Wavelet Galerkin Solutions of Ordinary Differential Equations

Wavelet Galerkin solutions 411
k
a x y
j,
k k j or equivalently AX = Y . (2.5)
Thus in Galerkin method, for each subset , we find an approximation u s in S to u
by solving (2.5) for X and then substituting its components in (2.2). It is expected
that as we increase in some systematic way, us converges to u, the actual solution.
Condition Number of a Matrix. We know that a linear system AX Y has a
unique solution X for every Y if a square matrix A is invertible. It is often observed
that for two close values of Y andY ',
X and X ' are far apart. Such a linear system
is called badly conditioned. Thus data Y is expected to be fairly accurate. Condition
number of A is given by
1
( A)
A A , C ( A)
1 .
C
#
#
Thus # ( ), A C is the measure of stability of the linear system under perturbation of the
data Y . Small condition number near 1 is desirable. In case it is high, replace the
system by equivalent system BAX BY , B is a preconditioning matrix such that
C# ( BA)
C#
( A)
.
To facilitate easy calculation, A is considered to be sparse, i.e., A should have high
proportion of entries 0. The best one is when A is in diagonal form.
Wavelet Galerkin method
j / 2 j
Let j,
k ( x)
2 ( 2 x k)
be a wavelet basis for
conditions
2
L ([ 0,
1])
with boundary
0)
( 1)
0.
j,
k ( j,
k
For each ( j , k)
,
j,
k is
The scale of approximates
2
C .
j
2 and is centralized near point k
j
2 and equates
to zero outside the interval centred at k
j
2 of length proportional to 2 .
j
In Wavelet Galerkin method equations (2.2) and (2.3) may thus be replaced by
x
and
u s
j,
k j,
k
j,
k
j,
k
L
j,
k , l,
m x
j,
k f , l,
m
( l,
m)
.
So that AX Y.
(2.6)
Where A= a , X ( x j,
k ) ( j,
k )
, Y ( yl
, m ) ( l,
m)
.
l,
m;
j,
k ( l,
m),
( j,
k )
In it a l,
m;
j,
k L
j,
k , l,
m , yl , m f , l,
m .
The pairs (l,m) and (j,k) represent respectively row and column of A.
Consider A to be sparse. Represent AX Y by equivalent