# Construction of Interval Wavelet Based on Restricted Variational

Authors: Shu-Li Mei; Hong-Liang Lv; Qin Ma
Journal: Mathematical Problems in Engineering
 (2008)

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## Abstract

Based on restricted variational principle, a novel method for interval wavelet construction is proposed. For the excellent local property of quasi-Shannon wavelet, its interval wavelet is constructed, and then applied to solve ordinary differential equations. Parameter choices for the interval wavelet method are discussed and its numerical performance is demonstrated.


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## Body

## 1. Introduction

Since the definition domain of wavelet transformation is an
infinite interval, the boundary effect would occur when applied for resolving
the engineering problems with bounded interval, for example, ordinary differential equations (ODEs). Consequently,
it will decrease the precision and computational efficiency of the solution. Nevertheless,
the boundary effect can be eliminated effectively by constructing an interval
wavelet using numerical methods.There are several ways available to construct an interval
wavelet. In general, the construction method is relative to the wavelet
function, that is, different interval wavelet has different construction
method. A simple solution is the even 2-periodical extensionf˜ of function f:[0,1]→ℝ,
which is usually used in image analysis. Unfortunately, this extension
generally produces discontinuities at the integers that are indicated by the
large wavelet coefficients near the endpoints 0 and 1. Thus, the constructed
wavelet cannot exactly analyze the boundary behavior of a given function. To
solve this problem, the popular method is based on special boundary and
interior scaling functions as well as wavelets to reduce the numerical problem
at the boundaries [1, 2].The aim of this paper is to introduce a general construction
method of interval wavelet based on the restricted variational principle. As an
example, the quasi-Shannon wavelet is firstly introduced in brief. Then, its
corresponding interval wavelet is constructed in detail based on the restricted
variational principle [3]. After that, the wavelet collocation method is
applied to obtain the interval wavelet discrete formulation of ODEs. The
performance of the quasi-Shannon interval wavelet is illustrated by comparing
the numerical results of Shannon, quasi-Shannon, and Shannon interval wavelets on a convection equation.


## 2. Quasi-Shannon Wavelet

The Shannon scaling function is smooth, this means that the function and all of its
derivatives exist and are continuous. But the Shannon function does not have compact support. In order to improve the localized and
asymptotic behavior of the Shannon scaling function, Wei [4] introduced
a regularization factor R(x) as follows:(2.1)w(x)=ϕ(x)R(x),ϕ(x)=sin(πx)πx. The advantage of the regularized Shannon scaling function is that its Fourier transformation is continuous [4],
resulting in an excellent local property. A common and important regularization
factor R(x) proposed by Wei [4] is the Gaussian function(2.2)Rσ(x)=exp(−x22σ2),σ>0, where σ is the width parameter (or called window
size). In practice, the best results are usually obtained, when σ=rΔ [4] (r is a parameter chosen in computations, Δ is the size of cell in discrete mesh). Substituting
(2.2) into (2.1), the Gaussian regularized orthogonal sampling scaling function can
be obtained as following:(2.3)w(x)=sin(πx)πxexp(−x22σ2),σ>0. The corresponding discrete formula is(2.4)w(x−xi)=sin(π/Δ)(x−xi)(π/Δ)(x−xi)exp(−(x−xn)22σ2).


## 3. Construction of Interval Wavelet Based on Restricted Variational Principle

### 3.1. Restricted Variational Principle

Defining a scalar fonctionelle∏ of u as(3.1)∏=∫ΩF(u,∂u∂x,…)dΩ+∫ΓE(u,∂u∂x,…)dΓ, where u is an unknown function, both F and E are specific operators, Ω is the definition domain, and Γ is the boundary of Ω.
The solution of a continuous medium problem is a stagnation point of the
functional equation ∏,
that is, its variation is equal to zero(3.2)δ∏=0. The method described above for solving a continuous medium
problem is called restricted variational principle [3].The unknown functionu could be approximately expressed as(3.3)u≈u˜=∑i=1nNiai=Na, where ai is the parameters to be determined and Ni is the trial function.If the unknown functionu is restricted by additional condition C(u)=0 in definition domain Ω,
a revised fonctionelle ∏* could be constructed as follows:(3.4)∏*=∏+∫ΩλTC(u)dΩ, where λ is called Lagrange multiplier, which is a
function vector with independent coordinates in Ω. The functions u and λ are two unknown variables within the revised
fonctionelle. Similarly as (3.3), λ could be approximately expressed by trial function N˜i as(3.5)λ˜=∑N˜ibi=N˜b. Let the variation of the revised fonctionelle equal to zero,
and a system of equations could be obtained as follows:(3.6)δ∏*δc={δ∏*δaδ∏*δb}=0,c={ab}. The values of a and b could be obtained by solving the above system
of equations.Let the Euler equation of the fonctionelle∏ is(3.7)A(u)=0, and the additional condition is(3.8)C(u)=L1(u)+C1=0. Substituting (3.5), (3.7), and (3.8) into (3.4), the following
function could be obtained:(3.9)δ∏*=δaT∫ΩNTA(u˜)dΩ+δbT∫ΩN˜[L1(u˜)+C1]dΩ+δaT∫ΩL1T(N)λ˜dΩ=0. Since (3.9) is satisfied for all variations of δa and δb,
the following two equations can be derived:(3.10)∫ΩNTA(u˜)dΩ+∫ΩL1T(N)λ˜dΩ=0,(3.11)∫ΩN˜T[L1(u˜)+C1]dΩ=0. The first item in (3.10) is the approximate equation of the
natural variation of the linear system of equations:(3.12)Ka=P. Thus, (3.10) and
(3.11) could be rewritten in the following form: (3.13)K0C+R=[KGGT0]{ab}-{PQ}=0, where(3.14)K=∫ΩNTA(N)dΩ,GT=∫ΩN˜TL1(N)dΩ,Q=∫ΩN˜TC1dΩ.


