Remote-Sensing Technologies and Data Processing Algorithms - New Ecoinformatics Tools in Environmental Science: Applications and Decision-making - page 218

K y ð x j ;

x i Þ

h i
2 k

2
3 k

l ji ¼ 1 þ K ð x i ; x i Þþ 1 þ h i K ð x i ; x i Þ

K ð x j ; x i Þ

K y ð x j ;

x i Þ

1
2 K ð x j ;

2
3 h i

c ji ¼ x i Þ 1 þ

K ð x j ; x i Þ

X

i 1

h s
2 u

d i ¼ f 0 ð x i Þþk h s K x ð x i ;

0 ð x s Þ

x s Þ uð x s Þþ

s¼0

Thus, starting the calculation by formula ( 2.80 ) from j = 1. We get the values of
solution u 1 ; u 2 ; ...; u m with the error (x i ). Let us estimate the value of modulus of

the error ʵ (x) from above. Let us introduce the following notation:

ʵ

0 ð x Þ 00 ð x Þ

M ¼ max

½a ; b uð x Þ

j j; N ¼ max

½a ; b u

j j; L ¼ max

½a ; b u

j j;

;

n o

K y ð x K x ð x Q ¼ max

R

; y Þ ; y Þ ; G ¼ max

R

j K ð x ; y Þ j;

n ; o

K 0 yy ð x K 0 xy ð x B ¼ max

R

;

y Þ

;

y Þ

;

h ¼ max
s

h j ;

m
6 BM þ

3h
4 N

3
4 Qh þ

3
10 Bh 2

l ¼ þ LG þ ;

h 2
4 L

n ¼ jj Bm
2

2h
3 N þ

h
6 BL

M þ þ

Then we have

X

j 1

jj hG þ

2
3 Qh

h
2 e

i

þ jj lh 3

e j ejjþ ;

i¼0

ejjþ jj Qh X

i 1

jj G

h
2 e

i s

þ nh 2

e ejjþ ;

s¼0

e i ¼ u 0 ð x i Þu 0 ð x i Þ . From these formulas we get roughly

where e j ¼ uð x j Þuð x j Þ ,

!

ejjþ hQ X

jj G

i 1
s¼0 ejj

i

Þ h 2

e þ n þ jj QNm ð ;

" #

X

X

jj hG þ

j 1 i 1

s¼0 ejj

2
3 Qh

1 þ jj Gh
2

ejjþ jj Gh
2

þ jj th 3 e j ;

i¼0