International Journal of Engineering Research and Development

International Journal of Engineering Research and Development
e-ISSN: 2278-067X, p-ISSN: 2278-800X, www.ijerd.com
Volume 10, Issue 5 (May 2014), PP.21-30
21
Wavelet– Galerkin Solution for Boundary Value Problems
D. Patel1
, M.K. Abeyratna2
, M.H.M.R. Shyamali Dilhani3*
1
Center for Industrial Mathematics, Faculty of Technology & Engineering,
The M S University of Baroda, Gujarat, India.
2
Department of Mathematics, Faculty of Science, University of Ruhuna, Matara, Sri Lanka.
3
Department of Interdisciplinary Studies, Faculty of Engineering, University of Ruhuna, Galle, Sri Lanka.
Abstract:- Wavelet-Galerkin technique has very important advantages over classical finite difference and finite
element method. In this paper, we have made an attempt to develop a technique for Wavelet-Galerkin solution
of Neumann and mixed boundary value problem in one dimension in parallel to the work of J. Besora [5]. The
Taylor’s approach has been used to include Neumann condition in Wavelet-Galerkin setup. The test examples
are given to validate techniques with exact solutions.
Keywords:- Boundary Value Problems, Wavelets, Scaling Function, Connection Coefficients, Wavelet
Coefficient ,Wavelet- Galerkin Method.
I. INTRODUCTION
Wavelet is an important area of mathematics and now a days it becomes an important tool for
applications in many areas of science and engineering.
The orthonormal bases of compactly supported wavelets for the space of square-integral function
𝐿2
ℝ was constructed by Daubecheis in 1988 (see [7]). Those wavelets are bases in a Galerkin method to solve
Dirichlet as well as Neumann mixed boundary value problem. The approximations of partial differential
equations with wavelet basis are more attentive, because the reality of orthogonality of compactly supported
wavelets. The concepts of Multiresolution Analysis based Fast wavelet transform algorithm have given great
momentum to make wavelet best approximations of ODE’s and PDE’s. Wavelet-Galerkin technique is
frequently used now a days, and its numerical solution of partial differential equations have been developed by
several researchers.
Many contributions have been done in the area of wavelets and differential equations by authors like
Beylkin et al.[4], Jawerth et al. [9], Qian et al.[14], Qian et al. [15], Williams et al. [16], Williams et al. [19].
Several researchers now a day’s uses Daubechies wavelets as bases in a Galerkin method to solve boundary
value problem. The contribution in this area is due to the remarkable work by Latto et al. [10], Xu et al. [21],
[22], Williams et al. [16], [17] , [18], Mishra et al [13], Jordi Besora [5] and Amartunga et al. [1], [2], [3]. It is
known that the problems with periodic boundary conditions have been handled successfully. Fictious boundary
approach with Dirichlet boundary conditions have been applied by Dianfeng et al. [8] in analyzing SH wave
equation. Latto et al. [10] presents a connection coefficients for j=0 and N=6. The problem with Distinct
boundary conditions is developed by Jordi Besora [5].
In this paper, we have proposed wavelet-Galerkin approximation to the Neumann and mixed boundary
value problem, using compactly supported wavelets as basis functions introduced by Daubechies [6], [7]. We
have used Taylors approach to deal with Neumann and mixed Boundary conditions.
II. WAVELETS
An oscillatory function 𝜓(𝑥) ∈ 𝐿2
ℝ is a wavelet if it has following properties:
 𝜓(𝑥) is n times differentiable and its derivatives are continuous.
 𝜓(𝑥)is well localized both in time and frequency domains, i.e. 𝜓(𝑥) and its derivatives must decay
very rapidly. For frequency localization 𝜓 𝜔 must decay sufficiently fast as 𝜔 ⟶ ∞ and that 𝜓 𝜔
becomes flat in the neighbourhood of 𝜔 = 0. The flatness is associated with number of vanishing
moments of 𝜓(𝑥), i.e.
𝑥 𝑘
𝜓 𝑥 𝑑𝑥 = 0
∞
−∞
or equivalently

