LEGENDRE MULTIWAVELET GALERKIN METHODS FOR DIFFERENTIAL EQUATIONS†

Journal of Applied Mathematics & Informatics.
2014.
Jan,
32(1_2):
267-284

- Received : February 10, 2013
- Accepted : August 13, 2013
- Published : January 28, 2014

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The multiresolution analysis for Legendre multiwavelets are given, anti-derivatives of Legendre multiwavelets are used for the numerical solution of differential equations, a special form of multilevel augmentation method algorithm is proposed to solve the disrete linear system efficiently, convergence rate of the Galerkin methods is given and numerical examples are presented.
AMS Mathematics Subject Classification : 42C15, 65N12, 65L20.
L
^{2}
[0, 1] be the Hilbert space equipped with the inner product
and the norm
We will define a multiresolution approximation of
L
^{2}
[0,1] of multiplicity r generated by Legendre multiwavelets (cf.
[2
,
3
,
9]
).
Clearly
P_{m}
(
x
) is a polynomial of degree m. It has the following properties on the interval [-1,1] (cf.
[1]
):
Legendre multiscaling functions are defined as follows:
Then they have the following properties on the interval [0,1]:
Now let
r
> 1 be an integer, and take the first r multiscaling functions
φ
^{1}
,
φ
^{2}
, · · ·,
φ^{r}
, which form an orthonormal bases of the function space
Let
be the vector of the multiscaling functions:
and denote for integers
j
> 0 and
k
= 0, 1, · · · , 2
^{j}
− 1
the dilates and translates of the multiscaling functions, where
is supported on the interval [2
^{−j}
k
, 2
^{−j}
(
k
+ 1)] ⊂ [0, 1]. For a fixed
j
> 0, all elements
of the vector
form an orthonormal bases of the 2
^{j}r
-dimensional space
then one has
and (iv) r functions
φ
^{1}
,
φ
^{2}
, · · · ,
φ^{r}
form an orthonormal bases of the space
V
_{0}
. Therefore, {
V_{j}
}
_{j}
_{≥0}
is an orthonormal multiresolution approximation of
L
^{2}
[0, 1] of multiplicity r.
V
_{0}
⊂
V
_{1}
, we denote
W
_{0}
the orthonormal complement of
V
_{0}
in
V
_{1}
,
V
_{1}
=
V
_{0}
⊕
W
_{0}
, then an orthonormal basis {
ψ
^{1}
, · · · ,
ψ^{r}
} of the space
W
_{0}
can be obtained by the Gram-Schmidt process: for
m
= 1, 2, · · · ,
r
we define inductively
For instance, when
r
= 3, we take Legendre polynomials
By formula (2.2), we have the multiscaling functions on their support [0, 1]
and through formula (2.10) the multiwavalet functions can be constructed as following
be the vector of the multiwavelet functions:
and denote for integers
j
≥ 0 and
k
= 0, 1, · · · , 2
^{j}
− 1
the dilates and translates of the multiwavelet functions, where
is supported on the interval [2
^{−j}
k
, 2
^{−j}
(
k
+ 1)] ⊂ [0, 1]. Let
ℓ
= 2
^{j}
− 1, and
then
W_{j}
is the orthonormal complement of
V_{j}
in
V_{j}
_{+1}
:
V_{j}
_{+1}
=
V_{j}
⊕
W_{j}
, so we inductively obtain the decomposition
and
Hence
is an orthonormal multiwavelet basis of
L
^{2}
[0, 1], and all elements of the vector
form another orthonormal bases of the space
V_{j}
(see (2.8)-(2.9)).
For any
f
(
x
) ∈
L
^{2}
[0, 1], we have
the projection of
f
(
x
) in
V_{n}
is
which can also be expanded by the orthonormal basis Փ
_{n}
(
x
) defined on (2.8):
where (
ℓ
= 2
^{n}
^{−1}
− 1,
ȷ
= 2
^{n}
− 1)
Since
ψ^{m}
(1 ≤
m
≤
r
) is orthogonal to {
φ
^{1}
,
φ
^{2}
, · · · ,
φ^{r}
},which is equivalent to the basis {1,
x
,
x
^{2}
, · · · ,
x
^{r−1}
}, the first
r
moments of
ψ^{m}
vanish:
Thus, if
f
(
x
) ∈
C^{r}
[0,1], then
[2]
T_{j}
between the two orthonormal basis Փ
_{j}
(
x
) and Ψ
_{j}
(
x
) of the space
V_{j}
defined on (2.8) and (2.15) respectively, such that
Rewrite formula (2.10) as
From the formula (2.6) and
V
_{0}
⊥
W
_{0}
one has
ψ^{m}
(1 −
x
) = (−1)
^{m}ψ^{m}
(
x
),
m
= 1, 2, · · · , then for
m
= 1, 2, · · · ,
r
Define matrices
and lower triangular matrices
then
The second equation in (2.21) is usually called dilation equation that leads to
or in its matrix form
where
Finally, from (2.15),(2.21), and (2.22) we obtain (by using induction)
