Solving Differential Equations of Second Order using
Quadratic Legendre Multi-wavelets (QLMW) with
Operational Matrix of Integration
Meenu Devi
S. R. Verma*
M. P. Singh
Department of Mathematics Department of Mathematics Department of Mathematics and Statistics, Faculty of Science, and Statistics, Faculty of Science, and Statistics, Faculty of Science,
Gurukula Kangri University Gurukula Kangri University Gurukula Kangri University Haridwar (UK), India. Haridwar (UK), India. Haridwar (UK), India.
*Corresponding Author:
ABSTRACT
In this paper is suggested an efficient method to solve differential equations. Using quadratic Legendre multi-wavelets approximation method, differential equations are converted into the system of algebraic equations with the help of operational matrix of integration and its product. Some illustrative examples are included to show the efficiency and applicability of the method.
AMS Subject Classifications:
65T60; 65L10;
65M70.
Keywords
Quadratic Legendre wavelets; Quadratic Legendre multi-wavelets; Operational matrix of integration; Differential equations.
1. INTRODUCTION
Wavelet is compactly supported [1] square integrable function in time domain and frequency domain. Firstly, wavelet was introduced by N. Ricker [2] in Seismology to provide a time dimension to seismic analysis. Also, it was repeated by E. A. Robinson [3, 4]. But finally wavelet was defined by J. Morlet et al [5, 6]. After that A. Grossman [7], a French theoretical physicist recognized the importance of wavelet transform and developed an inversion formula with the help of admissibility condition. Wavelet is new developing area which is being applied mathematics, medical science, technology and more widely in signal analysis. There is a reason of more success of wavelet and their transform to the comparison of Fourier analysis because Fourier analysis does not contain the local information of non-stationary signals but wavelets examine signals simultaneously in both time and frequency. In recent years, wavelet techniques are used to find out the numerical solution of linear and non-linear problems [10,13]. Current applications of wavelet contain climate analysis, financial denoising, denoising of astronomical images, fast solution of differential equations, computer graphic and so on.
The basic approach initiates from the integral of basis vector )
(t
which is defined by
, ) x ( P dt ) t (
x
0
(1)
where,
T n
t
)
[
,
,
...,
]
(
1
2
in which the elements
n
,...,
3
,
2
,
1
i
);
t
(
i
are orthonormal basis function on certain interval [0, 1], and the matrix
P
can be uniquely illustrated based on the particular orthonormal function. In section 2, quadratic Legendre multi-wavelet (QLMW) is introduced. Approximation of any function within interval [0, 1], using QLMW, followed by error analysis, is included in section 3. In section 4, the operational matrix of integration for QLMW is calculated. The product of operational matrix is produced in section 5. In section 6, these computations are used to derive a method for solving initial value problem. Finally, the advantages of present method with the help of two examples are shown in section 7.2. THE QUADRATIC LEGENDRE
MULTI-WAVELET (QLMW)
Let
ψ
be a function ofL
2(
),
called mother wavelet,satisfying the following properties [7, 8]:
.
1
dt
)
t
(
0
dt
)
t
(
2
and
This is also called unit energy function.
A family of such type of functions constructed from translation and dilation of a single function
ψ
,
wavelets [7, 8] can be defined as:
, 0 a , b , a , a
b t a 1 ) t (
b ,
a
(2)
where
a
andb
are dilation and translation parameters respectively.By discretization [1, 7] of these parameters
a
2
k
and
b
n
2
k, one gets
(
)
2
/2
2
,
,
t
t
n
k k n
k
Let
be a function ofL
2(
)
space. It is said to be scaling function forV
0,
if it satisfies the following condition
(
)
1
and
{
(
)}.
0
span
t
k
V
dt
t
k
(4)
2.1. Definition:
The nested sequence{
V
j}
j of the subspaces ofL
2(
)
with scaling function
is calledmulti-resolution analysis (MRA) if it satisfies the following conditions:
(i)
lim
V
L
(
)
2 jj
andlim
j
{
0
}
j
V
(ii)
j jt V
V
t) (2 )
( 0
(iii)
n
n
t
)}
(
{
is a Riesz basis ofV
0.For any orthogonal MRA with a multi-scaling function
, there exist [12] a multi-wavelet function
and a multi-scaling function
orthogonal to each other, given as below:
nn
n
(
2
t
n
)
b
2
)
t
(
(5)
n j n j,
}
,{
and
forms an orthonormal basis forL
2(
)
under certain condition [11].