Wavelet– Galerkin Solution for Boundary Value problems
22
𝑑 𝑘
𝑑𝜔 𝑘
𝜓 𝜔 = 0
for 𝑘 = 0,1, … 𝑛.
 in the sense that the larger number of vanishing moments more is the flatness when 𝜔 is small.
𝜓 𝜔 2
𝜔
∞
−∞
𝑑𝜔 < ∞
Daubechies wavelets are compactly supported functions. Since they have non zero values within a finite interval
and zero values elsewhere in some intervals and they can be used for representation of solution of BVP. In order
to have multiresolution properties, the scaling function is defined as 𝜙(𝑥), and it must satisfy,
𝜙 𝑥 = 𝑎 𝑘 𝜙(2𝑥 − 𝑘)
𝐿−1
𝑘=0
(1)
or
𝜙 𝑥 = 2 𝐶𝑘 𝜙 2𝑥 − 𝑘 ;𝑁−1
𝑘=0 when 𝑎 𝑘 = 2𝐶𝑘 with the property that 𝑎 𝑘
𝑁−1
𝑘=0 = 2.
where L denotes the genus of the Daubechies wavelet. The functions generated with these coefficients will have
supp 𝜙 = 0, 𝐿 − 1 and (
𝐿
2
− 1) vanishing wavelet moments.
Fig.1: Daubechies Scaling function with L=6 and j=0
The basic wavelet is defined in terms of scaling function by,
𝜓 𝑥 = (−1) 𝑘
𝑎1−𝑘 𝜙 𝑘(2𝑥 − 𝑘)1
𝑘=−𝐿 (2)
A family of scaling functions is generated by translation from
𝜙 𝑘 𝑥 = 𝜙 𝑥 − 𝑘 (3)
Similar a family of wavelets is denoted by translation and scaling from
𝜓𝑗 ,𝑘 𝑥 = 𝜓 2𝑗
𝑥 − 𝑘 (4)
Since 𝜙 𝑥 𝑑𝑥 = 1; the scaling function satisfies the three conditions.
1.
𝑎 𝑘 = 2 (5)𝐿−1
𝑘=0
2. The orthonormality condition of the scaling function
𝜙 𝑥 − 𝑘 𝜙 𝑥 − 𝑚 𝑑𝑥 = 𝛿 𝑘,𝑚
Implies that
𝑎 𝑘 𝑎 𝑘−2𝑚 = 𝛿0,𝑚 (6)𝐿−1
𝑘=0