From (2.11) we have
From the derivative formula (2.5) and the formula (2.7) it is readily seen that
Define the coefficients
then
where
L
is a
r
×
r
matrix and
vector,
Dilating and translating
we get
Let
then
where
P
is a (
ℓ
+ 1) × (
ℓ
+ 1) block matrix and
a (
ℓ
+ 1) × 1 block matrix,
Now we define
as the anti-derivative of Ψ
_{n}
(
x
):
then by (2.20) and (3.8)
Remark 3.1.
The matrix
P
in (3.9) is called operational matrix of integration for an approximate formula
[13]
. Therefore, (3.8) gives the exact definition of the operational matrix of integration.
and by (2.20) one has
so the integrals of the products of two basis functions are obtained. Next we want to evaluate the integrals of the products of three basis functions
For this purpose, we show that the product
φ^{m}
(
x
)
φ^{i}
(
x
) of two multiscaling functions can be expressed as a linear combination of finite number of functions among {
φ^{m}
(
x
),
m
≥ 1}. Note that
φ
_{1}
(
x
) = 1, one has
By defining the coefficients
the recurrence relation (2.4) can be rewritten as (note that
Multiplying this equation with
φ^{i}
(
x
) we obtain
Exchanging the index
m
and
i
we obtain
Then equalling right hand sides of above two equations, we get
This means if all products {
φ^{m}
(
x
)
φ
^{i−1}
(
x
),
m
≥
i
− 1} and {
φ^{m}
(
x
)
φ^{i}
(
x
),
m
≥
i
} are linear combinations of finite number of functions among {
φ^{m}
(
x
),
m
≥ 1}, then by induction all products {
φ^{m}
(
x
)
φ
^{i+1}
(
x
),
m
≥
i
+ 1} are also linear combinations of finite number of functions among {
φ^{m}
(
x
),
m
≥ 1}. Starting from (4.2) and (4.4), we inductively deduce the following equation
where the coefficients for
i
= 1, 2 are from (4.2) and (4.4)
and the coefficients for
i
≥ 2 are
Inserting (4.5) into (4.1), the integrals of the products of three basis functions are obtained:
Other integrals of the products of three basis functions can be obtained by combining above formula with one of the formula (2.20),(3.8) and (3.12).
with either one of the boundary conditions:
We assume that
f
∈
L
^{2}
[0, 1], the coefficients
p
(
x
) and
q
(
x
) are continuously differentiable in
I
= [0, 1] with
Let
H^{s}
(
I
) denote the standard Sobolev space with the norm ∥ · ∥
_{s}
and seminorm | · |
_{s}
given by
We define
It is well-known that the seminorm | · |
_{1}
is a norm in these two spaces and is equivalent to the norm ∥ · ∥
_{1}
.
where
a
(·, ·) is a bilinear form defined by
Clearly
a
(·, ·) is continuous and coercive on
It is well-known that, by Lax-Milgram lemma, (5.3) admits a unique weak solution
u
∈
H
^{1}
(
I
).
Now we apply Legendre multiwavelets to Galerkin methods for solving (5.3). According to the boundary condition, we construct a finite dimensional space
S_{n}
, then we solve numerically the Galerkin projection of the solution
u
on
S_{n}
defined by
It is also clear that (5.5) admits a unique solution
u_{n}
∈
S_{n}
such that
[14]
By (3.11) the elements of the vector
and a part of the functions of the set
where
We have
Lemma 5.1.
The three sets
{
q
^{2}
(
x
), · · · ,
q^{r}
(
x
)} ∪Ξ, {
q
^{1}
(
x
), · · ·,
q^{r}
(
x
)}∪Ξ,
and
{
q
^{1}
(
x
), · · ·,
q^{r}
(
x
)} ∪Ξ∪ {1}
form a basis for the spaces
respectively
.
Proof
. We only prove for space
the proofs for other two spaces are similar. First, it is easy to see that the functions of {
q
^{2}
(
x
), · · · ,
q^{r}
(
x
)} ∪ Ξ are linearly independent. Then, for any
Since
w
∈
L
^{2}
(
I
), by (2.16)-(2.17), there are numbers
such that
Since
and
we have
c
_{1}
= 0. Hence, if we take
then
and
As | · |
_{1}
is a norm on
the proof is completed.
Let the set
then
S_{n}
can be defined by its basis as
By (3.11)-(3.12) the basis of
S_{n}
consists of the elements of the vectors
respectively according to boundary conditions, where
is
T_{n}
with its first line deleted. So we also use these vectors to represent the basis of
S_{n}
.
From the proof of lemma 5.1 we know that for any
by (2.19)
This means for any
Combining (5.6) with (5.8), we have proved
