

∣∣∣∣f
(
−v + 2n − 1

2k

)
− f(t)

∣∣∣∣2 |ω(v)|dv
)1

2

,

v + 2n − 1
2k

∈
[
n − 1

2

2k−1
,
n

2k−1

)
and

−v + 2n − 1
2k

∈
(
n − 1
2k−1

,
n − 1

2

2k−1

]
|en(t)| ≤

1
√
π
O

(
1

2(k−1)

)α
+

1
√
π
O

(
1

2(k−1)

)α
=

2
√
π
O

(
1

2(k−1)

)α
= O

(
1

2(k−1)α

)
(6)

∥en∥22 =
∫ n

2k−1

n−1
2k−1

|en(t)|2|ωk(t)|dt

= O

(
1

2(k−1)

)2α ∫ n
2k−1

n−1
2k−1

|ωk(t)|dt

= O

(
1

22(k−1)α

)
π

2k

=
1

2k
O

(
1

22(k−1)α

)
≤

1

2k−1
O

(
1

22(k−1)α

)
∥en∥22 = O

(
1

2(k−1)(2α+1)

)
(7)

60



Shyam Lal and Deepak Kumar Singh

∥∥f − (S2k−1,0f)∥∥22 =
∫ 1
0


2k−1∑

n=1

en(t)


2 ωk(t)dt

=

∫ 1
0


2k−1∑

n=1

(en(t))
2ωk(t)


dt (8)

+
∑ ∑

1⩽ n̸=n′≤ 2k−1

∫ 1
0

en(t)en′(t)ωk(t)dt,

Due to disjointness of supports of en(t)&en′(t), equation(8) becomes,

=

∫ 1
0


2k−1∑

n=1

(en(t))
2ωk(t)


dt

=
2k−1∑
n=1

∫ 1
0

(en(t))
2ωk(t)

=
2k−1∑
n=1

∥en(t)∥22

=
2k−1∑
n=1

O

(
1

2(k−1)(2α+1)

)
= O

(
2k−1

2(k−1)(2α+1)

)
= O

(
1

22(k−1)α

)
∥∥f − (S2k−1,0f)∥∥2 = O

(
1

2(k−1)α

)
(9)

W

((
f − S2k−1,0f

)
,
1

2k

)
= sup

0<h≤ 1
2k

∥ (f − S2k−1,0f)(· + h) − (f − S2k−1,0f)(·)∥2

≤ ||
(
f − S2k−1,0f

)
||2 + ||

(
f − S2k−1,0f

)
||2

= 2||
(
f − S2k−1,0f

)
||2

= 2.O

(
1

2(k−1)α

)
= O

(
1

2(k−1)α

)
W

((
f − S2k−1,0f

)
,
1

2k

)
= O

(
1

2(k−1)α

)
. (10)

61



Moduli of continuity of functions in Hölder’s class....

(ii) Consider

f(x) =
∞∑
n=1

∞∑
m=0

cn,mΨ
(c1)

m,n(t)

cn,m = ⟨f,Ψ
(c1)

m,n(t)⟩ωk

=

∫ n
2k−1

n−1
2k−1

f(t)Ψ
(c1)

n,m(t)ωk(t)dt

=
2

k+1
2

√
π

(∫ n
2k−1

n−1
2k−1

f(t)Tm(2
kt − 2n + 1)ωk(t)dt

)

=
2

k+1
2

√
π

∫ n
2k−1

n−1
2k−1

(
f(t) − f

(
2n − 1
2k

)
+ f

(
2n − 1
2k

))
g(t)ωk(t)dt

=
2

k+1
2

√
π

(∫ n
2k−1

n−1
2k−1

(
f(t) − f

(
2n − 1
2k

))
g(t)ωk(t)dt

)

+
2

k+1
2

√
π
f

(
2n − 1
2k

)∫ n
2k−1

n−1
2k−1

g(t)ωk(t)dt (11)

where g(t) = Tm(2
kt − 2n + 1)∫ n

2k−1

n−1
2k−1

g(t)ωk(t)dt =

∫ n
2k−1

n−1
2k−1

Tm(2
kt − 2n + 1)ω(2kt − 2n + 1)dt

=

∫ 0
π

Tm(cosθ)ω(cosθ)
(− sinθ)

2k
dθ

=

∫ π
0

cosmθ

sinθ

(sinθ)

2k
dθ =

1

2k

[
sinmθ

m

]π
0

= 0 (12)

Therefore equation (11) reduces to,

cn,m =
2

k+1
2

√
π

(∫ n
2k−1

n−1
2k−1

(
f(t) − f

(
2n − 1
2k

))
g(t)ωk(t)dt

)

=
2

k+1
2

√
π

(∫ n
2k−1

n−1
2k−1

(
f(t) − f

(
2n − 1
2k

))
g(t)