23
where 𝛿 is a kroneker delta function.
3. The moment of the smooth wavelet function to be zero.
𝑥 𝑚
𝜓 𝑥 𝑑𝑥 = 0
implies that
(−1) 𝑘
𝑘 𝑚
𝑎 𝑘 = 0𝐿−1
𝑘=0 (7)
for 𝑚 = 0,1, …
𝐿
2
− 1
The relation between scaling function and wavelet function can be expressed as ;
𝑉𝑗+1 = 𝑉𝑗 ⨁𝑊𝑗
It implies that, for any integer m,
𝜙 𝑥 𝜓 𝑥 − 𝑚 𝑑𝑥 = 0 (8)
Where ⨁ denotes the direct sum and 𝑉𝑗 and 𝑊𝑗 be the subspaces generated, respectively as the 𝐿2
- closure of the
linear spans of
𝜙𝑗,𝑘 𝑥 = 2𝑗
𝜙 2𝑗
𝑥 − 𝑘 and
𝜓𝑗 ,𝑘 𝑥 = 2𝑗
𝜓 2𝑗
𝑥 − 𝑘 , 𝑘 ∈ ℤ
Using condition (8) we can write
𝑉0 ⊂ 𝑉1 ⊂ ⋯ ⊂ 𝑉𝑗 ⊂ 𝑉𝑗+1
And
𝑉𝑗+1 = 𝑉0⨁𝑊0⨁𝑊1⨁𝑊2⨁ ⋯ ⨁𝑊𝑗
where 𝑗 is the dilation parameter as scale. The support of the scale function 𝜙 2𝑗
𝑥 − 𝑘 for a certain value of 𝑗
and 𝐿 is given as
𝑠𝑢𝑝𝑝 𝜙 2𝑗
𝑥 − 𝑘 =
𝑘
2 𝑗 ,
𝐿+𝑘−1
2 𝑗
Each function𝑓(𝑥) ∈ 𝐿2
ℝ , can be written in the form
𝑓 𝑥 = 2𝑗
𝑐 𝑘 𝜙 2𝑗
𝑥 − 𝑘𝑘 (9)
and this should be satisfied the convergence property
𝑓 − 𝑐 𝑘 𝜙 2𝑗
𝑥 − 𝑘
𝑘
≤ 2−𝑗𝑝
𝐶 𝑓 𝑝
where
𝑐 𝑘 = 𝑓(𝑥)𝜙 2𝑗
𝑥 − 𝑘 𝑑𝑥
and C and p are constants.
III. MULTIRESOLUTION ANALYSIS
Construction of smooth wavelets with compact support is a major task in todays research scenario.
Meyer[6] and Mallat [11] found conditions which we called multiresolution analysis which is use by Daubchies
to construct wavelet of compact support having arbitrary smoothness.
A MRA of 𝐿2
ℝ is defined as a sequence of closed subspace of 𝑉𝑗 𝑗 ∈ℤ
of 𝐿2
ℝ and a scaling
function 𝜙 satisfy the following conditions.
1. The space 𝑉𝑗 are nested i.e.
⋯ ⊂ 𝑉−1 ⊂ 𝑉0 ⊂ 𝑉1 ⊂ ⋯,
2. The space 𝐿2
ℝ is a closure of the union of all 𝑉𝑗 and the intersection of all 𝑉𝑗 is empty. i.e.
⋃𝑗 𝑉𝑗 = 𝐿2
ℝ and ⋂𝑗 𝑉𝑗 = 0
3. 𝑓 𝑥 ∈ 𝑉𝑗 ⟺ 𝑓 2𝑥 ∈ 𝑉𝑗 +1 ; ∀ 𝑗 ∈ ℤ
4. 𝑓 𝑥 ∈ 𝑉0 ⟺ 𝑓 𝑥 − 𝑘 ∈ 𝑉0 ; ∀ 𝑘 ∈ ℤ
5. {𝜙(𝑥 − 𝑘)} is an orthonormal basis for the space 𝑉0.
IV. WAVELET-GALERKIN METHOD
Galerkin Method was introduced by V.I Galerkin. We can discuss this using a one-dimensional
differential equation.
𝐿𝑢 𝑥 = 𝑓 𝑥 ; 0 ≤ 𝑥 ≤ 1; (10)
where
𝐿 = −
𝑑
𝑑𝑥
𝑎 𝑥
𝑑𝑢
𝑑𝑥
+ 𝑏(𝑥)𝑢 𝑥
with boundary conditions, 𝑢 0 = 0 and 𝑢 1 = 0.