Theorem 5.2.
Let u and u_{n} be the solutions of (5.3) and (5.5) respectively. If
then we have
and
v
go through all elements of
in (5.5), then we have
If we denote the matrices
then a linear system of equations is obtained:
By assumption (5.2), the coefficient matrix
A_{n}
is a symmetric and positive definite matrix. Since the basis
is an orthonormal basis of
in the sense of the inner product <
u, v
>
_{1}
:=<
u′, v′
>, it is easy to prove that the condition number of
A_{n}
is bounded:
Proposition 5.3.
For any n
≥ 0,
Proof
. see
[4]
, page 166.
Now we discuss how to solve the linear system (5.10). In practise,we expand the functions
f
(
x
),
p
(
x
),
q
(
x
) in
V_{n}
through the basis Փ
_{n}
(
x
):
By virtue of (2.19), the above approximation of
p
(
x
) and
q
(
x
) still keep their property (5.2) for sufficiently large
n
, so
A_{n}
is still a symmetric and positive definite matrix. We have (
P
is defined in (3.9))
Let
Then
D_{n}
is a block diagonal matrix by virtue of (4.1) and (4.6), thus the inverses of
D_{n}
and
B_{n}
can be computed very efficiently.
In recent years a special multilevel augmentation method (MAM) has been developed
[5]
to solve some linear system of equations arising from discretizing differential equations that requires the use of special multiscale bases and leads to an efficient, stable and accurate solver for the discrete linear system. Our basis
is just a basis of this type that meets all requirements for this MAM algorithm, but it is different from those bases in
[5]
. Besides, The MAM algorithm in an example in
[5]
can be applied to (5.10) only when
p
(
x
) =
q
(
x
) = 1 is valid. Here we propose a special form of the MAM algorithm for (5.10) as follows:
MAM algorithm for (5.10)
Let
m
_{0}
> 0 be a fixed integer.
Step 1
Solve
u
_{m0}
∈
R
^{N1}
(
N
_{1}
= 2
^{m0}
r
) from equation
A
_{m0}
u
_{m0}
=
f
_{m0}
.
Step 2
Set
u
_{m0}
,
_{0}
:=
u
_{m0}
and split the matrix
where
Step 3
For integer
m
≥ 1, suppose that
u
_{m0}
,
_{m-1}
∈
R
^{N2}
(
N
_{2}
= 2
^{m0+m-1}
r
) has been obtained and do the following:
where
and
respectively;
(ii):
Augment
u
_{m0}
,
_{m-1}
by setting
(iii):
Solve
from the algebraic equations
Noting that
D
_{m0,m}
is a block diagonal matrix,
and
T_{n}
is an orthogonal matrix, the computation of the inverse of
can be reduced to the computation of the inverse of one 2
^{m0}
r
× 2
^{m0}
r
matrix by using inverse operation for block matrices in basic linear algebra. Therefore this algorithm is very efficient. As for its accuracy, the following proposition ensures that the approximate solution
u
_{m0,m}
generated by the MAM has the same order of approximation as that of the subspaces
S_{n}
.
Proposition 5.4.
Let u and
be the solutions of (5.3) and (5.5) respectively, where u_{m0,m} is the solution of (5.10). If
hen there exists m
_{0}
≥ 1,
such that
Proof
. From the assumption (5.2) and theorem 5.2, it is obvious that the operator T of (5.1) satisfies the conditions (
H
_{1}
) − (
H
_{5}
) in
[5]
. Then the conclusion comes from theorem 2.3 of
[5]
.
^{2}
Let
L
^{2}
(Ω) be the Hilbert space equipped with the inner product
and the norm
We assume that
f
∈
L
^{2}
(Ω). Let
H
^{1}
(Ω) denotes the standard Sobolev space with the norm ∥ · ∥
_{1}
and semi-norm | · |
_{1}
given by
We define
It is well-known that | · |
_{1}
is equivalent to ∥ · ∥
_{1}
in
The variational form of (6.1) is
where
a
(·, ·) is a bilinear form defined by
Clearly
a
(·, ·) is continuous and coercive on
and (6.2) admits a unique weak solution
by Lax-Milgram lemma.
Now we apply Legendre multiwavelets to Galerkin methods to solve (6.2). Some results in section 2 can be extended to
L
^{2}
(Ω). Firstly, the tensor products {
V_{n}
[0, 1]⊗
V_{n}
[0, 1]} form a multiresolution analysis (MRA) of the space
L
^{2}
(Ω) =
L
^{2}
[0, 1] ⊗
L
^{2}
[0, 1], the elements of Փ
_{n}
(
x
) ⊗ Փ
_{n}
(
y
) and the elements of Ψ
_{n}
(
x
) ⊗ Ψ
_{n}
(
y
) are two equivalent bases of {
V_{n}
[0, 1] ⊗
V_{n}