For constructing the quadratic Legendre multi-wavelet, when the single scaling function
(
t
)
is replaced with vector of scaling function
(
t
)
yields:
(
t
)
(
t
),
(
t
),
(
t
)
,
T 2 10
]
[
where
.
1
0
),
1
6
6
(
5
)
(
and
)
1
2
(
3
)
(
,
1
)
(
2 2
1 0
t
t
t
t
t
t
t
(6)
Since T
2 1 0
(t),
ψ
(t),
ψ
(t)
ψ
Ψ(t)
[
]
are the corresponding mother wavelet functions [11], using equation (2) one gets the quadratic Legendre wavelet (QLW) as givenbelow:
, t ), t t (
t ), t t ( (t) ψ
1 2 1 55 168 120 3 1
2 1 0 7 72 120 3 1
2 2
0
, t ), t t (
t ), t t ( (t) ψ
1 2 1 17 46 30 3
2 1 0 1 14 30 3
2 2
1
(7)
, t ), t t (
t ), t t ( (t)
ψ
1 2 1 31 78 48 3
5
2 1 0 1 18 48 3
5
2 2
2
Translating and dilating the quadratic Legendre wavelet
(
t
),
we get
(
)
2
(
2
)
,
2
,
t
t
n
k j k j
n
k
k
,
n
,
j
.
(8) The family{
kj,n(
t
)}
forms an orthonormal basis forL
2(
)
and subfamily is orthonormal in
L
2[0,
1]
for, 1 , 0
n
2
,...,
2
k
1
;
,...
2
,
1
,
0
k
and j0, 1, 2. If n0,1;;
2
,
1
,
0
j
1
,
0
k
and
the scaling functions and subfamily members are plotted in the figures 1-12. Here, variable t andfamily members
}
{ 2
11 1 11 0 11 2 10 1 10 0 10 2 00 1 00 0 00 2 1
0, , , , , , , , , , ,
are shown
along horizontal and vertical axes respectively.
3. FUNCTION APPROXIMATIONS
The function
f
(
x
),
using quadratic Legendre multi-wavelets in the interval [0, 1], is approximated as given below:( ) ( ),
, 0
2
0 1 2
0 , 2
0
x c c
x
f kjn
k j n j
n k i
i i
k
(9)
where,
.
)
x
(
),
x
(
g
c
)
x
(
),
x
(
f
c
jn , k j
n , k i
i
and
After truncating the infinite series, equation (9) takes the following form:
), x ( C ) x ( c c
) x (
f j T
n , k M
0 k
2
0 j
1 2
0 n
j n , k 2
0 i
i i
k
(10) where,
T M M
M
M M
M
c
c
c
c
c
c
c
c
c
C
]
,...,
,
....
,...,
,
,...,
,
,
,
[
2 ) 1 2 ( 2 00 1
) 1 2 (
1 00 0
) 1 2 ( 0
00 2 , 1 0
(11)
and
. ,
... , , ,....,
, ,...,
, , , ) x (
T 2
) 2 ( M 2 00 1
) 2 ( M 1
00
0 ) 2 ( M 0 00 2 1 0
1 M 1
M
1 M
] [
(12)
Theorem 3.1
Let the functionf
:
[0,
1]
and1].
[0,
3
C
f
ThenC
T
approximatesf
with mean error bound as follows:sup f (x),
2 ! 3
1 C
f '''
x k 3 T
[0,1]
where
.
denotes the norm inL
2(
)
space.Proof. The proof of this theorem can be seen in [2]. It can be seen from this theorem; the factor
k
3
2
!
3
1
shows bestapproximation of function because if
k
increases then error decreases rapidly.4. OPERATIONAL MATRIX OF
INTEGRATION OF QLMW
With the help of equation (12), equation (1) takes theform:
, ) x ( P dt ) t (
x
0
12 12
where,
P
1212 an operational matrix of integration of QLMW[11] is given below:
In general
P
,) 2 . 3 ( ) 2 . 3 ( ) 2 . 3 ( ) 2 . 3 (
) 2 . 3 ( ) 2 . 3 ( ) 2 . 3 ( ) 2 . 3 (
n n n n
n n n
n
C B
B A
T (13) where order of
P
,
given by [11], is(
3
.
2
n1)
(
3
.
2
n1)
.