√
ωk(t)ωk(t)dt

)

|cn,m| ≤
2

k+1
2

√
π

(∫ n
2k−1

n−1
2k−1

∣∣∣∣f(t) − f
(
2n − 1
2k

)∣∣∣∣2 |ωk(t)|dt
)1

2

×

( ∫ n
2k−1

n−1
2k−1

∣∣Tm(2kt − 2n + 1)∣∣2 |ωk(t)| dt
)1

2

62



Shyam Lal and Deepak Kumar Singh

=
2

k+1
2

√
π

(∫ n
2k−1

n−1
2k−1

∣∣∣∣f(t) − f
(
2n − 1
2k

)∣∣∣∣2 |ω(2kt − 2n + 1)|dt
)1

2

×

(∫ n
2k−1

n−1
2k−1

∣∣Tm(2kt − 2n + 1)∣∣ |ω(2kt − 2n + 1)|dt
)

=
2

k+1
2

√
π

(∫ 1
−1

∣∣∣∣f
(
v + 2n − 1

2k

)
− f

(
2n − 1
2k

)∣∣∣∣2 |ω(v)|dv2k
)1

2

×
(∫ π

0

|Tm(cosθ)| |ω(cosθ)|
sinθ

2k
dθ

)

=
1

2k

√
2

π

(∫ 1
−1

∣∣∣∣f
(
v + 2n − 1

2k

)
− f

(
2n − 1
2k

)∣∣∣∣2 |ω(v)|dv
)1

2

(∫ π
0

|cosmθ|
∣∣∣∣ 1sinθ

∣∣∣∣sinθdθ
)

=
1

2k

√
2

π

(∫ 1
−1

∣∣∣∣f
(
v + 2n − 1

2k

)
− f

(
2n − 1
2k

)∣∣∣∣2 |ω(v)|dv
)1

2

(∫ π/2
− π

2

∣∣∣cos(mπ
2

− mt)
∣∣∣ dt

)

=

√
2
π

2k
(∫ 1

−1

∣∣∣∣f
(
v + 2n − 1

2k

)
− f

(
2n − 1
2k

)∣∣∣∣2 |ω(v)|dv)12∫ π
2

− π
2

∣∣∣cos mπ
2

cosmt + sin
mπ

2
sinmt

∣∣∣dt
≤

√
2
π

2k−1

(∫ 1
−1

∣∣∣∣f
(
v + 2n − 1

2k

)
− f

(
2n − 1
2k

)∣∣∣∣2 |ω(v)|dv
)1

2

(∫ π/2
0

(|cos(mt)| + |sin(mt)|) dt

)

≤
1

2k−1

√
2

π

(∫ 1
−1

∣∣∣∣f
(
v + 2n − 1

2k

)
− f

(
2n − 1
2k

)∣∣∣∣2 |ω(v)|dv
)1

2

×
[
|sin(mt)|

m
+

|cos(mt)|
m

]π
2

0

=
1

m.2k

√
2

π

((
1

2k

)α
+

(
1

2k

)α)
=

1

m.2k−1

√
2

π

(
1

2kα

)

63



Moduli of continuity of functions in Hölder’s class....

|cn,m| ≤
1

m.2k−1

√
2

π

(
1

2kα

)
. (13)

Let I(t) = f(x) − S2k−1,M(f)(t)

I(t) =
∞∑
n=1

∞∑
m=0

cn,mΨ
(c1)

n,m(t) −
2k−1∑
n=1

M∑
m=0

cn,mΨ
(c1)

n,m(t)

=
(2k−1∑

n=1

+
∞∑
n=1

2k−1+1

)( M∑
m=0

+
∞∑
m=
M+1

)
cn,mΨ

c1
n,m(t) −

2k−1∑
n=1

M∑
m=0

cn,mΨ
c1
n,m(t)

=
2k−1∑
n=1

∞∑
m=M+1

cn,mΨ
(c1)

n,m(t), by def
n of Ψ

(c1)

n,m

(I(t))2 =


2k−1∑

n=1

∞∑
m=M+1

cn,mΨ
(c1)

n,m(t)


2 = 2k−1∑

n=1

∞∑
m=M+1

c2n,m

(
Ψ

(c1)

n,m(t)
)2

+
2k−1∑
n=1

∑ ∑
M+1≤m̸=m′≤∞

cn,mcn,m′Ψ
(c1)

n,m(t)Ψ
(c1)

n,m′(t)

+
∑ ∑

1⩽ n̸=n′≤ 2k−1

∞∑
m=M+1

cn,mcn′,mΨ
(c1)

n,m(t)Ψ
(c1)

n′,m(t)