24
Where 𝑎, 𝑏 and 𝑓 are given real valued continuous function on [0,1]. We also assume that 𝐿 is a elliptic
differential operator.
Consider, 𝑣𝑗 is a complete orthonormal basis of 𝐿2
[0,1] and every 𝑣𝑗 𝐶2
∈ ( 0,1 ) such that,
𝑣𝑗 0 = 0 𝑣𝑗 1 = 0
We can select the finite set Λ of indices j and then consider the subspace S,
𝑆 = 𝑠𝑝𝑎𝑛{𝑣𝑗 ; 𝑗 ∈ Λ}.
Approximate solution 𝑢 𝑠 can be written in the form,
𝑢 𝑠 = 𝑥 𝑘 𝑣 𝑘 ∈ 𝑆𝑘∈Λ (11)
where each 𝑥 𝑘 is scalar. We may determine 𝑥 𝑘 by seeing the behaviour of 𝑢 𝑠 as it look like a true solution on 𝑆.
i.e.
𝐿 𝑢 𝑠 , 𝑣𝑗 = 𝑓, 𝑣𝑗 ∀ 𝑗 ∈ Λ, (12)
such that the boundary conditions 𝑢 𝑠 0 = 0 and 𝑢 𝑠 1 = 0 are satisfied. Substituting 𝑢 𝑠 values into the
equation (12),
𝐿 𝑣 𝑘 , 𝑣𝑗𝑘∈Λ 𝑥 𝑘 = 𝑓, 𝑣𝑗 ∀ 𝑗 ∈ Λ (13)
Then this equation can be reduced in to the linear system of equation of the form
𝑎𝑗𝑘 𝑥 𝑘 = 𝑦𝑖 (14)
or AX=Y
where 𝐴 = 𝑎𝑗𝑘 𝑗 ,𝑘∈Λ
and 𝑎𝑗𝑘 = 𝐿 𝑣 𝑘 , 𝑣𝑗 , x denotes the vector 𝑥 𝑘 𝑘∈Λ and y denotes the vector 𝑦 𝑘 𝑘∈Λ . In
the Galerkin method, for each subset Λ, we obtain an approximation 𝑢 𝑠 ∈ S by solving linear system (14).
If 𝑢 𝑠 converges to 𝑢 then we can find the actual solution.
Our main concern is the method of linear system (14) by choosing a wavelet Galerkin method. The
matrix A should have a small condition number to obtain stability of solution and A should sparse to perform
calculation fast.
Similarly we can do the same thing in above set up.
Let 𝜓𝑗,𝑘 𝑥 = 2𝑗
𝜓 2𝑗
𝑥 − 𝑘 (15)
is a basis for 𝐿2
[0,1] with boundary conditions
𝜓𝑗,𝑘 0 = 𝜓𝑗,𝑘 1 = 0 ∀ 𝑗, 𝑘 ∈ Λ and 𝜓𝑗 ,𝑘 is 𝐶2
.
We can replace equations (12) and (13) by
𝑢 𝑠 = 𝑥𝑗 ,𝑘 𝜓𝑗 ,𝑘𝑗 ,𝑘∈Λ
and
𝐿 𝜓𝑗 ,𝑘 , 𝜓𝑙,𝑚𝑗,𝑘∈Λ 𝑥𝑗 ,𝑘 = 𝑓, 𝜓𝑙,𝑚 ∀ 𝑙, 𝑚 ∈ Λ
So that AX=Y.
Where 𝐴 = 𝑎𝑙,𝑚;𝑗,𝑘 𝑙,𝑚 ,(𝑗,𝑘)∈Λ
; 𝑋 = 𝑥𝑗 ,𝑘 (𝑗,𝑘)∈Λ
𝑌 = 𝑦𝑙,𝑚 (𝑙,𝑚)∈Λ
Then
𝑎𝑙,𝑚;𝑗,𝑘 = 𝐿 𝜓𝑗 ,𝑘 , 𝜓𝑙,𝑚
𝑦𝑙,𝑚 = 𝑓, 𝜓𝑙,𝑚
Where 𝑙, 𝑚 and 𝑗, 𝑘 represent respectively row and column of A.
This is an accurate method to find the solution of partial differential equation.
V. CONNECTION COEFFICIENT
To find the solution of differential equation using the Wavelet Galerkin technique we have to find the
connection coefficients which is also explored in Latto et al.([10]),
Ωℓ1ℓ2
d1d2
= Φℓ1
d1
x
∞
−∞
Φℓ2
d2
(x)dx (16)
Taking derivatives of the scaling function 𝑑 times, we get
𝜙 𝑑
𝑥 = 2 𝑑
𝑎 𝑘 𝜙 𝑘
𝑑
(2𝑥 − 𝑘)𝐿−1
𝑘=0 (17)
We can simplify equation (16) then for all Ωℓ1ℓ2
d1d2
gives a system of linear equation with unknown vector Ωd1d2
𝑇Ωd1d2
=
1
2 𝑑−1 Ωd1d2
(18)
where 𝑑 = d1 + d2 and 𝑇 = 𝑎𝑖 𝑎 𝑞−2𝑙+𝑖𝑖 . These are so called scaling equation.
But this is the homogeneous equation and does not have a unique nonzero solution. In order to make the system
inhomogeneous, one equation is added and it derived from the moment equation of the scaling function 𝜙. This
is the normalization equation,
𝑑! = −1 𝑑
𝑀𝑙
𝑑
𝑙
Ω𝑙
0,𝑑
Connection coefficient Ω𝑙
0,𝑑
can be obtained very easily using Ω𝑙
d1d2
,