[0, 1]}. (Here and afterward we use tensor product, or Kronecker product, of two matrices. For its definition and properties we refer readers to
[8]
). Secondly, each function
f
∈
L
^{2}
(Ω) can be approximated by
and the error of the approximation for
f
∈
C
^{r+1,r+1}
(Ω) is
i.e., the rate of convergence is of order r/2 (see
[2]
).
Following the line of section 5, we want to seek the approximate solution on a finite dimensional space
i.e., we will solve numerically the Galerkin projection
u_{n}
of the solution
u
on
S_{n}
defined by
It is easy to see that
is a basis of
S_{n}
. We suppose that
and let
in (6.5), then we have a linear system of equations
where (
P
is defined in (3.9))
This linear system can be solved by using the method of separation of variables. Since
K_{n}
is a symmetric matrix with positive eigenvalues, there exists an orthonormal matrix
G_{n}
and an invertible diagonal matrix Λ such that
then the solution of (6.6) is given by
where
H
=
I
⊗ Λ + Λ ⊗
I
+ Λ ⊗ Λ is an invertible diagonal matrix. Thus the problem of solving (6.6) is reduced to an one-dimensional eigenvalue problem for a symmetric matrix
K_{n}
, which size is (2
^{n}
r
− 1) × (2
^{n}
r
− 1).
If
then we have the error estimate
Remark 6.1.
The operator
T
_{1}
of (6.1) satisfies the conditions (
T
_{1}
) − (
T
_{5}
) in
[5]
so the MAM algorithm can be applied to solve (6.6). The elements of
must be reordered according to the standard construction of the bases of
L
^{2}
[0,1] ⊗
L
^{2}
[0,1] (see
[2]
for an outline of the construction). However, further investigations are needed to see how to split
A_{n}
of (6.6) such that the inverse of the matrix
in Step 3 of MAM algorithm in section 5 can be computed rapidly.
Example 7.1.
Two-point boundary value problem
The exact solution is
u
(
x
) = |2
x
−1|
^{3}
−1. We take
r
= 2, this means the Legendre multiwavelet basis is a linear basis. And we take
m
_{0}
= 2 to apply the special form of MAM algorithm of section 5. The numerical results are summarized in
Table 1
.
Example 7.2.
Two-point boundary value problem
The exact solution is
u
(
x
) =
sin
(
πx
). We take
r
= 3, this means the Legendre multiwavelet basis is a quadratic basis. And we take
m
_{0}
= 1 to apply the special form of MAM algorithm of section 5. The numerical results are also summarized in
Table 1
.
Numerical results for (7.1) and (7.2)
Example 7.3.
Dirichlet problem for the elliptic equation
The exact solution is
u
(
x, y
) =
sin
(
πx
)
y
(
y
− 1). We make use of linear basis (
r
= 2) which means each basis function is a product of two linear functions
g
_{1}
(
x
) and
g
_{2}
(
y
), and cubic basis (
r
= 4) respectively. The numerical results are summarized in
Table 2
.
Numerical results for (7.3)
Conclusions.
Legendre multiwavelets together with its anti-derivatives are suitable for constructing orthonormal bases to solve some boundary value problems of ODEs and PDEs using Galerkin methods. In one dimensional case, applying multilevel augmentation method leads to an efficient, stable, and accurate algorithm, the rate of convergence in Galerkin methods is of order r. In two dimensional case, it is convenient to use tensor product of Legendre multiwavelet bases to form a basis for boundary value problems on a rectangle, the rate of convergence in Galerkin methods is of order r/2, and further investigations are needed to see how to apply the MAM algorithm to solve the discrete linear system more efficiently.
Xiaolin Zhou received M.S. from Fudan University in China and Ph.D at University of Toledo in USA. He worked at Xi’an Jiaotong University from 1998 to 2006. Since 2006 he has been at Xiamen University of China. His research interests include numerical methods for differential equations and wavelet analysis.
Department of Information and Computational Sciences, Tan Kah Kee College of Xiamen University, Development Zone of Zhangzhou Investment Promotion Bureau, Fujian 363105, P.R.China.
e-mail: xlzhou@xujc.com