### 3.2. Construction of Quasi-Shannon Interval Wavelet

Considering a one-dimensional problemu(x),
whose domain of definition Ω is [m,n].
Assuming that the number of discrete point is 2j+1,j∈Z,
the ith discrete point of variable x could be written as(3.15)xi=m+n−m2j⋅i,i∈Z. If the quasi-Shannon wavelet with interpolation property is
employed as the trial function Nj,
that is,(3.16)Nj=wj(x−xi)=sin(2jπ/(n−m))(x−xi)(2jπ/(n−m))(x−xi)exp(−22j−1(x−xi)2r2(n−m)2).Any element in the matrices K and GT could be expressed, respectively, as(3.17)Kk,n=A(wj(k−n)),(3.18)Gk,nT=∫ΩN˜kTL1(Nn)dΩ, where N˜k is the Lagrange basic function.As the definition domain of wavelet
transformation is a double infinite interval, the wavelet coefficients are
large near the endpoints of the bounded signal, which increases the
computational error. To obtain a high computational precision, the additional
condition of scalar fonctionelle in boundaryΓ could be changed to(3.19)u=u¯. Thus,(3.20)L1(Nn)=∑iNiu¯i, where u¯i is a known value. Recomputing (3.13) with this
additional condition, it could be rewritten as follows:(3.21)K0C+R=[KGGT0]{ab}-{P0}=0. Correspondingly, (3.18) changed to the
form(3.22)Gk,nT=N˜kTNn. The interpolation properties of N˜ and N imply that a and b are the vectors consisting of the approximate values
of u in discrete points xi and extended discrete points outside the
domain of [m,n],
respectively. Typically, periodic, symmetry, and zero extension methods could
be employed. Assuming that there are R=2j+1 discrete points in [m,n],
that is, x0,x1,…,x2j,
and L extended points in both outsides of the domain [m,n],
respectively, that is, x−L−1,x−L,…,x−1 and xR,xR+1,…,xR+L,
the function u can be extended outside of the definition
domain using the symmetry method, period method, or the zero extension method.
Here, we take the symmetry method as example to illustrate the interval wavelet
construction method as  follows: (3.23)u(x−1)=u(x0)u(x−2)=u(x1)⋮u(x−1−i)=u(xi)⋮u(x−L−1)=u(xL),u(xR)=u(xR−1)u(xR+1)=u(xR−2)⋮u(xR+i)=u(xR−i−1)⋮u(xR+L)=u(xR−L−1)
Substituting (3.23) into (3.21) results
in(3.24)[G1AG2]{u−L−1u−L⋮u−1u0u1⋮u2juRuR+1⋮uR+L}={P'0}, where(3.25)A=[A(wj(0−0))A(wj(0−1))⋯A(wj(0−2j))A(wj(1−0))A(wj(1−1))⋯A(wj(1−2j))⋮⋮⋮⋮A(wj(2j−0))A(wj(2j−1))⋯A(wj(2j−2j))],G1=[a−L−1,−L−1wj(0)a−L−1,−Lwj(−1)⋯a−L−1,−1wj(−L)a−L,−L−1wj(1)a−L,−Lwj(0)⋯a−L,−1wj(1−L)⋮⋮⋮⋮a−1,−L−1wj(L)a−1,−Lwj(L−1)⋯a−1,−1wj(0)],G2=[bR,Rwj(0)bR,R+1wj(−1)⋯bR,R+Lwj(−L)bR+1,Rwj(1)bR+1,R+1wj(0)⋯bR+1,Rwj(1−L)⋮⋮⋮⋮bR+L,Rwj(L)bR+L,R+1wj(L−1)⋯bR+L,R+Lwj(0)].
In the matrices G1 and G2, ank and bnk could be calculated, respectively, as (3.26)ank=ljk1(xjn),bnk=ljk2(xjn),(3.27)lj,k1=∏i=−L−1i≠k−1x−xj,ixj,k−xj,i,lj,k2=∏i=2j+1i≠k2j+1+Lx−xj,ixj,k−xj,i. According to GTb=0 and the relational expression (3.23), (3.24) could
be condensed in the following way:(3.28)[G1+A11A12A13A21A22A23A31A32G2+A33]{u0u1⋮u2j}={P0}. Subsequently, as interval
interpolation basic functions, the quasi-Shannon interval wavelet can be
obtained from (3.28) as follows:(3.29)wjk=w(2jx−k)+∑n=−N+1−1ankw(2jx−n),k=0,…,L,wjk=w(2jx−k),k=L+1,…,2j−L−1,wjk=w(2jx−k)+∑n=2j+12j+N−1bnkw(2jx−n),k=2j−L,…,2j, where L is the number of the external points, N is the support domain of the wavelet function,
that is, supϕ=[−N,N].It is easy to know that the quasi-Shannon interval wavelet is
a linear combination of the quasi-Shannon scaling functionwj,k(x).
Therefore, the quasi-Shannon interval wavelet function possesses all the
properties of the quasi-Shannon scaling function.It should be noted that the interval
wavelet function (3.29) is similar to the result obtained in [5] using different
methods. But we can construct different interval
wavelets possessed similar format based on different extension technologies by
this method. Otherwise, this method reveals the close relationship between the
restricted variational principle and the interval interpolation wavelet.


