Geoscience Reference
In-Depth Information
2.12.5.1 Method of Solution and Estimation of Error
Let us consider the equation
k
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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