1. Introduction

The idea of applying wavelet bases to discretize differential equations in wavelets-Galerkin methods has been explored by many authors
[6
,
10
,
11
,
14]
. Among them, Xu and Shann
[14]
presented a thorough study of one dimensional problems. Instead of using wavelets directly, they took the anti-derivatives of wavelets as trial functions. In this way, singularity of wavelets is smoothed, the boundary condition can be treated easily. Since the introduction of multiwavelets into the numerical solution of the integral equation
[2]
, multiwavelets bases have also been applied to discretize the differential equations because some difficulties of using wavelets for the representation of differential operators may be eliminated by using multiwavelets that may possess all the following properties: orthogonality on a finite interval, symmetry, compact support without overlap and high order of vanishing moments. In recent years, Legendre wavelets and multiwavelets, which are not differentiable on [0,1], has drawn a lot of attention in this direction
[3
,
9
,
12
,
13]
. Recent developments include a special multilevel augmentation method (MAM), which was proposed by Chen, Wu and Xu
[5]
to solve some linear system of equations arising from discretizing differential equations that requires the use of special multiscale bases. Under certain conditions it leads to an efficient, stable and accurate solver for the discrete linear system. The most important point is that this MAM is not an iterative method.
In this paper we shall study Legendre multiwavelets-Galerkin methods based on variational principles. We shall take anti-derivatives of multiwavelets as trial functions as in
[14]
, and then we show that the problem of evaluating the integrals of multiwavelets (cf.
[7
,
11]
) can be resolved by using the properties of Legendre polynomials. The convergence rate of the method is given, and MAM algorithm
[5]
is applied for the two-point boundary value problem. The paper is organized as follows. Section 2 reviews the constructions of Legendre multiscaling functions and multiwavelets. Section 3 defines the anti-derivatives of Legendre multiscaling functions and multiwavelets, and a general operational matrix of integration is derived by using a derivative formula of Legendre polynomials. Section 4 discusses how to calculate the product matrix of basis functions and resolve the problem of evaluating the integrals of multiwavelets. In section 5 we use the antiderivatives of Legendre multiwavelets as trial functions in the Galerkin methods for the two-point boundary value problems of ODEs, and propose a special form of MAM algorithm to solve the discrete linear system of equations. In section 6 we use the tensor product of Legendre multiwavelet basis to solve a Dirichlet problem for the elliptic equation on a rectangle. Finally, section 7 presents some numerical examples and conclusion.
2. Legendre multiwavelets

Let
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- 2.1. Legendre multiscaling functions

The Legendre polynomial is given by the following Rodrigues formula:
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- (i) orthogonality:

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- (ii) recurrence relation:

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- (iii) derivative formula:

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- (iv) symmetry(or asymmetry) aboutx= 0:Pm(−x) = (−1)mPm(x);
- (v) function values at endpoints:Pm(1) = 1,Pm(−1) = (−1)m.