## 3.1. Restricted Variational Principle

Defining a scalar fonctionelle∏ of u as(3.1)∏=∫ΩF(u,∂u∂x,…)dΩ+∫ΓE(u,∂u∂x,…)dΓ, where u is an unknown function, both F and E are specific operators, Ω is the definition domain, and Γ is the boundary of Ω.
The solution of a continuous medium problem is a stagnation point of the
functional equation ∏,
that is, its variation is equal to zero(3.2)δ∏=0. The method described above for solving a continuous medium
problem is called restricted variational principle [3].The unknown functionu could be approximately expressed as(3.3)u≈u˜=∑i=1nNiai=Na, where ai is the parameters to be determined and Ni is the trial function.If the unknown functionu is restricted by additional condition C(u)=0 in definition domain Ω,
a revised fonctionelle ∏* could be constructed as follows:(3.4)∏*=∏+∫ΩλTC(u)dΩ, where λ is called Lagrange multiplier, which is a
function vector with independent coordinates in Ω. The functions u and λ are two unknown variables within the revised
fonctionelle. Similarly as (3.3), λ could be approximately expressed by trial function N˜i as(3.5)λ˜=∑N˜ibi=N˜b. Let the variation of the revised fonctionelle equal to zero,
and a system of equations could be obtained as follows:(3.6)δ∏*δc={δ∏*δaδ∏*δb}=0,c={ab}. The values of a and b could be obtained by solving the above system
of equations.Let the Euler equation of the fonctionelle∏ is(3.7)A(u)=0, and the additional condition is(3.8)C(u)=L1(u)+C1=0. Substituting (3.5), (3.7), and (3.8) into (3.4), the following
function could be obtained:(3.9)δ∏*=δaT∫ΩNTA(u˜)dΩ+δbT∫ΩN˜[L1(u˜)+C1]dΩ+δaT∫ΩL1T(N)λ˜dΩ=0. Since (3.9) is satisfied for all variations of δa and δb,
the following two equations can be derived:(3.10)∫ΩNTA(u˜)dΩ+∫ΩL1T(N)λ˜dΩ=0,(3.11)∫ΩN˜T[L1(u˜)+C1]dΩ=0. The first item in (3.10) is the approximate equation of the
natural variation of the linear system of equations:(3.12)Ka=P. Thus, (3.10) and
(3.11) could be rewritten in the following form: (3.13)K0C+R=[KGGT0]{ab}-{PQ}=0, where(3.14)K=∫ΩNTA(N)dΩ,GT=∫ΩN˜TL1(N)dΩ,Q=∫ΩN˜TC1dΩ.


## 3.2. Construction of Quasi-Shannon Interval Wavelet

Considering a one-dimensional problemu(x),
whose domain of definition Ω is [m,n].
Assuming that the number of discrete point is 2j+1,j∈Z,
the ith discrete point of variable x could be written as(3.15)xi=m+n−m2j⋅i,i∈Z. If the quasi-Shannon wavelet with interpolation property is
employed as the trial function Nj,
that is,(3.16)Nj=wj(x−xi)=sin(2jπ/(n−m))(x−xi)(2jπ/(n−m))(x−xi)exp(−22j−1(x−xi)2r2(n−m)2).Any element in the matrices K and GT could be expressed, respectively, as(3.17)Kk,n=A(wj(k−n)),(3.18)Gk,nT=∫ΩN˜kTL1(Nn)dΩ, where N˜k is the Lagrange basic function.As the definition domain of wavelet
transformation is a double infinite interval, the wavelet coefficients are
large near the endpoints of the bounded signal, which increases the
computational error. To obtain a high computational precision, the additional
condition of scalar fonctionelle in boundaryΓ could be changed to(3.19)u=u¯. Thus,(3.20)L1(Nn)=∑iNiu¯i, where u¯i is a known value. Recomputing (3.13) with this
additional condition, it could be rewritten as follows:(3.21)K0C+R=[KGGT0]{ab}-{P0}=0. Correspondingly, (3.18) changed to the
form(3.22)Gk,nT=N˜kTNn. The interpolation properties of N˜ and N imply that a and b are the vectors consisting of the approximate values
of u in discrete points xi and extended discrete points outside the
domain of [m,n],
respectively. Typically, periodic, symmetry, and zero extension methods could
be employed. Assuming that there are R=2j+1 discrete points in [m,n],
that is, x0,x1,…,x2j,
and L extended points in both outsides of the domain [m,n],
respectively, that is, x−L−1,x−L,…,x−1 and xR,xR+1,…,xR+L,
the function u can be extended outside of the definition
domain using the symmetry method, period method, or the zero extension method.
Here, we take the symmetry method as example to illustrate the interval wavelet
construction method as  follows: (3.23)u(x−1)=u(x0)u(x−2)=u(x1)⋮u(x−1−i)=u(xi)⋮u(x−L−1)=u(xL),u(xR)=u(xR−1)u(xR+1)=u(xR−2)⋮u(xR+i)=u(xR−i−1)⋮u(xR+L)=u(xR−L−1)
Substituting (3.23) into (3.21) results
in(3.24)[G1AG2]{u−L−1u−L⋮u−1u0u1⋮u2juRuR+1⋮uR+L}={P'0}, where(3.25)A=[A(wj(0−0))A(wj(0−1))⋯A(wj(0−2j))A(wj(1−0))A(wj(1−1))⋯A(wj(1−2j))⋮⋮⋮⋮A(wj(2j−0))A(wj(2j−1))⋯A(wj(2j−2j))],G1=[a−L−1,−L−1wj(0)a−L−1,−Lwj(−1)⋯a−L−1,−1wj(−L)a−L,−L−1wj(1)a−L,−Lwj(0)⋯a−L,−1wj(1−L)⋮⋮⋮⋮a−1,−L−1wj(L)a−1,−Lwj(L−1)⋯a−1,−1wj(0)],G2=[bR,Rwj(0)bR,R+1wj(−1)⋯bR,R+Lwj(−L)bR+1,Rwj(1)bR+1,R+1wj(0)⋯bR+1,Rwj(1−L)⋮⋮⋮⋮bR+L,Rwj(L)bR+L,R+1wj(L−1)⋯bR+L,R+Lwj(0)].
In the matrices G1 and G2, ank and bnk could be calculated, respectively, as (3.26)ank=ljk1(xjn),bnk=ljk2(xjn),(3.27)lj,k1=∏i=−L−1i≠k−1x−xj,ixj,k−xj,i,lj,k2=∏i=2j+1i≠k2j+1+Lx−xj,ixj,k−xj,i. According to GTb=0 and the relational expression (3.23), (3.24) could
be condensed in the following way:(3.28)[G1+A11A12A13A21A22A23A31A32G2+A33]{u0u1⋮u2j}={P0}. Subsequently, as interval
interpolation basic functions, the quasi-Shannon interval wavelet can be
obtained from (3.28) as follows:(3.29)wjk=w(2jx−k)+∑n=−N+1−1ankw(2jx−n),k=0,…,L,wjk=w(2jx−k),k=L+1,…,2j−L−1,wjk=w(2jx−k)+∑n=2j+12j+N−1bnkw(2jx−n),k=2j−L,…,2j, where L is the number of the external points, N is the support domain of the wavelet function,
that is, supϕ=[−N,N].It is easy to know that the quasi-Shannon interval wavelet is
a linear combination of the quasi-Shannon scaling functionwj,k(x).
Therefore, the quasi-Shannon interval wavelet function possesses all the
properties of the quasi-Shannon scaling function.It should be noted that the interval
wavelet function (3.29) is similar to the result obtained in [5] using different
methods. But we can construct different interval
wavelets possessed similar format based on different extension technologies by
this method. Otherwise, this method reveals the close relationship between the
restricted variational principle and the interval interpolation wavelet.