+
∑ ∑

1⩽ n̸=n′≤ 2k−1

∑ ∑
M+1≤m̸=m′≤∞

cn,mcn′,m′Ψ
(c1)

n,m(t)Ψ
(c1)

n′,m′(t)

||I||22 =
∫ 1
0

|f(t) − S2k−1,M(f)(t)|2ωn(t)dt

=
2k−1∑
n=1

∞∑
m=M+1

|cn,m|2
∫ 1
0

|Ψ
(c1)

n,m(t)|
2ωn(t)dt

+
2k−1∑
n=1

∑ ∑
M+1≤m̸=m′≤∞

cn,mcn,m′

∫ 1
0

(
Ψ

(c1)

n,m(t)Ψ
c
n,m′(x)

)
ωn(t)dt

+
∑ ∑

1⩽ n̸=n′≤ 2k−1

∞∑
m=M+1

cn,mcn′,m

∫ 1
0

(
Ψ

(c1)

n,m(t)Ψ
(c1)

n′,m(t)
)
ωn(t)dt

+
∑ ∑

1⩽ n̸=n′≤ 2k−1

∑ ∑
M+1≤m̸=m′≤∞

cn,mcn′,m′

∫ 1
0

Ψ
(c1)

n,m(t)Ψ
(c1)

n′,m′(t)ωn(t)dt

=
2k−1∑
n=1

∞∑
m=M+1

|cn,m|2, by orthonormality of {Ψ
(c1)

n,m}.

64



Shyam Lal and Deepak Kumar Singh

||f − S2k−1,M(f)||22 ≤
2k−1∑
n=1

∞∑
m=M+1

[
1

m.2k−1

√
2

π

(
1

2kα

)]2

=
2

π
.2k−1.

1

22(k−1)
.
1

22kα

∞∑
m=M+1

1

m2

=
22

π 2k 22kα

∞∑
m=M+1

1

m2

≤
22

π 2k 22kα

[
1

(M + 1)2
+

∫ ∞
M+1

1

m2
dm

]
By Cauchy integral test,

≤
22

π 2k 22kα

[
1

(M + 1)
+

(
−1
m

)∞
M+1

]

=
22

π 2k 22kα

[
1

(M + 1)
+

1

M + 1

]
∥f − S2k−1,M(f)∥2 ≤ 2

√
2

π

1

2k(α+1/2)
√
M + 1

= O

(
1

2k(α+1/2)
√
M + 1

)

W

((
f − S2k−1,Mf

)
,
1

2k

)
= sup

0<h≤ 1
2k

∥
(
f − S2k−1,Mf

)
(· + h) −

(
f − S2k−1,Mf

)
(·)∥2

≤ ||
(
f − S2k−1,0f

)
||2 + ||

(
f − S2k−1,Mf

)
||2

= 2||
(
f − S2k−1,Mf

)
||2

= 2.O

(
1

2k(α+1/2)
√
M + 1

)
= O

(
1

2k(α+1/2)
√
M + 1

)
.

Thus, this theorem is completely established.

65



Moduli of continuity of functions in Hölder’s class....

5 Corollary

Corolary 5.1. If f ∈ Hα,2ωk [0,1) and its first kind Chebyshev wavelet expansion be

f(t) =
∞∑
n=1

∞∑
m=0

cn,mΨ
(c1)

n,m(t)

having (2k−1,M + 1)th partial sums

(
S2k−1,M(f)

)
(t) =

2k−1∑
n=1

M∑
m=0

cn,mΨ
(c1)

n,m(t)

Then the first kind Chebyshev wavelet approximation E2k−1,M(f) of f is given by

(i)E2k−1,0(f) = min||f − (S2k−1,0f)||2 = min||f −
2k−1∑
n=1

cn,0Ψ
(c1)

n,0 (t)||2

= O

(
1

2(k−1)α

)
(ii)E2k−1,M(f) = min||f − (S2k−1,Mf)||2 = min||f −

2k−1∑
n=1

M∑
m=0

cn,mΨ
(c1)

n,m(t)||2

= O

(
1

2k(α+1/2)
√
M + 1

)
The proof of corollary (5.1) can be developed parallel to the proof of theorem

(3.1) independently.

Remark
If f ∈ Hα,2ωk [0,1), then the moduli of continuity W

((
f − S2k−1,Mf

)
, 1
2k

)
=

O
(

1
2k(α+1/2)

√
M+1

)
and approximation E2k−1,M(f) = O

(
1

2k(α+1/2)
√
M+1

)
tends

to 0 as k → ∞ , M→ ∞ . Hence W
((
f − S2k−1,Mf

)
, 1
2k

)
and E2k−1,M(f) are

best possible modulus of continuity and approximation of functions respectively
in wavelet analysis.It is also observed that

W

((
f − S2k−1,Mf

)
,
1

2k

)
≤ E2k−1,M(f).