5. THE PRODUCT OF QLMW BASIS
VECTOR AND IT’S TRANSPOSE
In this section the product of operational integration matrix of QLMW is obtained, which is useful in solving non
homogeneous differential and integral equations. From equation (12) we have
, I I
) x ( ) x (
2
0 j
M
0 i
1 2
0 k
j k , i j
k , i 2
0 j
j j T
M
(14) where,
.
)
''
'
(
]
)
''
[(
)
''
(
)
'
(
,
)
''
'
(
]
)
''
[(
)
''
(
)
'
(
,
)
''
'
(
]
)
''
[(
)
''
(
)
'
(
,
)
''
'
(
]
)
''
[(
)
''
(
)
'
(
,
]
[
,
]
[
) 2 . 3 ( ) 2 . 3 ( , ,
, ,
.
) 2 . 3 ( ) 2 . 3 (
) 2 . 3 ( ) 2 . 3 ( , ,
, ,
.
) 2 . 3 ( ) 2 . 3 (
) 2 . 3 ( ) 2 . 3 ( , ,
, , ,
) 2 . 3 ( ) 2 . 3 (
(15)
11 1 1
n n
n n
n n
n n
n n
n n
j k i T j
k i
j k i j
k i j
k i
j T j
j j
j
j k i T j
k i
j k i j
k i j
k i
j T
j
j j
j
j k i T j
k i
j k i j
k i j
k i
j T j
j j j
c
c
c
c
C
c
c
c
c
C
a
a
a
a
A
a
a
a
a
A
C
B
B
A
I
C
B
B
A
I
The order of the elements of the matrices j
A
,j
B , j
C ,
A
ij.k,j
1
,
0
k
;
2
,
1
,
0
j
;
1
,
0
i
and
M
1
,
the matricesj
I
andj k i
I
, are of order 1212. Also, the matricesI
0,
A
0 areidentities and
A
11,1,A
12,1 are null matrices. Thus, the entries of matricesA
j,
A
ij,k are given below:, 1
)
'
a
(
0 0 0 1 0 1 0 5 2 5 2 , 1)
'
'
a
(
0 0 5 8 3 9 0 0 0 0 0 0 , 1)
''
'
a
(
0 5 3 2 0 5 3 2 0 24 17 0 24 17 0 , 2)
'
a
(
7 5 2 0 1 0 5 2 0 1 0 0 , 2)
'
'
a
(
0 7 3 3 0 0 0 5 8 3 9 0 0 0 , 2)
'
'
'
a
(
63 5 26 0 504 173 0 5 7 4 0 504 173 0 5 63 58 , 0 0 , 0)
'
a
(
0 5 8 3 9 0 5 8 3 9 0 0 0 0 0 , 504 173 0 5 63 58 0 24 17 0 0 0 1 0 0 , 0)
'
'
a
(
, 0 0 , 0)
''
'
a
(
0 15 168 31 0 15 168 31 0 3 21 2 0 3 21 2 0 , 1 0 , 0)
'
a
(
7 3 3 0 0 0 0 0 0 0 0 , 1 0 , 0)
'
'
a
(
0 5 7 4 0 5 3 2 0 24170 1 0
, 1 0 , 0 ) ' ' ' a ( 3 21 52 0 15 168 31 0 3 7 3 0 0 0 3 21 2 , , 2 0 , 0 2 0 ,
0
(
a
'
'
)
)
'
a
(
63 5 23 0 504 173 0 5 3 2 0 1 0 0 0 0 0 0 0 0 0 0 0 . 2 0 , 0)
''
'
a
(
0 3 21 52 0 3 21 52 0 15 168 31 10 16 3 15 168 31 0Remaining twelve matrices are the generalization of the following six matrices X1, X2, X3, X4, X5, X6:
, X
,
X1 2
2 64 27 10 64 3 63 1) ( 10 16 24 10 8 3 3 1) ( 2 64 27 2 16 3 3 1) ( 0 0 0 2 16 27 1) ( 10 64 3 9 0 10 64 3 9 0 0 0 0 0 1 l 1 l 1 l 1 l , X ,
X3 4
2 112 3 3 0 0 0 0 0 0 0 0 2 4 3 1) ( 2 64 3 57 10 16 3 1) ( 2 64 3 57 2 64 27 1) ( 2 16 3 3 10 16 3 1) ( 2 16 3 3 2 4 3 1) ( l l l l 1 l . X ,
X5 6
3 7 2 2 2 14 5 3 3 7 10 2 14 5 3 2 112 3 45 2 28 15 1) ( 3 7 10 2 28 15 1) ( 6 7 5 2 14 3 1) ( 2 112 5 9 2 28 15 1) ( 0 0 0 0 0 0 1 l 1 l 1 l 1 l , 1 l , (a''') 0 l , (a''') X , 1 l , (a'') 0 l , (a'') X , 1 l , (a') 0 l , (a') X , 1 l , (a''') 0 l , (a''') X , 1 l , (a'') 0 l , (a'') X , 1 l , (a') 0 l , (a') X 0 1 , 1 2 0 , 1 6 0 1 , 1 2 0 , 1 5 0 1 , 1 2 0 , 1 4 1 0 , 1 0 0 , 1 3 1 0 , 1 0 0 , 1 2 1 0 , 1 0 0 , 1 1 that such