## 4. Interval Wavelet Numerical Method for Ordinary Deferential Equations (ODEs)

### 4.1. Interval Wavelet Discrete Formulation of
ODEs [6, 7]

Considering the following ODE:(4.1)A(x)u′′(x)+B(x)u′(x)+C(x)u(x)=f(x),x∈[a,b]. According to the collocation method, its approximate solution
can be expressed as(4.2)uj(x)=∑n=02juj(xn)w(2jx−n),n=0,1,2,…,2j. Substituting (4.2) into (4.1), the system of algebraic
equations can be obtained as follows:(4.3)∑n=02juj(xn)[A(xk)22jw′′(k−n)+B(xk)2jw′(k−n)+C(xk)w(k−n)]=f(xk),k=0,1,2,…,2j. The solution of (4.3) is an approximate one of (4.1) at the
collocation point xk.
Equation (4.3) could be rewritten in the new form below as follows:(4.4)MU=F, where U is the vector of solution, F is right side vector, and M is the stiffness matrix. Any element of the
matrix M could be expressed as follows:(4.5)mk,n=A(xk)22jw′′(k−n)+B(xk)2jw′(k−n)+C(xk)w(k−n), where (4.6)w′(k−n)={2jcos[π(k−n)]exp[−(k−n)2/(2r2)](k−n)(b−a),k≠n0,k=n,w′′(k−n)={−22j+1cos[π(k−n)]exp[−(k−n)2/2r2](b−a)2[1(k−n)2+1r2],k≠n−22j(3+π2r2)3r2(b−a)2,k=n.   In the solution of the above differential equations, there are several types of boundary conditions commonly
encountered, such as Dirichlet and Neumann boundary conditions. For simple, we only consider the Dirichlet boundary in the
following discussions.Dirichlet boundary conditions of differential
equations specify the value of function on a surfaceS:(4.7)u(x)|S=β(x),x∈S. That is, the solution u(x) on the boundary S is known. Assuming that the definition
interval of the solution is [0,1],
the approximate solution will be(4.8)uj(x)=uj(0)w(2jx)+∑n=12j−1uj(xn)w(2jx−n)+uj(1)w(2jx−2j). Substituting (4.8) into (4.4), the system of
algebraic equations satisfying the boundary conditions could be obtained.


### 4.2. Numerical Example

Considering the following two-point boundary
problem as a convection equation,(4.9)−εu′′(x)+u′(x)+u(x)=1,0<x<1,u(0)=u(1)=0. Its analytical solution as shown in
Figure 1 is(4.10)u(x)=1−eλ2eλ2−eλ1eλ1x−1−eλ1eλ2−eλ1eλ2x+1, where λ1 and λ2 are Eigenvalues:(4.11)λ1=1+1+4ε2ε,λ2=1−1+4ε2ε. Obviously, the boundary layer phenomenon is
evident near the boundary x=1, which
would increase with the decrease of the parameter ε.
This could be used to test the effectiveness of the interval wavelet.Figure 1
Analytical
solution of the convection (4.9).Substituting (4.5) into (4.9) results in its
wavelet discrete formulation as the following: (4.12)∑n=12j−1uj*(xn)[−22jεw′′(k−n)+2jw′(k−n)+w(k−n)]=1,k=1,2,…,2j-1. The
weight function w in (4.12) can be set as the Shannon, the
quasi-Shannon, or the interval wavelet functions. Their numerical solutions and
corresponding errors are shown in Figure 2, where the count of collocation
points is 65, and the points are distributed evenly in space. Obviously, the
calculation precision of the interval wavelet is higher than others.The comparison of the Shannon (a), the
quasi-Shannon (b), and the interval wavelet (c), and their errors of numerical
results (e), (f), and (g), respectively.
(a)
Shannon wavelet(b)
Quasi-Shannon wavelet(c)
Interval wavelet(d)(e)(f)Besides, the regularized width parameterσ which has been discussed in [8], the
parameters L,ε,andj affect the calculation precision significantly.
Therefore, appropriate values should be selected when implementing the numerical
method developed.


#### 4.2.1.  Total Number of External Collocation PointsL

In fact, the definition of interval wavelet in
(3.29) indicates that the values of u in external collocation points are obtained by
Lagrange interpolation method. So, the parameter L is relative with the smoothness of the solution, which is
difficult to be determined. With the increasing of L,
the Runge phenomena could occur in the Lagrange interpolation [9]. As a result,
it could decrease the computational precision, which is proved by the numerical
results shown in Table 1. When L=1,
the maxima of both the relative error and the absolute error are apparently
less than that of the other two methods. However, opposite results were
obtained, when L=3.
Actually, the solution of ODE with the Dirichlet boundary condition is not
smooth on the boundary in most cases, and so the bigger L cannot make any contribution to improve the computational
precision except increasing the time complexity of the numerical method. Consequently, L=1∼3  might be the best choice.Table 1
Influence of the total
number of external collocation pointsL(j=6,ε=1).
LMaximum of absolute error(10−3)Maximum of relative error(10−1)Quasi-waveletWaveletQuasi-waveletWavelet10.2939090.2311540.3967740.21403420.5015392.8244440.6123043.44822131.2575812.6414091.5353173.85599142.0801912.0205582.5396011.273226