Hence modulus of continuity is sharper than the approximation of function in
H

α,2

ωk
[0,1) by first kind Chebyshev wavelet method.

66



Shyam Lal and Deepak Kumar Singh

6 First kind Chebyshev wavelet operational matrix
of integration

In this section, the operational matix P of integration of order 2k−1(M + 1) ×
2k−1(M + 1)has been derived. This plays a great role in dealing with the problem
of Lane-Emden differential equations ( Babolian [5]) .
First the matrix P8×8 has been obtained for M = 3 and k = 2 . There are eight
basis wavelet functions given below:

Ψ1,0(t) =

{
2√
π
, 0 ≤ t < 1

2
,

0, 1
2
≤ t < 1,

Ψ1,1(t) =

{
2
√

2
π
(4t − 1), 0 ≤ t < 1

2
,

0, 1
2
≤ t < 1,

Ψ1,2(t) =

{
2
√

2
π
(32t2 − 16t + 1), 0 ≤ t < 1

2
,

0, 1
2
≤ t < 1,

Ψ1,3(t) =

{
2
√

2
π
(256t3 − 192t2 + 36t − 1), 0 ≤ t < 1

2
,

0, 1
2
≤ t < 1,

Ψ2,0(t) =

{
0, 0 ≤ t < 1

2
,

2√
π
, 1

2
≤ t < 1,

Ψ2,1(t) =

{
0, 0 ≤ t < 1

2
,

2
√

2
π
(4t − 3), 1

2
≤ t < 1,

Ψ2,2(t) =

{
0, 0 ≤ t < 1

2
,

2
√

2
π
(32t2 − 48t + 17), 1

2
≤ t < 1,

Ψ2,3(t) =

{
0, 0 ≤ t < 1

2
,

2
√

2
π
(256t3 − 576t2 + 420t − 99), 1

2
≤ t < 1.

Let,

Ψ(t) = [Ψ10,Ψ11,Ψ12,Ψ13,Ψ20,Ψ21,Ψ22,Ψ23]
T (14)

By integrating the basis functions from 0 to t and expressing them in term of
Chebyshev wavelet series ,

67



Moduli of continuity of functions in Hölder’s class....

∫ t
0

Ψ1,0(x) =

{
2√
π
t, 0 ≤ t < 1

2
,

1√
π
, 1

2
≤ t < 1,

=

[
1

4
,

1

4
√
2
,0,0,

1

2
,0,0,0

]
Ψ(t)

∫ t
0

Ψ1,1(x) =

{
2
√

2
π
(4t − 1), 0 ≤ t < 1

2
,

0, 1
2
≤ t < 1,

=

[
−

1

8
√
2
,0,

1

16
,0,0,0,0,0

]
Ψ(t)

∫ t
0

Ψ1,2(x) =

{
2
√

2
π
(32t2 − 16t + 1), 0 ≤ t < 1

2
,

0, 1
2
≤ t < 1,

=

[
−

1

6
√
2
,−

1

8
,0,

1

24
,−

1

3
√
2
,0,0,0

]
Ψ(t)

∫ t
0

Ψ1,3(x) =

{
2
√

2
π
(256t3 − 192t2 + 36t − 1), 0 ≤ t < 1

2
,

0, 1
2
≤ t < 1,

=

[
1

16
√
2
,0,−

1

16
,0,0,0,0,0

]
Ψ(t)∫ t

0

Ψ2,0(x) =

{
0, 0 ≤ t < 1

2
,

2√
π
, 1

2
≤ t < 1,

=

[
0,0,0,0,

1

4
,

1

4
√
2
,0,0

]
Ψ(t)

∫ t
0

Ψ2,1(x) =

{
0, 0 ≤ t < 1

2
,

2
√

2
π
(4t − 3), 1

2
≤ t < 1,

=

[
0,0,0,0,−

1

8
√
2
,0,

1

16
,0

]
Ψ(t)

∫ t
0

Ψ2,2(x) =

{
0, 0 ≤ t < 1

2
,

2
√

2
π
(32t2 − 48t + 17), 1

2
≤ t < 1,

=

[
0,0,0,0,−

1

6
√
2
,−

1

8
,0,

1

24

]
Ψ(t)

∫ t
0

Ψ2,3(x) =

{
0, 0 ≤ t < 1

2
,

2
√

2
π
(256t3 − 576t2 + 420t − 99), 1

2
≤ t < 1.