Thus, the entries of
C
j and j k , iC
are achieved. Also, it can be seen that0
C is obtained as identity matrix,
(
c
'
)
11,0,1 1 , 1
)
'
'
'
c
(
,(
c
'
'
'
)
10,0,(
c
'
)
12,1 as null matrices and the following as non-zero matrices:
,
, 1
1
(
c
'
'
)
)
'
c
(
0 5 3 1 0 0 0 48 170 0 02 3 0 48 17 0 2 3 0 48 17 0 2 3 , , 0 0 , 1
1
(
c
'
)
)
'
'
'
c
(
0 0 3 21 2 2 0 0 0 3 21 2 2 0 0 2 3 0 5 3 1 0 2 3 0 5 3 1 0 2 3 , 1 0 , 1 0 0 ,1
(
c
'
'
)
)
'
'
c
(
0 0 0 0 0 2 21 3 2 0 0 0 , 0 30 84 31 0 0 0 0 0 0 0 , 0 0 0 0 3 21 2 2 0 0 0 0 , 7 6 3 0 0 0 0 0 0 0 3 21 2 2 0 1 , 1 2 0 ,1
(
c
'
)
, , 0 1 , 1 0 1 ,
1
(
c
'
'
'
)
)
'
'
c
(
3 21 2 52 0 0 0 0 0 0 0 7 6 3 0 0 0 30 84 31 0 0 0 0 0,
)
'
'
c
(
)
'
'
c
(
)
'
'
'
c
(
2 T1 , 1 2 0 , 1 2 1 ,
1 ,
[
]
0 0 0 0 0 0 0 30 84 31 0 0 0 3 21 2 52 0 0 0 3 21 2 52 0 0 . , 1 0 , 1 1 1 , 1 2 0 , 1 1 1 ,
1
(
c
'
''
)
(
c
'
)
(
c
'
''
)
)
''
c
(
0 0 30 84 31 0 0 0 30 84 31 0 0 0 0 0 0 3 21 2 52 0 0 0 0Similarly, corresponding to the entries of j
C and j k , i C the twelve matrices are the generalization of the following six matrices Z1,Z2,Z3,Z4,Z5,Z6:
, 5 7 1 0 15 1 3 32 5 17 0 5 126 29 0 15 1 3 32 5 17 0 5 126 29 ) 15 ( Z l l 1 l , Z l 2 0 15 1 3 2 1 0 2016 173 0 15 1 3 32 5 17 0 2016 173 0 ) 15 ( 1 l l , Z l 3 l l l l (5) 5 126 13 0 15 1 3 2 1 0 (5) 5 126 13 0 15 1 3 2 1 0 5 7 1 ) 15 ( , 5 2 1 Z l 4 l l l l l 11 19 21 11 0 3 24 ) 17(5 0 5 547 189 5 0 3 24 ) 17(5 0 5 547 189 5 , 0 5 2 3 24 17 0 5 2 1512 5 173 0 5 2 3 24 17 0 5 2 1512 5 173 0 Z l l l l 5
,Z6 l
l l l l l 386 41 189 193 0 2 1 15 3 2 0 386 41 189 193 0 2 1 15 3 2 0 110 19 21 11 5 such that , 1 l , ) ' ' ' c ( 0 l , ) ' ' ' c ( Z , 1 l , ) ' ' c ( 0 l , ) ' ' c ( Z , 1 l , ) ' c ( 0 l , ) ' c ( Z , 1 l , ) ' ' ' c ( 0 l , ) ' ' ' c ( Z , 1 l , ) ' ' c ( 0 l , ) ' ' c ( Z , 1 l , ) ' c ( 0 l , ) ' c ( Z 2 0 , 0 0 0 , 0 6 2 0 , 0 0 0 , 0 5 2 0 , 0 0 0 , 0 4 1 0 , 0 2 3 1 0 , 0 2 2 1 0 , 0 2 1
,
0 5 3
1 0 5 42
19 1) ( 0 3 48
5 17 1) (
0 5 6
1) ( 0 7
3 0 96
17 1) (
0 15 3
2 0 21
11 1) ( 0 3 24
17
0 3 2
1) ( 0 5 7
1 0 3 32
17 1) (
0 5 3
1 0 2
3 1) ( 0 48 17
0 0 0 1 0 0
=
l l
1 l 1
l
1 l
l l
l
3
Y
,
4
Y
0 378
5 41 1) ( 0 3 3
1 0 756
173 1) (
0 3 42
65 0
5 6
1) ( 0 2 672
5 173
0 189
193 1) ( 0 15 3
2 0 1512
5 173 1) (
0 126
5 13 0 3 2
1) ( 0 2016
173
0 2
3 1) ( 0 5 3
1 0 0
0 1 0 0 0 0
1 l l
1 l
1 l l
l
l
.