25
Ω𝑙
0,𝑑
= 𝜙d1 𝜙𝑙
d2
𝑑𝑥
= [𝜙d1−1
𝜙𝑙
d2
]−∞
∞
− 𝜙d1−1
𝜙d2+1
𝑑𝑥
∞
−∞
As a result of compact support wavelet basis functions exhibit, the above equation becomes
Ωd1d2
= − 𝜙d1−1
𝜙𝑙
d2+1
𝑑𝑥
∞
−∞
(19)
After d1 integration ,
Ω𝑙
d1d2
= −1 𝑑
𝜙d1−1
𝜙𝑙
d2+d3
𝑑𝑥 = −1 𝑑∞
−∞
Ω𝑙
0,𝑑
The moments 𝑀𝑖
𝑘
of 𝜙𝑖 are defined as
𝑀𝑖
𝑘
= 𝑥 𝑘
𝜙𝑖(𝑥)𝑑𝑥
∞
−∞
With 𝑀0
0
= 1
Latto et al derives a formula as
𝑀𝑖
𝑚
=
1
2(2 𝑚 −1)
𝑚
𝑡
𝑚
𝑡=0 𝑖 𝑚−𝑡 𝑡
𝑙
𝑡−1
𝑙=0 𝑎𝑖 𝑖 𝑡−𝑙𝐿−1
𝑖=0 (20)
Where 𝑎𝑖’s are the Daubechies wavelet coefficients. Finally, the system will be
𝑇 −
1
2 𝑑−1 𝐼
𝑀 𝑑
Ωd1d1
=
0
d!
(21)
Matlab software is used to compute the connection coefficient and moments at different scales. Latto et al [10]
computed the coefficients at j=0 and L=6 only. The computation of connection coefficients at different scales
have been done by using the program given in Jordi Besora [5]. The scaling function at j=0 and L=6 connection
coefficient prepared by Latto et al [10] is given in Table 1 and Table 2 respectively.
Table I: Scaling function at j=0 and L=6 have been provided by Latto et al. [10]
x Phi(x) x Phi(x)
0.000 0 2.500 -0.014970591
0.125 0.133949835 2.625 -0.03693836
0.250 0.284716624 2.750 -0.040567571
0.375 0.422532739 2.875 0.037620632
0.500 0.605178468 3.000 0.095267546
0.625 0.743571274 3.125 0.062104053
0.750 0.89811305 3.250 0.02994406
0.875 1.090444005 3.375 0.011276602
1.000 1.286335069 3.500 -0.031541303
1.125 1.105172581 3.625 -0.013425276
1.250 0.889916048 3.750 0.003025131
1.375 0.724108826 3.875 -0.002388515
1.500 0.441122481 4.000 0.004234346
1.625 0.30687191 4.125 0.001684683
1.750 0.139418882 4.250 -0.001596798
1.875 -0.125676646 4.375 0.000149435
2.000 -0.385836961 4.500 0.000210945
2.125 -0.302911152 4.625 -7.95485E-05
2.250 -0.202979935 4.750 1.05087E-05
2.375 -0.158067602 4.875 5.23519E-07
5.000 -3.16007E-20

