Wavelet Galerkin Solutions of Ordinary Differential Equations

Wavelet Galerkin solutions 415
simultaneous equations in wavelet space and not in physical space. The solution in
wavelet space is then transformed back to physical space by FTT.
Let the wavelet expansion u(x) at scale j be
j / 2 j
u(
x)
c 2 ( 2 x k)
, k Z,
(4.2)
k
k
where ck s are periodic wavelet coefficients of u, periodic in x.
j
Put y 2 x so that
U ( y)
u(
x)
C ( y k),
C 2
k
k
k
If d is the period of u, then U (y)
and so also Ck is periodic in y with period 2 d.
j
j / 2
j
Let us discretize U(y ) at all dyadic points x y y Z
2 ,
U i C k
ik
C
ik
k , i 0,
1,
2,....,
n 1.
k
k
The matrix representation is U k
* C , where k is the convolution kernal, i.e. the
first column of the scaling function matrix.
Similarly the wavelet expansion for f (x),
j / 2 j
f ( x)
d 2 ( 2 x k)
, k Z.
(4.3)
k
k
j / 2
f ( x)
Dk
( y k),
Dk
2 dk
.
k
F(
y)
And the matrix representation is
F k
* D.
Substitute the expansions of u(x) and f (x)
in (4.1) and then take inner product on
both sides with ( y j),
j Z .
( n)
Use j k
( y k)
(
y j)
dy and jk (
y k)
(
y j)
dx , we obtain
k . C g.
Now take Fourier Transforms
Uˆ kˆ
. Cˆ
.
Fˆ kˆ
. Dˆ
.
kˆ . Cˆ
gˆ
.
Subsequently, Uˆ
Fˆ
/ kˆ
. Inverse FT will give U.
Wherein . and / denote component by component multiplication and division
respectively.
c
k
.