#### 4.2.2. Parameterε

Table2 shows the influence of parameter ε on the calculation precision. It can be
observed that the solution is smooth as ε=1.
Although the quasi-Shannon wavelet has local property, its computational
precision is lower than that of the Shannon wavelet without local property. With ε decreasing, the superiority of quasi-Shannon
wavelet method to Shannon wavelet method
exhibits gradually. When ε=0.001,
the maximum of the relative error of the quasi-Shannon wavelet method is only
0.33, while that of Shannon wavelet method is
as high as 26.6. The precision of the interval wavelet method keeps higher than
the other two methods in the whole process. When j=6 and ε≥0.1,
the maxima of both the relative error and the absolute error of the interval
wavelet method are the lowest among the three methods.Table 2
Influence of parameterε.
jεMaximum of absolute error(10−3)Maximum of relative error(10−1)WaveletQuasi-waveletInterval waveletWaveletQuasi-waveletInterval wavelet6100.1069140.1116780.0289171.3803311.4417190.37694611.1236561.1742480.2939091.3783191.4392170.3967740.111.37610511.9413662.0100261.4202791.4834180.6241550.01222.14112823.40834757.4687845.0433181.9729641.8879830.001412.488802207.321583411.521362266.3243723.3120586.5742457100.0532600.0565180.0140501.3637061.4248850.36157610.5630710.5884910.1459691.3627341.4236540.3716810.16.0852026.3746301.3221551.3860661.4464750.4915320.0131.34584133.2978241.5657851.3476941.7402751.3628280.001164.839419151.45704533.225063185.3495152.5525345.2854458100.0265820.0277790.0069151.3555281.41656670.35355010.2818600.2945930.0726721.3550511.41595660.3586530.13.1466383.2926530.7457931.3672551.4275460.4202380.0123.22431824.5353147.9647101.4744501.5880040.9476080.00156.19513563.509684206.24791657.7476682.3382093.339211100.00131.43806233.34413823.7640240.7985181.8074921.523272However, when the parameterε further decreases, the relative error of the
interval wavelet method is slightly larger than that of quasi-Shannon wavelet
method, and the absolute error of the interval wavelet method approximately
identical to that of quasi-Shannon wavelet method. This is due to the boundary
layer phenomena of the two-point boundary problem appearing at the points
adjacent to the boundary x=1.
The boundary layer phenomena result in the large gradient of solution, and thus
the Gibbs phenomena [10] are unavoidable. Since the construction of the
interval wavelet is based on the Lagrange continuation, the Gibbs phenomena
would decrease the precision of continuation. Hence, the computational
precision of quasi-Shannon interval wavelet method is decreased.


#### 4.2.3. Influence ofj

The
parameterj is used to control the total number of
collocation points in the discrete formula of wavelet collocation method. The
number of collocation points is the exponential function of the parameter j,
that is, 2j+1.
According to the explanation in Section 4.2.2, if there is no Gibbs phenomena
or the Gibbs phenomena are weak enough, the precision of the interval wavelet
method should be slightly higher than the quasi-Shannon wavelet method, even
the value of ε is small. The Gibbs phenomena would be
continuously weakening up to none with the increment of the collocation points.
Table 3 indicates that calculation errors of all three methods decrease with
number of the collocation points increasing. When j=10,
the maximum absolute error of the interval wavelet method is less than the
other two methods. In addition, the maximum relative error of the interval
wavelet method is less than the error of the quasi-Shannon wavelet method, but
larger than the error of Shannon wavelet
method.Table 3
Influence ofj(ε=0.001).
jMaximum of absolute error(10−3)Maximum of relative error(10−1)WaveletQuasi-waveletInterval waveletWaveletQuasi-waveletInterval wavelet6412.488802207.321583411.521362266.3243723.3120586.5742457164.839419151.45704533.225063185.3495152.5525345.285445856.19513563.509684206.24791657.7476682.3382093.3392111031.43806233.34413823.7640240.7985181.8074921.523272


### 4.3. Comparison of Calculation Precision Obtained by Shannon Interval Wavelet Method and Quasi-Shannon Interval Wavelet Method

In this work, the Shannon interval wavelet is also constructed to further evaluate the performance of the
quasi-Shannon interval wavelet for numerical analysis. Instead of (3.16), the
following weight function is used to construct the Shannon interval wavelet:(4.13)wj(x−xi)=sin[(2jπ/(b−a))(x−xi)](2jπ/(b−a))(x−xi). The first and second derivatives of (4.13) are
expressed, respectively, as follows: (4.14)wj'(x−xi)={cos[(2jπ/(b−a))(x−xi)]x−xi−b−a2jπ⋅sin[(2jπ/(b−a))(x−xi)](x−xi)2,x≠xi,0,x=xi,(4.15)wj′′(x−xi)={2sin[(2jπ(x−xi)/(b−a))]2jπ(x−xi)3/(b−a)−2cos[(2jπ(x−xi)/(b−a))](x−xi)2−2jπsin[(2jπ(x−xi)/(b−a))](b−a)(x−xi),x≠xi,−22jπ23(b−a)2,x=xi. The
collocation method of Shannon interval wavelet
method could be obtained by substituting (4.13)–(4.15) into (4.5).Table4 indicates that the maxima of both the relative error and the absolute error
obtained by the Shannon interval wavelet
method are less than those obtained by the quasi-Shannon interval wavelet
method when the solution curve of the two-point problem is smooth. As ε decreasing, the precision of quasi-Shannon
interval method become higher than that of the Shannon interval wavelet method. Especially when ε≤0.01,
the calculation precision obtained by quasi-Shannon interval wavelet method is
higher than that obtained by Shannon interval
wavelet method in the whole process with the collocation number varying from a
smaller number to 1025 (i.e., j=10).Table 4
Comparison of calculation precision obtained by Shannon
and quasi-Shannon interval wavelet methods.
jεMaximum of absolute error(10−5)Maximu of relative errorShannon
Interval waveletQuasi-Shannon
Interval waveletShannon
Interval waveletQuasi-Shannon
Interval wavelet6101.9573982582.8917249950.017645884840.0376945532123.11540816529.3909054410.021403366310.03967738570.1465.352264175201.0026205540.068863650660.06241552547100.9003345031.4050002060.018737618400.036157587911.0175604461.4596918360.020591293930.03716813640.1175.721016219132.2154673650.043247889420.04915319010.016008.1247399311565.7850305940.673809620160.1362827615100.01221.30267150.796680.058257343640.0519607606Figure3 also shows that when the equation is
solved by the quasi-Shannon interval wavelet, the numerical error mainly lies
in the boundary, while the errors of other parts are very little. On the contrast,
when the Shannon interval wavelet method is
utilized, the numerical error almost evenly distributes in the all domain.
Consequently, the sum of the error is much larger than that of the
quasi-Shannon interval wavelet.Comparison of the quasi-Shannon interval
wavelet and the Shannon interval wavelet (j=7, ε=0.4, and L=1).
(a)
quasi-Shannon
interval wavelet(b)
Shannon interval wavelet