=

[
0,0,0,0,

1

16
√
2
,0,−

1

16
,0

]
Ψ(t)

68



Shyam Lal and Deepak Kumar Singh

Thus,
∫ t
0

Ψ(x)dx = P8×8Ψ(t) ( M. A. Fariborzi[13]) (15)

where P8×8 =




1
4

1
4
√
2

0 0 1
2

0 0 0

− 1
8
√
2

0 1
16

0 0 0 0 0

− 1
6
√
2

−1
8

0 1
24

− 1
3
√
2

0 0 0
1

16
√
2

0 − 1
16

0 0 0 0 0

0 0 0 0 1
4

1
4
√
2

0 0

0 0 0 0 − 1
8
√
2

0 1
16

0

0 0 0 0 − 1
6
√
2

−1
8

0 1
24

0 0 0 0 1
16

√
2

0 − 1
16

0




(16)

7 The Product Operation matrix (POM)
In this section , the product operation matrix has been obtained. This has

important role in solving differential equations ( Babolian [5]).
By Equation (14),

Ψ ΨT =


Ψ1,0Ψ1,0 Ψ1,0Ψ1,1 Ψ1,0Ψ1,2 Ψ1,0Ψ1,3 Ψ1,0Ψ2,0 Ψ1,0Ψ2,1 Ψ1,0Ψ2,2 Ψ1,0Ψ2,3
Ψ1,1Ψ1,0 Ψ1,1Ψ1,1 Ψ1,1Ψ1,2 Ψ1,1Ψ1,3 Ψ1,1Ψ2,0 Ψ1,1Ψ2,1 Ψ1,1Ψ2,2 Ψ1,1Ψ2,3
Ψ1,2Ψ1,0 Ψ1,2Ψ1,1 Ψ1,2Ψ1,2 Ψ1,2Ψ1,3 Ψ1,2Ψ2,0 Ψ1,2Ψ2,1 Ψ1,2Ψ2,2 Ψ1,2Ψ2,3
Ψ1,3Ψ1,0 Ψ1,3Ψ1,1 Ψ1,3Ψ1,2 Ψ1,3Ψ1,3 Ψ1,3Ψ2,0 Ψ2,1Ψ2,1 Ψ1,3Ψ2,2 Ψ1,3Ψ2,3
Ψ2,0Ψ1,0 Ψ2,0Ψ1,1 Ψ2,0Ψ1,2 Ψ2,0Ψ1,3 Ψ2,0Ψ2,0 Ψ2,0Ψ2,1 Ψ2,0Ψ2,2 Ψ2,0Ψ2,3
Ψ2,1Ψ1,0 Ψ2,1Ψ1,1 Ψ2,1Ψ1,2 Ψ2,1Ψ1,3 Ψ2,1Ψ2,0 Ψ2,1Ψ2,1 Ψ2,1Ψ2,2 Ψ2,1Ψ2,3
Ψ2,2Ψ1,0 Ψ2,2Ψ1,1 Ψ2,2Ψ1,2 Ψ2,2Ψ1,3 Ψ2,2Ψ2,0 Ψ2,2Ψ2,1 Ψ2,2Ψ2,2 Ψ2,2Ψ2,3
Ψ2,3Ψ1,0 Ψ2,3Ψ1,1 Ψ2,3Ψ1,2 Ψ2,3Ψ1,3 Ψ2,3Ψ2,0 Ψ2,3Ψ2,1 Ψ2,3Ψ2,2 Ψ2,3Ψ2,3




Substituting the values of each entry of above symmetric matrix , and after sim-
plification expressing them in term of Chebyshev wavelet series ,

Ψ ΨT =


 A O4×4
O4×4 B




where O4×4 is a 4 × 4 zero matrix and

A =




2√
π
Ψ1,0

2√
π
Ψ1,1

2√
π
Ψ1,2

2√
π
Ψ1,3

2√
π
Ψ1,1

2√
π
Ψ1,0 +

√
2
π
Ψ1,2

√
2
π
Ψ1,1 +

√
2
π
Ψ1,3

√
2
π
Ψ1,2

2√
π
Ψ1,2

√
2
π
Ψ1,1 +

√
2
π
Ψ1,3

2√
π
Ψ1,0

√
2
π
Ψ1,1

2√
π
Ψ1,3

√
2
π
Ψ1,2

√
2
π
Ψ1,1

2√
π
Ψ1,0




69



Moduli of continuity of functions in Hölder’s class....