5
Y
0 756
173 1) ( 0 3 48
5 17 0 5 378
547 1) (
0 3 673
5 173 0 96
17 1) ( 0 3 42
29
0 1512
5 173 1) ( 0 3 24
17 0 189
5 1) (
0 2016
173 0 3 32
5 17 1) ( 0 5 26
29
0 0 0 48 17 0 2
3 1) (
0 0 0 0 0 1
l l
1 l
l 1
l
l l
When
l
0
then Y 2 , 0 , 13B
1 1 , 1
B
4
Y and 0
0 , 1 5
Y B and if
l
1
then
Y3,Y4,Y5 are related to 10 , 1 2
1 , 1 0
1 , 1 ,B ,B B
respectively with cyclic operation of columns
5
4,
3,
2,
1,
=
,
1c
s
c
s
s and. 1 l , B
0 l , B Y , 1 l , B
0 l , B
Y 2
0 , 0 0 0 , 0 2 1
0 , 0 2 1
With the help of equation (11) and the matrices
I
jandI
ij,k, the equation (34) of [9] takes the form:C (x).
~ C ) x ( ) x
( T
(16)
6. METHOD OF SOLUTION OF INITIAL
VALUE PROBLEMS
In this section is suggested an approximate solution of second order differential equation with initial conditions using QLMW and operational matrix of integration.
Consider the equation
,
)
0
(
'
and
)
0
(
with
)
(
)
(
)
(
'
)
(
'
'
)
(
y
y
x
f
x
y
x
c
y
x
b
y
x
a
(17)
where,
a
,
b
,
c
andf
are function ofx
or constants. First, approximatinga
,
b
,
c
and f by using equation (10) we get,
)
x
(
C
)
x
(
c
,
)
x
(
B
)
x
(
b
),
x
(
A
)
x
(
a
T
T
T
).
x
(
D
)
0
(
y
)
x
(
E
)
0
(
'
y
),
x
(
F
)
x
(
f
T0 T
0
T
and
(18)Now
)
x
(
Y
)
x
(
'
'
y
T
and
y
'
(
x
)
y
'
'
(
x
)
dx
y
'
(
0
)
x0
Y P (x) E (x)
T 0
T
(19)
(
)
'
(
)
(
0
)
0
y
dx
x
y
x
y
x
Y P (x) E P (x) D (x) T 0 T
0 2
T
By using equations (18) and (19), the equation (17) yields
0 T T 0 T
T T T
0 T
T T T
T T T
D ) x ( E P ) x (
Y P ) x ( ) x ( C ) E ) x (
Y P ) x ( ( ) x ( B Y ) x ( ) x ( A F ) x (
2
] [
. D ) x ( ) x ( C E P ) x ( ) x ( C
Y P ) x ( ) x ( C E ) x ( ) x ( B
Y P ) x ( ) x ( B Y ) x ( ) x ( A
0 T T T 0 T T
2 T T T
0 T T
T T T
T T
] [
(20)
Now using the product operation matrix C~ of equation (16) we obtained
CD F
~ E P C~ Y P C~ E B~ Y P B~ Y A ~
0 T 0 T
2 0
T
.
D C~ E P C~ E B~ F Y P C~ Y P B~ Y A ~
0 T 0 0
T 2
T
or
After finding
Y
the approximated solution of differential equation can be obtained.7. EXAMPLES
The following two examples have been solved by proposed method in this paper.
