Remote-Sensing Technologies and Data Processing Algorithms - New Ecoinformatics Tools in Environmental Science: Applications and Decision-making - page 210

From ( 2.59 ) and ( 2.60 ) we get:

Z

x k þ 1

k h k þ h k X

X

n 1
s¼2 e

n

h s 2
k
s

0

ð s Þ
k

½P i y ð n i Þ P i y ð n i Þ dx e k þ 1 ¼ e k þ e

!

i¼1

x k

n

Z Z

x k þ 1 x k þ 1

½f if k dx þ k

if W½y W k ½y g dx þ

x k x k

n n

We know that

Z

x k þ 1

Z

x k þ 1

if k dx ¼ if k h k
n !

W k ½ y dx ¼ W k ½y
n !

;

x k x k

n n

Let us denote

;

;

ð j Þ
k

y ð n i Þ

E k ¼ max
j

e h max ¼ max

k

h k ; p i ¼ max

½a ; b

P j ; M n i ¼ max

½a ; b

; ; ;

P ik if k

y ð n i Þ

L i ¼ max
½a ; b

N n i ¼ max
½a ; b

F ¼ max
½a ; b

j ; G 0 ¼ max

½a ; b

j X

;

r

M j max
G

W k ½y

T ¼ max
G

s ¼ b a j max

G

K j ð x ; nÞ j W½y j;

j¼0

" #

X

n

1
n

l ¼ ð p i M n i þ L i N n i Þ þ F þ G 0 þ jjð T þ s Þ ;

!

i¼1

g ¼ 1 þ X

n 1

h s 1
ax
s

;

!

s¼2

and M n − i ≈

N n − i . Then we get the following recurrent error estimation

E k þ 1 ð 1 þ gh max Þ E k þ lh n max

ð 2 : 61 Þ

Hence we get

k k

e 0 þ lh n 1 g 1

E k ð 1 þ gh max Þ ½ ð 1 þ gh max Þ 1 ð 2 : 62 Þ

ax

where ʵ 0 is the maximum error in the initial data. Obviously, if ʵ 0 = 0, then from

( 2.62 ) it follows that if h max → 0 then E k → 0, i.e. ~ y ð x k Þ! y ð x k Þ .