26
Table II: Connection coefficients at j=0 and L=6 have been provided by Latto et al. [13] using 𝛀 𝒏 −
𝒌=𝝓′′𝒙−𝒌𝝓(𝒙−𝒏)𝒅𝒙
VI. TEST PROBLEMS
Consider
𝑑2 𝑢(𝑥)
𝑑𝑥2 + 𝛽𝑢 𝑥 = 𝑓 (22)
Now we use Wavelet-Galerkin method solution
Here, we consider 𝐿 = 6 𝑎𝑛𝑑 𝑗 = 0
We can write the solution of the differential equation (22) is,
𝑢 𝑥 = 𝑐 𝑘 2
𝑗
2Φ 2𝑗
𝑥 − 𝑘 , 𝑥 ∈ [0,1]
2 𝑗
𝑘=𝐿−1
= 𝑐 𝑘 Φ 𝑥 − 𝑘 , 𝑥 ∈ [0,1]1
𝑘=−5 (23)
where 𝑐 𝑘 are the unknown constant co-efficient
Substitute (23) in (22) we get
𝑑2
𝑑𝑥2
𝑐 𝑘Φ 𝑥 − 𝑘 + 𝛽 𝑐 𝑘 Φ 𝑥 − 𝑘 = 0
1
𝑘=−5
1
𝑘=−5
𝑐 𝑘 𝜙′′
𝑥 − 𝑘 + 𝛽 𝑐 𝑘 𝜙 𝑥 − 𝑘 = 0
1
𝑘=−5
1
𝑘=−5
Taking inner product with 𝜙 𝑥 − 𝑘
We have
𝑐 𝑘 𝜙′′
𝑥 − 𝑘 𝜙(𝑥 − 𝑛)
𝐿−1+2 𝑗
2 𝑗
1−𝐿
2 𝑗
𝑑𝑥 + 𝛽 𝑐 𝑘 𝜙 𝑥 − 𝑘 𝜙(𝑥 − 𝑛)
𝐿−1+2 𝑗
2 𝑗
1−𝐿
2 𝑗
𝑑𝑥 = 01
𝑘=−5
1
𝑘=−5
⟹ 𝑐 𝑘 Ω 𝑛 − 𝑘 + 𝛽 𝑐 𝑘 𝛿 𝑛,𝑘 = 01
𝑘=−5
1
𝑘=−5 (24)
𝑛 = 1 − 𝐿, 2 − 𝐿, … 2𝑗
i.e; 𝑛 = −5, −4, … ,0 1
where
Ω 𝑛 − 𝑘 = 𝜙′′
𝑥 − 𝑘 𝜙(𝑥 − 𝑛)𝑑𝑥
𝛿 𝑛,𝑘 = 𝜙 𝑥 − 𝑘 𝜙(𝑥 − 𝑛)𝑑𝑥
By using Neumann Boundary conditions
𝑢 0 = 1; 𝑢′ 1 = 0
Considering left and right boundary conditions we can write
𝑢 0 = 𝑐 𝑘 𝜙 −𝑘 =1
𝑘=−5 1 (25)
𝑢′
1 = 𝑐 𝑘 𝜙 1 − 𝑘 =1
𝑘=−5 0 (26)
We use Taylors method to approximate derivative on the right side
Ω[−3] 5.357142857141622e-003
Ω[−2] 1.142857142857171e-001
Ω[−1] -8.761904761905105e-001
Ω[0] 3.390476190476278e+000
Ω[1] -5.267857142857178e+000
Ω[2] 3.390476190476152e+000
Ω[2] -8.761904761904543e-001
Ω[3] 1.142857142857135e-001
Ω[4] 5.357142857144167e-003

