Geoscience Reference
In-Depth Information
2.12.5.1 Method of Solution and Estimation of Error
Let us consider the equation
L½
k
W½
¼fx
ð ;
;
ð
2
:
49
Þ
L[y] being the differential operator
L½y
¼
X
n
y
0
y
ð
m
i
Þ
Þ
y
ð
n
i
Þ
P
i
ð
x
;
y
;
; ...;
;
ð
m
i
\
n
Þ
ð
2
:
50
Þ
i¼0
and W[y] the generalized Volterra operator
Z
x
X
r
; nÞ
y
ð
j
Þ
ðnÞ
d
W½y
¼
K
j
ð
x
n;
ð
r
n
Þ
ð
2
:
51
Þ
\
j
¼
0
a
y
0
; ...;
y
ð
m
i
Þ
and f(x,y)
ʻ—
a real number, P
i
ð
x
;
y
;
—
continuous functions with
respect to their arguments in the
finite interval [a,b], P
0
≠
0 and kernels K
j
(x,
ʾ
),
j =0,1,
…
, r are continuous functions in the region G{a
≤ ʾ ≤
x
≤
b}.
The initial conditions are
y
ð
s
Þ
ð
a
Þ
¼y
ð
s
Þ
0
;
s ¼ 0
;
1
; ...;
n
1
ð
2
:
52
Þ
Assuming that Eq. (
2.49
) with the initial conditions (
2.52
) has a unique con-
tinuous solution y(x), let us construct an approximate solution
~
y
ð
x
Þ
in [a,b]. Let us
divide the interval [a,b] by a sequence of points x
0
= a, x
1
,
…
, x
m
= b, h
k
= x
k+1
−
x
k
.
On each subinterval [x
k
,x
k+1
], k =0,1,
1 let us replace Eq. (
2.49
) by the
following linear differential equation of the nth order with constant coef
…
,m
−
cients
L
k
½y
¼
k
W
k
½y
þ
f
ð
x
k
; ~
y
k
Þ
ð
2
:
53
Þ
with the initial conditions:
y
ð
s
Þ
k
y
ð
s
Þ
ð
x
k
Þ
¼
~
;
s ¼ 0
;
1
; ...;
n
1
;
ð
2
:
54
Þ
where
L
k
½y
¼
X
n
P
i
ð
x
k
;
y
k
;
y
0
k
; ...;
y
ð
m
i
Þ
y
ð
n
i
Þ
;
ð
2
:
55
Þ
k
i¼0
W
k
½y
¼
X
r
j¼0
ð
K
j
;
k
;
0
~
y
ð
j
0
h
0
þ
K
j
;
k
;
1
~
y
ð
j
1
h
1
þþ
K
j
;
k
;
k
~
y
ð
j
k
h
k
Þ
ð
2
:
56
Þ
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