## 4.1. Interval Wavelet Discrete Formulation of
ODEs [6, 7]

Considering the following ODE:(4.1)A(x)u′′(x)+B(x)u′(x)+C(x)u(x)=f(x),x∈[a,b]. According to the collocation method, its approximate solution
can be expressed as(4.2)uj(x)=∑n=02juj(xn)w(2jx−n),n=0,1,2,…,2j. Substituting (4.2) into (4.1), the system of algebraic
equations can be obtained as follows:(4.3)∑n=02juj(xn)[A(xk)22jw′′(k−n)+B(xk)2jw′(k−n)+C(xk)w(k−n)]=f(xk),k=0,1,2,…,2j. The solution of (4.3) is an approximate one of (4.1) at the
collocation point xk.
Equation (4.3) could be rewritten in the new form below as follows:(4.4)MU=F, where U is the vector of solution, F is right side vector, and M is the stiffness matrix. Any element of the
matrix M could be expressed as follows:(4.5)mk,n=A(xk)22jw′′(k−n)+B(xk)2jw′(k−n)+C(xk)w(k−n), where (4.6)w′(k−n)={2jcos[π(k−n)]exp[−(k−n)2/(2r2)](k−n)(b−a),k≠n0,k=n,w′′(k−n)={−22j+1cos[π(k−n)]exp[−(k−n)2/2r2](b−a)2[1(k−n)2+1r2],k≠n−22j(3+π2r2)3r2(b−a)2,k=n.   In the solution of the above differential equations, there are several types of boundary conditions commonly
encountered, such as Dirichlet and Neumann boundary conditions. For simple, we only consider the Dirichlet boundary in the
following discussions.Dirichlet boundary conditions of differential
equations specify the value of function on a surfaceS:(4.7)u(x)|S=β(x),x∈S. That is, the solution u(x) on the boundary S is known. Assuming that the definition
interval of the solution is [0,1],
the approximate solution will be(4.8)uj(x)=uj(0)w(2jx)+∑n=12j−1uj(xn)w(2jx−n)+uj(1)w(2jx−2j). Substituting (4.8) into (4.4), the system of
algebraic equations satisfying the boundary conditions could be obtained.


## 4.2. Numerical Example

Considering the following two-point boundary
problem as a convection equation,(4.9)−εu′′(x)+u′(x)+u(x)=1,0<x<1,u(0)=u(1)=0. Its analytical solution as shown in
Figure 1 is(4.10)u(x)=1−eλ2eλ2−eλ1eλ1x−1−eλ1eλ2−eλ1eλ2x+1, where λ1 and λ2 are Eigenvalues:(4.11)λ1=1+1+4ε2ε,λ2=1−1+4ε2ε. Obviously, the boundary layer phenomenon is
evident near the boundary x=1, which
would increase with the decrease of the parameter ε.
This could be used to test the effectiveness of the interval wavelet.Figure 1
Analytical
solution of the convection (4.9).Substituting (4.5) into (4.9) results in its
wavelet discrete formulation as the following: (4.12)∑n=12j−1uj*(xn)[−22jεw′′(k−n)+2jw′(k−n)+w(k−n)]=1,k=1,2,…,2j-1. The
weight function w in (4.12) can be set as the Shannon, the
quasi-Shannon, or the interval wavelet functions. Their numerical solutions and
corresponding errors are shown in Figure 2, where the count of collocation
points is 65, and the points are distributed evenly in space. Obviously, the
calculation precision of the interval wavelet is higher than others.The comparison of the Shannon (a), the
quasi-Shannon (b), and the interval wavelet (c), and their errors of numerical
results (e), (f), and (g), respectively.
(a)
Shannon wavelet(b)
Quasi-Shannon wavelet(c)
Interval wavelet(d)(e)(f)Besides, the regularized width parameterσ which has been discussed in [8], the
parameters L,ε,andj affect the calculation precision significantly.
Therefore, appropriate values should be selected when implementing the numerical
method developed.


### 4.2.1.  Total Number of External Collocation PointsL

In fact, the definition of interval wavelet in
(3.29) indicates that the values of u in external collocation points are obtained by
Lagrange interpolation method. So, the parameter L is relative with the smoothness of the solution, which is
difficult to be determined. With the increasing of L,
the Runge phenomena could occur in the Lagrange interpolation [9]. As a result,
it could decrease the computational precision, which is proved by the numerical
results shown in Table 1. When L=1,
the maxima of both the relative error and the absolute error are apparently
less than that of the other two methods. However, opposite results were
obtained, when L=3.
Actually, the solution of ODE with the Dirichlet boundary condition is not
smooth on the boundary in most cases, and so the bigger L cannot make any contribution to improve the computational
precision except increasing the time complexity of the numerical method. Consequently, L=1∼3  might be the best choice.Table 1
Influence of the total
number of external collocation pointsL(j=6,ε=1).
LMaximum of absolute error(10−3)Maximum of relative error(10−1)Quasi-waveletWaveletQuasi-waveletWavelet10.2939090.2311540.3967740.21403420.5015392.8244440.6123043.44822131.2575812.6414091.5353173.85599142.0801912.0205582.5396011.273226