27
𝑢 𝑥 = 𝑢 𝑎 + ℎ𝑢′
𝑎 +
ℎ2
2!
𝑢′′
𝑎 + ⋯ ⋯
which gives the approximation of derivative at the boundary using forward, backward or central differences.
Equation (25) and (26) represent the relation of the coefficient 𝑐 𝑘 .
We can replace first and last equations of (24) using (25) and (26) respectively. Then we can get the following
matrix with L=6
TC=B
𝑇 =
0 𝜙(4) 𝜙(3) 𝜙(2) 𝜙(1) 0 0
Ω[1] Ω 0 Ω[−1] Ω[−2] Ω[−3] Ω[−4] Ω[−5]
Ω[2] Ω[1] Ω 0 Ω[−1] Ω[−2] Ω[−3] Ω[−4]
Ω[3] Ω[2] Ω 1 Ω 0 Ω[−1] Ω[−2] Ω[−3]
Ω[4] Ω[3] Ω[2] Ω 1 Ω 0 Ω[−1] Ω[−2]
Ω[5] Ω[4] Ω[3] Ω[2] Ω 1 Ω 0 Ω[−1]
0 0 𝑝 1 𝑝 2 𝑝 3 𝑝 4 0
𝑝 1 = 𝜙 4 − 𝜙 4 − ℎ
𝑝 2 = 𝜙 3 − 𝜙 3 − ℎ
𝑝 3 = 𝜙 2 − 𝜙 2 − ℎ
𝑝 4 = 𝜙 1 − 𝜙 1 − ℎ
𝐶 =
𝑐−5
𝑐−4
𝑐−3
𝑐−2
𝑐−1
𝑐0
𝑐1
and 𝐵 =
1
0
0
0
0
0
0
1. Suppose given Boundary Value Problem is,
𝑢 𝑥𝑥 = −2 (27)
with boundary conditions 𝑢 0 = 1 𝑎𝑛𝑑 𝑢′
(1) = 0
The exact solution is,
𝑢 𝑥 = −𝑥2
+ 2𝑥 + 1
𝑐−5 =-0.997475258266510
𝑐−4 = -0.877910629139006
𝑐−3 =0.127857696605495
𝑐−2 =1.054541418880270
𝑐−1 =1.087386803133082
𝑐0 = 0.247680298288321
𝑐1 = -0.506012992523331

28
Fig.2: Wavelet-Galerkin Solution for 𝒖 𝒙𝒙 = −𝟐, 𝒖 𝟎 = 𝟏 𝐚𝐧𝐝 𝒖′
(𝟏) = 𝟎
2. The given boundary value Problem is,
𝑢′′
+ 𝑢 = 0; 𝑢 0 = 1 𝑎𝑛𝑑 𝑢′
1 = 0
𝑢 𝑥 = cos 𝑥 + tan 1 sin 𝑥
𝑐−5 =-16.784099470868910
𝑐−4 =-15.622648376150911
𝑐−3 =-8.883687788040305
𝑐−2 = -3.026098858023991
𝑐−1 = 0.579267070363675
𝑐0 = 1.960144587700136
𝑐1 = 1.418474146775382
Fig.3: Wavelet-Galerkin Solution for 𝒖′′
+ 𝒖 = 𝟎; 𝒖 𝟎 = 𝟏 𝐚𝐧𝐝 𝒖′
𝟏 = 𝟎
3. The given boundary value Problem is,
𝑢′′
+ 𝑢 = 0; 𝑢 0 = 1 𝑎𝑛𝑑 𝑢′
1 + 𝑢(1) = 0
𝑢 𝑥 = cos 𝑥 + (tan 2 − sec 2) sin 𝑥

29
𝑐−5 =-13.541294401791857
𝑐−4 =-12.126299651408310
𝑐−3 = -6.346329455749815
𝑐−2 = -1.657781810857689
𝑐−1 =0.790083941059115
𝑐0= 1.133626147467154
𝑐1= 0.186034129844783
Fig.4: Wavelet-Galerkin Solution for 𝒖′′
+ 𝒖 = 𝟎; 𝒖 𝟎 = 𝟏 𝐚𝐧𝐝 𝒖′
𝟏 + 𝒖(𝟏) = 𝟎
VII. CONCLUSIONS
Wavelet method has shown a very powerful numerical technique for the stable and accurate solution of
given boundary value problem. The comparison shows that the exact solution correlates with numerical solution,
using Daubechies wavelets.
REFERENCES
[1]. K. Amaratunga, J.R. William, S. Qian and J. Weiss, wavelet Galerkin solution for one dimensional
partial differential equations, Int. J. numerical methods eng. 37,2703-2716 (1994).
[2]. K. Amaratunga, J.R. Williams, Qian S. and J. Weiss, Wavelet-Galerkin solutions for one Dimensional
Partial Differential Equations, IESL Technical Report No. 92-05, Intelligent Engineering Systems
Laboratory, M. I. T ., 1992.
[3]. K. Amaratunga and J.R. William, Wavelet-Galerkin Solutions for One dimensional Partial Differential
Equations, Inter. J. Num. Meth. Eng. 37(1994), 2703-2716.
[4]. J. Besora, Galerkin Wavelet Method for Global Waves in 1D, Master Thesis,Royal Inst. of Tech.
Sweden, 2004.
[5]. G. Beylkin, R. Coifman, and V. Rokhlin, 1991,Fast Wavelet Transfermation and Numerical Algorithm,
Comm. Pure Applied Math.,44,141-183.
[6]. I. Daubechis, 1992 , Ten Lectures on Wavelet, Capital City Press, Vermont.
[7]. I. Daubechies, 1988, Orthonormal bases of compactly supported wavelets, Commun. Pure Appl. Math.,
41, 909-996.
[8]. L.U. Dianafeng, Tadashi Ohyoshi and Lin ZHU, Treatment of Boundary condition in the Application
of Wavelet-Galerkin Method to a SH Wave Problem, 1996, Akita Uni. (Japan)