B =




2√
π
Ψ2,0

2√
π
Ψ2,1

2√
π
Ψ2,2

2√
π
Ψ2,3

2√
π
Ψ2,1

2√
π
Ψ1,0 +

√
2
π
Ψ1,2

√
2
π
Ψ2,1 +

√
2
π
Ψ2,3

√
2
π
Ψ2,2

2√
π
Ψ2,2

√
2
π
Ψ2,1 +

√
2
π
Ψ2,3

2√
π
Ψ2,0

√
2
π
Ψ2,1

2√
π
Ψ2,3

√
2
π
Ψ2,2

√
2
π
Ψ2,1

2√
π
Ψ2,0




8 Solution of the Lane -Emden differential equation
by first kind Chebyshev wavelet

Many problems in the literature of mathematical physics can be distinctively
formulated as equations of Lane-Emden type as follows:

d2y

dt2
+

2

t

dy

dt
+ L(y) = 0 ( A-M Wazwaz[12])

where L(y) is some given function of y.
A difficult element in the analysis of this type of equations is the singularity

behaviour that occurs at t = 0. Most algorithms currently in use for handling
the Lane-Emden type problems are based on either series solution or perturbation
techniques. In recent years, a lot of attention has been devoted to the study of
wavelet theory. The first kind Chebyshev wavelet method accurately computes
the solution of differential equations and it is of great interest to solve this type of
problem.

The most popular form of L(y) attracted by the scientific community is

L(y) = yn

where n is constant parameter.
The standard Lane-Emden equation of index n is of the form

d2y

dt2
+

2

t

dy

dt
+ yn = 0, with y(0) = 1, y′(0) = 0 (17)

It is a basic equation in the theory of stellar structure. The equation describes the
temperature variation of a spherical gas cloud under the mutual attraction of its
molecules .

It was physically shown that interesting values of n lie in the interval [0,5] .
In addition, exact solutions exist only for n = 0,1,5. Notice that equation (17) is
linear for n = 0,1 and non-linear otherwise.
Example (1). For n = 0, the Lane-Emden differential equation (17) reduces to

ty′′ + 2y′ + t = 0 with y(0) = 1,y′(0) = 0 (18)

70



Shyam Lal and Deepak Kumar Singh

The exact solution of the equation (18) is y(t) = 1 − t
2

6
.

The equation (18) has been solved using first kind Chebyshev wavelet. First it is
assumed that y′′(t) can be expanded in terms of Chebyshev wavelet as

y′′(t) = CT Ψ(t) (19)

where Ψ(t) is given by equation (14). By integrationg equation (19) twice with
respect to t from 0 to t and using conditions of equation (18) & (15) following
equations are obtained ,

y′(t) = CT P Ψ(t) (20)

and, y(t) = CT P2 Ψ(t) + dT Ψ(t) (21)

where, dT = [
√
π

2
,0 , 0 ,0 ,

√
π

2
,0 ,0, 0] (22)

Using equation (14), t is expressed as

t = eT Ψ(x) (23)

where, eT = [
√
π

8
,

√
π
2

8
, 0, 0,

3
√
π

8
,

√
π
2

8
, 0, 0] (24)

Substituting values from Equation (19) to (24) in Equation (18),

eT Ψ(t)ΨT (t)C + 2CTPΨ(t) + eT Ψ(t) = 0 (25)

The following property of the product of two first kind Chebyshev wavelet vectors
will also be used:

eT Ψ(t)ΨT (t) ≃ ΨT (t)ẽ (26)

where ẽ is 8×8 matrix. Let us establish equation (26) & find ẽ,

eT ΨΨT =
1

4

[
Ψ1,0+

1
√
2
Ψ1,1,

1
√
2
Ψ1,0+Ψ1,1+

1

2
Ψ1,2, Ψ1,2+

1

2
Ψ1,1+

1

2
Ψ1,3, Ψ1,3

+
Ψ1,2
2
, 3Ψ2,0+

Ψ2,1√
2
, 3Ψ2,1+

1
√
2
Ψ2,0+

Ψ2,2
2
,
Ψ2,1
2

+3Ψ2,2+
1

2
Ψ2,3, 3Ψ2,3+

1

2
Ψ2,2

]

eT ΨΨT =




Ψ1,0
Ψ1,1
Ψ1,2
Ψ1,3
Ψ2,0
Ψ2,1
Ψ2,2
Ψ2,3




·
1

4




1 1√
2

0 0 0 0 0 0
1√
2

1 1
2

0 0 0 0 0

0 1
2

1 1
2

0 0 0 0
0 0 1

2
1 0 0 0 0

0 0 0 0 3 1√
2

0 0

0 0 0 0 1√
2

3 1
2

0

0 0 0 0 0 1
2

3 1
2

0 0 0 0 0 0 1
2

3




71



Moduli of continuity of functions in Hölder’s class....

where

ẽ =
1

4




1 1√
2

0 0 0 0 0 0
1√
2

1 1
2

0 0 0 0 0

0 1
2

1 1
2

0 0 0 0
0 0 1

2
1 0 0 0 0

0 0 0 0 3 1√
2

0 0

0 0 0 0 1√
2

3 1
2

0

0 0 0 0 0 1
2

3 1
2

0 0 0 0 0 0 1
2

3




8×8

(27)