### 4.2.2. Parameterε

Table2 shows the influence of parameter ε on the calculation precision. It can be
observed that the solution is smooth as ε=1.
Although the quasi-Shannon wavelet has local property, its computational
precision is lower than that of the Shannon wavelet without local property. With ε decreasing, the superiority of quasi-Shannon
wavelet method to Shannon wavelet method
exhibits gradually. When ε=0.001,
the maximum of the relative error of the quasi-Shannon wavelet method is only
0.33, while that of Shannon wavelet method is
as high as 26.6. The precision of the interval wavelet method keeps higher than
the other two methods in the whole process. When j=6 and ε≥0.1,
the maxima of both the relative error and the absolute error of the interval
wavelet method are the lowest among the three methods.Table 2
Influence of parameterε.
jεMaximum of absolute error(10−3)Maximum of relative error(10−1)WaveletQuasi-waveletInterval waveletWaveletQuasi-waveletInterval wavelet6100.1069140.1116780.0289171.3803311.4417190.37694611.1236561.1742480.2939091.3783191.4392170.3967740.111.37610511.9413662.0100261.4202791.4834180.6241550.01222.14112823.40834757.4687845.0433181.9729641.8879830.001412.488802207.321583411.521362266.3243723.3120586.5742457100.0532600.0565180.0140501.3637061.4248850.36157610.5630710.5884910.1459691.3627341.4236540.3716810.16.0852026.3746301.3221551.3860661.4464750.4915320.0131.34584133.2978241.5657851.3476941.7402751.3628280.001164.839419151.45704533.225063185.3495152.5525345.2854458100.0265820.0277790.0069151.3555281.41656670.35355010.2818600.2945930.0726721.3550511.41595660.3586530.13.1466383.2926530.7457931.3672551.4275460.4202380.0123.22431824.5353147.9647101.4744501.5880040.9476080.00156.19513563.509684206.24791657.7476682.3382093.339211100.00131.43806233.34413823.7640240.7985181.8074921.523272However, when the parameterε further decreases, the relative error of the
interval wavelet method is slightly larger than that of quasi-Shannon wavelet
method, and the absolute error of the interval wavelet method approximately
identical to that of quasi-Shannon wavelet method. This is due to the boundary
layer phenomena of the two-point boundary problem appearing at the points
adjacent to the boundary x=1.
The boundary layer phenomena result in the large gradient of solution, and thus
the Gibbs phenomena [10] are unavoidable. Since the construction of the
interval wavelet is based on the Lagrange continuation, the Gibbs phenomena
would decrease the precision of continuation. Hence, the computational
precision of quasi-Shannon interval wavelet method is decreased.


### 4.2.3. Influence ofj

The
parameterj is used to control the total number of
collocation points in the discrete formula of wavelet collocation method. The
number of collocation points is the exponential function of the parameter j,
that is, 2j+1.
According to the explanation in Section 4.2.2, if there is no Gibbs phenomena
or the Gibbs phenomena are weak enough, the precision of the interval wavelet
method should be slightly higher than the quasi-Shannon wavelet method, even
the value of ε is small. The Gibbs phenomena would be
continuously weakening up to none with the increment of the collocation points.
Table 3 indicates that calculation errors of all three methods decrease with
number of the collocation points increasing. When j=10,
the maximum absolute error of the interval wavelet method is less than the
other two methods. In addition, the maximum relative error of the interval
wavelet method is less than the error of the quasi-Shannon wavelet method, but
larger than the error of Shannon wavelet
method.Table 3
Influence ofj(ε=0.001).
jMaximum of absolute error(10−3)Maximum of relative error(10−1)WaveletQuasi-waveletInterval waveletWaveletQuasi-waveletInterval wavelet6412.488802207.321583411.521362266.3243723.3120586.5742457164.839419151.45704533.225063185.3495152.5525345.285445856.19513563.509684206.24791657.7476682.3382093.3392111031.43806233.34413823.7640240.7985181.8074921.523272


## 4.2.1.  Total Number of External Collocation PointsL

In fact, the definition of interval wavelet in
(3.29) indicates that the values of u in external collocation points are obtained by
Lagrange interpolation method. So, the parameter L is relative with the smoothness of the solution, which is
difficult to be determined. With the increasing of L,
the Runge phenomena could occur in the Lagrange interpolation [9]. As a result,
it could decrease the computational precision, which is proved by the numerical
results shown in Table 1. When L=1,
the maxima of both the relative error and the absolute error are apparently
less than that of the other two methods. However, opposite results were
obtained, when L=3.
Actually, the solution of ODE with the Dirichlet boundary condition is not
smooth on the boundary in most cases, and so the bigger L cannot make any contribution to improve the computational
precision except increasing the time complexity of the numerical method. Consequently, L=1∼3  might be the best choice.Table 1
Influence of the total
number of external collocation pointsL(j=6,ε=1).
LMaximum of absolute error(10−3)Maximum of relative error(10−1)Quasi-waveletWaveletQuasi-waveletWavelet10.2939090.2311540.3967740.21403420.5015392.8244440.6123043.44822131.2575812.6414091.5353173.85599142.0801912.0205582.5396011.273226


## 4.2.2. Parameterε

Table2 shows the influence of parameter ε on the calculation precision. It can be
observed that the solution is smooth as ε=1.
Although the quasi-Shannon wavelet has local property, its computational
precision is lower than that of the Shannon wavelet without local property. With ε decreasing, the superiority of quasi-Shannon
wavelet method to Shannon wavelet method
exhibits gradually. When ε=0.001,
the maximum of the relative error of the quasi-Shannon wavelet method is only
0.33, while that of Shannon wavelet method is
as high as 26.6. The precision of the interval wavelet method keeps higher than
the other two methods in the whole process. When j=6 and ε≥0.1,
the maxima of both the relative error and the absolute error of the interval
wavelet method are the lowest among the three methods.Table 2
Influence of parameterε.
jεMaximum of absolute error(10−3)Maximum of relative error(10−1)WaveletQuasi-waveletInterval waveletWaveletQuasi-waveletInterval wavelet6100.1069140.1116780.0289171.3803311.4417190.37694611.1236561.1742480.2939091.3783191.4392170.3967740.111.37610511.9413662.0100261.4202791.4834180.6241550.01222.14112823.40834757.4687845.0433181.9729641.8879830.001412.488802207.321583411.521362266.3243723.3120586.5742457100.0532600.0565180.0140501.3637061.4248850.36157610.5630710.5884910.1459691.3627341.4236540.3716810.16.0852026.3746301.3221551.3860661.4464750.4915320.0131.34584133.2978241.5657851.3476941.7402751.3628280.001164.839419151.45704533.225063185.3495152.5525345.2854458100.0265820.0277790.0069151.3555281.41656670.35355010.2818600.2945930.0726721.3550511.41595660.3586530.13.1466383.2926530.7457931.3672551.4275460.4202380.0123.22431824.5353147.9647101.4744501.5880040.9476080.00156.19513563.509684206.24791657.7476682.3382093.339211100.00131.43806233.34413823.7640240.7985181.8074921.523272However, when the parameterε further decreases, the relative error of the
interval wavelet method is slightly larger than that of quasi-Shannon wavelet
method, and the absolute error of the interval wavelet method approximately
identical to that of quasi-Shannon wavelet method. This is due to the boundary
layer phenomena of the two-point boundary problem appearing at the points
adjacent to the boundary x=1.
The boundary layer phenomena result in the large gradient of solution, and thus
the Gibbs phenomena [10] are unavoidable. Since the construction of the
interval wavelet is based on the Lagrange continuation, the Gibbs phenomena
would decrease the precision of continuation. Hence, the computational
precision of quasi-Shannon interval wavelet method is decreased.