Example 1.1
Consider.
0
)
0
(
'
y
,
0
)
0
(
y
0
x
6
'
y
2
'
'
xy
with
(21) Putting,
y
'
'
(
x
)
Y
T
(
x
),
(22) where
T
y
y
y
y
y
y
y
y
y
y
y
y
Y
[
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]
(23) and.
,
,
,
,
,
,
,
,
,
,
,
)
x
(
2 T11 1 11 0 11 2 10 1 10 0 10 2 00 1 00 0 00 2 1
0
]
[
(24) Using initial condition, we can get
and
xET
(x), 2FT
(x), 6GT
(x). (26) Now inserting equations (22)-(26) into equation (21) we obtain T(E~F~PT)YTG~E.Which is hold for each
x
in defined interval, therefore we get (E~F~PT)Y G~E,so yields
[2,0,0,0,0,0,0,0,0,0,0,0] .
T
Y (27)
By virtue of equation (25) we get exact solution
y
(
x
)
x
2.
(28)Example 1.
2 Consider
y
''
(
x
)
1
, withy
(
0
)
0
,
y
'
(
0
)
0
.
(29) Taking,
y
'
'
(
x
)
Y
T
(
x
),
(30) where
T 11 , 10 , 9 , 8 , 7 , 6 , 5 , 4 , 3 , 2 , 1 ,
0
y
y
y
y
y
y
y
y
y
y
y
]
y
[
Y
(31) and
.
,
,
,
,
,
,
,
,
,
,
,
)
x
(
2 T11 1 11 0 11 2 10 1 10 0 10 2 00 1 00 0 00 2 1
0
]
[
(32) From initial condition, we have
y'(x) Y P (x), y(x) Y P (x) 2 T
T
(33)
and
1FT
(x). (34) Using equations (28)-(33) into equation (28) we get
TY
TFwhich is satisfy for each
x
in finite interval, therefore we getY
F
and hence
Y
[1,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0]
T.
(35) Using equation (33) and we get exact solution
.
2
)
(
2
x
x
y
(36)8. CONCLUSION
The aim of the present work is to suggest an efficient method for solving non homogeneous differential equation with IVP by reducing an integral equation into a set of algebraic equations with the help of operational matrix. It is also shown the QLMW provides an exact solution. This work shall facilitate in solving
real-world problems, such as related to the mathematical physics and digital electronics etc.
9. REFERENCES
[1] Daubechies, I., Ten Lectures on Wavelets, SIAM, Philadelphia, PA, 1992.
[2] Ricker, N., The Form and Laws of Propagation of Seismic Wavelets, Geophysics, 18(1953), 10-40.
[3] Robinson, E. A., Random Wavelets and Cybernetic Systems, Griffin and Co., London, 1962.
[4] Robinson, E. A., Wavelet composition of time series, in H. O. A. Wold ed., Econometric model building, essays on the causal chain approach: North Holland Publishing CO., (1964a), 37-106.
[5] Morlet, J., et al, Wave Propagation and Sampling Theory, Part-1, Complex Signal Land Scattering In Multilayer Media, J. Geophys, 47(1982a), 203-221.
[6] Morlet, J., et al, Wave Propagation and Sampling Theory, Part 2: Sampling Theory and Complex Waves, J Geophys, 47(1982b), 222-236.
[7] Debnath, L., Wavelet Transforms and Their Applications, Birkhauser, Boston, 2002.
[8] Resnikoff, H. L. and Wells, R. O. Jr., Wavelet Analysis, Springer-Verlag New York, Inc. 2004.
[9] Babolian, E. and Fattahzdeh, F., Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration, Applied Mathematics and Computation, 188(2007), 417-426.
[10] Yousefi, S. A., Legendre wavelet method for solving differential equations of Lane- Emden type, Applied Mathematics and Computation, 181(2006), 1417-1422. [11] Pathak, A., Numerical solution of linear
integro-differential equations by using quadratic Legendre multi-wavelet direct method, Advanced Research in Scientific Computing, 4(2012), 1943-2364.
[12] Soman, K. P. and Ramachandran, K. I., Insights Into Wavelets From Theory to Practice, PHI learning, 2010. [13] Razzaghi, M. and Yousefi, S., The Legendre wavelets
operational matrix of integration, International Journal of System Science, 32(2001), 495-502.