Using Equation (26), Equation (25) becomes

(ΨT (t)ẽ)C + ΨT (t)(2PTC) + ΨT (t)e = 0

ΨT (t)
[
ẽC + (2PT C) + e

]
= 0

ẽC + (2PT C) + e = 0 (28)

Equation (28) is a set of algebraic equations which is solved for C in following
form

C =
(
ẽ + 2PT

)−1
(−e)

C =




−0.295408975150919
0.000000000000000
−0.000000000000000
−0.000000000000000
−0.295408975150919
0.000000000000000
−0.000000000000000
0.000000000000000




(29)

By substituting the Chebyshev wavelet coefficients C from equation (29) into
equation (21), the explicit form of the approximate solution of equation (18) is,

y(t) =
[
0.872379629742559,−0.013055355597036,−0.003263838899259,0,

0.798527385954829,−0.039166066791109,−0.003263838899259,0
]
Ψ(t)

Using the simplified value of Ψ(t) from equation (14), y(t) becomes

y(t)=

{
1 − 0.166666666666660t2, 0 ≤ t < 1

2
,

1 − 6.357719098970234.10−15 t − 0.166666666666661t2, 1
2
≤ t < 1.

The exact solution and the solution obtained by Chebyshev wavelet method of
differential equation (18) at different values of t are given in Table (1):

72



Shyam Lal and Deepak Kumar Singh

Variable (t) Exact solution Chebyshev solution Absolute error
0 1.000000000000000 1.000000000000000 0

0.1 0.998333333333333 0.998333333333333 0
0.2 0.993333333333333 0.993333333333334 0.000000000000001
0.3 0.985000000000000 0.985000000000001 0.000000000000001
0.4 0.973333333333333 0.973333333333334 0.000000000000001
0.5 0.958333333333333 0.958333333333333 0
0.6 0.940000000000000 0.939999999999999 0.000000000000001
0.7 0.918333333333333 0.918333333333333 0
0.8 0.893333333333333 0.893333333333333 0
0.9 0.865000000000000 0.865000000000000 0

Table(1): Comparison table for the exact and Chebyshev wavelet solution of
Lane-Emden equation of index 0.
This table shown that the solution of equation (18) obtained by first kind Cheby-
shev wavelet method nearly coincides with its exact solution.The graphs of exact
and first kind Chebyshev wavelet solutions of Lane -Emden differential equation
for n = 0 are drawn in figure (1).

73



Moduli of continuity of functions in Hölder’s class....

Example (2). For n = 1, the Lane-Emden differential equation (17) reduces to

ty′′ + 2y′ + ty = 0 with y(0) = 1,y′(0) = 0 (30)

The exact solution of the equation (30) is y(t) =

{
sin(t)

t
, t ̸= 0

1 t = 0.

Substituting values from Equation (19) − (24) in equation (30) ,

eT Ψ(t)ΨT (t)C + 2CTPΨ(t) + eT Ψ(t)(CT P2 + dT )Ψ(t) = 0

C =
[
ẽ + 2PT + ẽ(P2)T

]−1
(−ẽd)

C = [−0.2871902685,0.0077317973,0.0019076755,−0.0000134466,
−0.2449764764,0.0218383818,0.0015733213,−0.0000401720]T (31)

By substituting the Chebyshev wavelet coefficients C from equation (31) into
equation (21), the explicit form of the approximate solution of equation (30) is,

y(t) =
[
0.872505218190184,−0.012913303990151,−0.003192905781766,

0.000020169906045,0.801594934932526,−0.036835535048721,
−0.002723022009203, 0.000056975401025

]
Ψ(t)

or, y(t)=



0.9999960361853 + 2.5435469133605.10−4 t − 0.1692243050960t2+
0.0082397473925t3, 0 ≤ t < 1

2

0.9979746687982 + 0.0116372814448t − 0.1914197434042t2+
0.0232754139255t3 1

2
≤ t < 1.

Table (2): Comparisons between the exact solution and numerical solutions for
various values of t

Variable (t) Exact solution Chebyshev solution Absolute error
0 1 0.999996036185 0.0396381467104×10−4

0.1 0.99833416646 0.998337468350 0.0330188261299×10−4
0.2 0.99334665397 0.993343852898 0.0280107641192×10−4
0.3 0.98506735553 0.985064628313 0.0272722411598×10−4
0.4 0.97354585577 0.973549233079 0.0337730799104×10−4
0.5 0.95885107720 0.958847800410 0.0327679807199×10−4
0.6 0.94107078899 0.941073419447 0.0263045585602×10−4
0.7 0.92031098176 0.920308558518 0.0242325006294×10−4
0.8 0.89669511362 0.896692870105 0.0224351906097×10−4
0.9 0.87036323291 0.870366006692 0.0277377353607×10−4

74



Shyam Lal and Deepak Kumar Singh

This table shown that the solution of equation (30) obtained by first kind Cheby-
shev wavelet method nearly coincides with its exact solution.The graphs of exact
and first kind Chebyshev wavelet solutions of Lane -Emden differential equation
for n = 1 are drawn in figure (2).