30
[9]. B. Jawerth and W. Sweldens, Wavelets Multiresolution Analysis Adapted for Fast Solution of
Boundary Value Ordinary Differential Equations, Proc. 6th Cop.Mount Multi. Conf., April 1993,
NASA Conference Pub., 259--273.
[10]. A. Latto, H.L Resnikoff and E. Tenenbaum, 1992 The Evaluation of connection coefficients of
compactly Supported Wavelet: in proceedings of the French-USA Workshop on Wavelet and
Turbulence, Princeton, New York, June 1991,Springer- Verlag,
[11]. S. Mallat, Multiresolution approximation and wavelets, Trans. Amer. Math. Soc., 315,69-88(1989).
[12]. S.G. Mallat, 1989, A Theory for Multiresolution Signal Decomposition: The Wavelet Representation.
IEEE Transactions on Pattern Analysis and Machine Intelligence,Vol II, No 7, July 1989.
[13]. V. Mishra and Sabina, Wavelet-Galerkin solution of ordinary differential equations, Int. Journal of
Math. Analysis, Vol.5, 2011, no.407-424
[14]. S. Qian, and J. Weiss,1993 Wavelet and the numerical solution of boundary value problem, pl. Math.
Lett, 6, 47-52
[15]. S.Qian and J. Weiss,1993 Wavelet and the numerical solution of partial differential equation ‘, J.
Comput. Phys, 106, 155-175
[16]. J. R. Williams and K. Amaratunga, Wavelet Based Green’s Function Approach to 2D PDEs, Engg.
Comput.10 (1993), 349-367.
[17]. J. R. Williams and K. Amaratunga, High Order wavelet Extrapolation Schemes for Initial Problems
and Boundary Value Problems, July 1994, IESL Tech. Rep., No.94-07, Intelligent Engineering
Systems Laboratory, MIT.
[18]. J.R. Williams and K. Amaratunga, Simulation Based Design using Wavelets, Intelligent Engineering
Systems Laboratory, MIT (USA).
[19]. Williams,J.R.and Amaratunga,K., 1992, Intriduction to Wavelet in engineering, IESL Tech.Rep.No.92-
07,Intelegent Engineering system labrotary,MIT.
[20]. J.R. Williams and K. Amaratunga,1994, High order Wavelet extrapolation schemes for initial problem
and boundary value problem,IESL Tech.Rep.No.94-07,Inteligent Engineering systems Labrotary,MIT.
[21]. J.-C. Xu and W.-C. Shann, Wavelet-Galerkin Methods for Two-point Boundary Value Problems, Num.
Math. Eng. 37(1994), 2703-2716.
[22]. J.C. Xu, and W.-C. Shann, Galerkin-Wavelet method for two point boundary value problems, Number.
Math.63 (1992) 123-144.

International Journal of Engineering Research and Development

More Related Content

What's hot (17)

Viewers also liked (20)

Similar to International Journal of Engineering Research and Development (20)

More from IJERD Editor (20)

Recently uploaded (20)

International Journal of Engineering Research and Development