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- (i) orthonormality:

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- (ii) recurrence relation:

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- (iii) derivative formula:

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- (iv) symmetry(or asymmetry) about

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- (v) function values at endpoints:

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- 2.2. Legendre multiwavelet functions

Since
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- 2.3. Legendre multiwavelet basis

Let
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- 2.4. The transformation matrix between two bases

Now we give out the transformation matrix
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3. Anti-derivatives of Legendre multiwavelets

Let
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4. Evaluation of the connection coefficients

The integrals of the products of several basis functions are called the connection coefficients
[7
,
11]
. By orthonormality of the multiscaling functions and multiwavelet functions one has
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5. Applications to two-point boundary value problems

In this section, we apply Legendre multiwavelets to numerical solutions of a two-point boundary value problem of ODE
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- 5.1. Error estimates

The variational form of (5.1) is
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- 5.2. The solution of the linear system of equations

In this subsection we suppose that Dirichlet condition is imposed on the boundary. Let
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- (i):Augment the matricesto form

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6. Application to Dirichlet problem for the elliptic equation on a rectangle

In this section we discuss the application of Legendre multiwavelets to numerical solution of the Dirichlet problem for the elliptic equation on the rectangle Ω = [0, 1]
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7. Numerical Examples and Conclusions

We present three numerical examples.
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Numerical results for (7.1) and (7.2)

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Numerical results for (7.3)

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Acknowledgements

The author wishes to express the sincere thanks to the anonymous referees for valuable suggestions that helped improve the quality of this paper.

BIO

Abramowitz M.
,
Stegun I. A.
Handbook of Mathematical Functions
Applied Mathematics Series
(Natl. Bur. of Standards, Washington, DC, 1972)
55

Albert B. K.
(1993)
A class of bases in L2 for the sparse representation of integral operators
SIAM J. Math. Anal.
24
(1)
246 -
262
** DOI : 10.1137/0524016**

Alpert B.
,
Beylkin G.
,
Gines D.
,
Vozovoi L.
(2002)
Adaptive Solution of Partial Differential Equations in Multiwavelet Bases
Journal of Computational Physics
182
149 -
190
** DOI : 10.1006/jcph.2002.7160**

Chen Z
,
Wu B
(2007)
Wavelet Analysis
Science Press
Beijing
166 -

Chen Z
,
Wu B
,
Xu Y
(2006)
Multilevel augmentation methods for differential equations
Advances in computational mathematics
24
213 -
238
** DOI : 10.1007/s10444-004-4092-6**

Dahmen W.
,
Kunoth A.
,
Urban K.
(1996)
A Wavelet-Galerkin method for the Stokes-equations
Computing
56
(3)
259 -
302
** DOI : 10.1007/BF02238515**

Dahmen W.
,
Micchelli C.A.
(1993)
Using the refinement equation for evaluating the integrals of wavelets
SIAM J. Num. Anal.
30
507 -
537
** DOI : 10.1137/0730024**

Horn R. A.
,
johnson C. R.
1991
Topics in Matrix Analysis
Cambridge University Press
Cambridge

Ling Jie
(1998)
Regularization-wavelet method for solving the integral equation of the first kind and its numerical experiments
Numerical mathematics, a journal of Chinese universities
12
(3)
158 -
179

Lin En-Bing
,
Zhou Xiaolin
(1997)
Coiflet interpolation and approximate solutions of elliptic partial differential equations
Numerical methods for partial differential equations
13
303 -
320

Lin E. B.
,
Zhou X.
(2001)
Connection coefficients on an interval and wavelet solutions of Burgers equation
J Comput Appl Math
135
(1)
63 -
78
** DOI : 10.1016/S0377-0427(00)00562-8**

Liu Nanshan
,
Lin En-Bing
(2010)
Legendre wavelet method for numerical solutions of partial differential equations
Numerical methods for partial differential equations
26
81 -
94
** DOI : 10.1002/num.20417**

Mohammadi F.
,
Hosseini M.M.
,
Mohyud-Din Syed Tauseef
(2011)
Legendre wavelet Galerkin method for solving ordinary differential equations with non-analytic solution
International Journal of Systems Science
42
(4)
579 -
585
** DOI : 10.1080/00207721003658194**

Xu J. S.
,
Shann W. C.
(1992)
Galerkin-wavelet methods for two-point boundary value problems
Numerische Mathematik
63
123 -
144
** DOI : 10.1007/BF01385851**

Citing 'LEGENDRE MULTIWAVELET GALERKIN METHODS FOR DIFFERENTIAL EQUATIONS†
'

@article{ E1MCA9_2014_v32n1_2_267}
,title={LEGENDRE MULTIWAVELET GALERKIN METHODS FOR DIFFERENTIAL EQUATIONS†}
,volume={1_2}
, url={http://dx.doi.org/10.14317/jami.2014.267}, DOI={10.14317/jami.2014.267}
, number= {1_2}
, journal={Journal of Applied Mathematics & Informatics}
, publisher={Korean Society of Computational and Applied Mathematics}
, author={ZHOU, XIAOLIN}
, year={2014}
, month={Jan}