Example (3). For n = 2 , The Lane-Emden differential equation is

ty′′ + 2y′ + ty2 = 0 with y(0) = 1,y′(0) = 0 (32)

The Taylor series expansion of function L(y) about a point y = 1 is given as,

L(y) =
∞∑

m=0

(y − 1)m

m!

dmL(y)

dym

∣∣∣∣
y=1

y2 = 1 + 2 (y − 1) + (y − 1)2

From equation (21) & (22),

= dT Ψ(t) + 2CT P2 Ψ(t) + CT P4Ψ(t)

Substituting the Taylor series exapansion of y2 and values from equation (19) to
(24) in equation (32) ,

eT Ψ(t)ΨT (t)C + 2CTPΨ(t) + eT Ψ(t)ΨT (t)
[
d + 2 (PT )2 C + (PT )4 C

]
= 0

75



Moduli of continuity of functions in Hölder’s class....

C =
[
ẽ + 2PT + 2 ẽ(PT )2 + ẽ(PT )4

]−1
(−ẽd) (33)

C = [−0.279001698770205,0.015429503002548,0.003798307476540,
− 0.000031441187989,−0.195293820350412,0.043103314900479,
0.003011315345934,−0.000095144707040]T (34)

By substituting the Chebyshev wavelet coefficients C from equation (34) into
equation (21), the explicit form of the approximate solution of equation (32) is,

y(t) =
[
0.872630597117721,−0.012771501223632,−0.003122128096052,

0.000040262875496,0.804642516403461,−0.034524067315646,

−0.002189080129024,0.000112495988561
]
Ψ(t)

or, y(t)=




0.9999921077841 + 5.0666971784118.10−4 t − 0.1717663081650t2

+0.016448064886675t3, 0 ≤ t < 1
2

0.996061399263954 + 0.022704430920508t − 0.215186679101439t2

+0.045956511961640t3 1
2
≤ t < 1.

9 Conclusion

(i) In theorem (3.1) , the moduli of continuity has been estimated as following:

W

((
f − S2k−1,0f

)
,
1

2k

)
= O

(
1

2(k−1)α

)
→ 0 as k → ∞,

W

((
f − S2k−1,Mf

)
,
1

2k

)
= O

(
1

2k(α+1/2)
√
M + 1

)
→ 0 as k → ∞,M → ∞.

(ii) In corollary (5.1) ,

E2k−1,0(f) = O

(
1

2(k−1)α

)
→ 0 as k → ∞,

E2k−1,M(f) = O

(
1

2k(α+1/2)
√
M + 1

)
→ 0 as k → ∞,M → ∞.

Thus W
((
f − S2k−1,0f

)
, 1
2k

)
, W

((
f − S2k−1,Mf

)
, 1
2k

)
, E2k−1,0(f) & E2k−1,M(f)

are best possible estimators in wavelet analysis.

76



Shyam Lal and Deepak Kumar Singh

(iii) From theorem (3.1) and corollary (5.1) , it is observed that

W

((
f − S2k−1,0f

)
,
1

2k

)
≤ 2E2k−1,0(f)

W

((
f − S2k−1,Mf

)
,
1

2k

)
≤ 2E2k−1,M(f)

Hence moduli of continuity W
((
f − S2k−1,0f

)
, 1
2k

)
, W

((
f − S2k−1,Mf

)
, 1
2k

)
are better and sharper than approximations E2k−1,0(f), E2k−1,M(f) respectively.

(iv) Solution of Lane-Emden differential equations by first kind Chebyshev wavelet
is approximately same as exact solution of differential equations. This is the sig-
nificant achievement of this research paper . Calculations performed in section(8)
demonstrate that the accuracy of Chebyshev wavelet method is quite high. In this
method , there is no complex integral or methodology. Application of this pro-
posed method is very simple & gives the explicite form of approximate solutions
to the Lane - Emden differential equations. These are the main advantages of
the method. This method is also very convenient for solving the boundary value
problems. Hence, this proposed method is very reliable, simple, fast & computa-
tionally efficients method.

Acknowledgments

Shyam Lal, one of the authors, is thankful to DST - CIMS for encouragement
to this work.

Deepak Kumar Singh, one of the authors, is grateful to U.G.C. (India) for
providing financial assistance in the form of UGC Non-NET Fellowship for his
research work.

Authors are greatful to the referee for his valuable comments and suggestions,
to improve the quality of this research paper.

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78