## 4.2.3. Influence ofj

The
parameterj is used to control the total number of
collocation points in the discrete formula of wavelet collocation method. The
number of collocation points is the exponential function of the parameter j,
that is, 2j+1.
According to the explanation in Section 4.2.2, if there is no Gibbs phenomena
or the Gibbs phenomena are weak enough, the precision of the interval wavelet
method should be slightly higher than the quasi-Shannon wavelet method, even
the value of ε is small. The Gibbs phenomena would be
continuously weakening up to none with the increment of the collocation points.
Table 3 indicates that calculation errors of all three methods decrease with
number of the collocation points increasing. When j=10,
the maximum absolute error of the interval wavelet method is less than the
other two methods. In addition, the maximum relative error of the interval
wavelet method is less than the error of the quasi-Shannon wavelet method, but
larger than the error of Shannon wavelet
method.Table 3
Influence ofj(ε=0.001).
jMaximum of absolute error(10−3)Maximum of relative error(10−1)WaveletQuasi-waveletInterval waveletWaveletQuasi-waveletInterval wavelet6412.488802207.321583411.521362266.3243723.3120586.5742457164.839419151.45704533.225063185.3495152.5525345.285445856.19513563.509684206.24791657.7476682.3382093.3392111031.43806233.34413823.7640240.7985181.8074921.523272


## 4.3. Comparison of Calculation Precision Obtained by Shannon Interval Wavelet Method and Quasi-Shannon Interval Wavelet Method

In this work, the Shannon interval wavelet is also constructed to further evaluate the performance of the
quasi-Shannon interval wavelet for numerical analysis. Instead of (3.16), the
following weight function is used to construct the Shannon interval wavelet:(4.13)wj(x−xi)=sin[(2jπ/(b−a))(x−xi)](2jπ/(b−a))(x−xi). The first and second derivatives of (4.13) are
expressed, respectively, as follows: (4.14)wj'(x−xi)={cos[(2jπ/(b−a))(x−xi)]x−xi−b−a2jπ⋅sin[(2jπ/(b−a))(x−xi)](x−xi)2,x≠xi,0,x=xi,(4.15)wj′′(x−xi)={2sin[(2jπ(x−xi)/(b−a))]2jπ(x−xi)3/(b−a)−2cos[(2jπ(x−xi)/(b−a))](x−xi)2−2jπsin[(2jπ(x−xi)/(b−a))](b−a)(x−xi),x≠xi,−22jπ23(b−a)2,x=xi. The
collocation method of Shannon interval wavelet
method could be obtained by substituting (4.13)–(4.15) into (4.5).Table4 indicates that the maxima of both the relative error and the absolute error
obtained by the Shannon interval wavelet
method are less than those obtained by the quasi-Shannon interval wavelet
method when the solution curve of the two-point problem is smooth. As ε decreasing, the precision of quasi-Shannon
interval method become higher than that of the Shannon interval wavelet method. Especially when ε≤0.01,
the calculation precision obtained by quasi-Shannon interval wavelet method is
higher than that obtained by Shannon interval
wavelet method in the whole process with the collocation number varying from a
smaller number to 1025 (i.e., j=10).Table 4
Comparison of calculation precision obtained by Shannon
and quasi-Shannon interval wavelet methods.
jεMaximum of absolute error(10−5)Maximu of relative errorShannon
Interval waveletQuasi-Shannon
Interval waveletShannon
Interval waveletQuasi-Shannon
Interval wavelet6101.9573982582.8917249950.017645884840.0376945532123.11540816529.3909054410.021403366310.03967738570.1465.352264175201.0026205540.068863650660.06241552547100.9003345031.4050002060.018737618400.036157587911.0175604461.4596918360.020591293930.03716813640.1175.721016219132.2154673650.043247889420.04915319010.016008.1247399311565.7850305940.673809620160.1362827615100.01221.30267150.796680.058257343640.0519607606Figure3 also shows that when the equation is
solved by the quasi-Shannon interval wavelet, the numerical error mainly lies
in the boundary, while the errors of other parts are very little. On the contrast,
when the Shannon interval wavelet method is
utilized, the numerical error almost evenly distributes in the all domain.
Consequently, the sum of the error is much larger than that of the
quasi-Shannon interval wavelet.Comparison of the quasi-Shannon interval
wavelet and the Shannon interval wavelet (j=7, ε=0.4, and L=1).
(a)
quasi-Shannon
interval wavelet(b)
Shannon interval wavelet


## 5. Conclusions

A novel general method for construction of interval wavelet
based on the restricted variational principle is proposed in this paper. Both
Shannon and quasi-Shannon interval wavelets could be
constructed using this new method. With appropriate values of parametersL,ε, and j  selected, the numerical results for a
differential equation show that the quasi-Shannon interval wavelet outperforms
than Shannon and quasi-Shannon wavelets. Furthermore, the capability of
quasi-Shannon interval wavelet for numerical analysis is better than that of Shannon interval wavelet.It should be noted that combining the
corresponding interval wavelet with the classical Galerkin method can obtain an
excellent wavelet-Galerkin method for engineering problems. Since quasi-Shannon
wavelet is orthogonal, this guarantees the stability of the matrix equations in
Galerkin method.



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*Source: 629253-2008-05-06.xml*