Remote-Sensing Technologies and Data Processing Algorithms - New Ecoinformatics Tools in Environmental Science: Applications and Decision-making - page 217

Neglecting the small quantity R j in this expression, we get a recurrent formula
for determining , x j − 1 .By

differentiating ( 2.75 ) we get a formula to calculate the values of the derivative

ˆ (x j ) from the values of the function ˆ (x)atx = x 0 , x 1 , …

ˆ′ (x):

Z

x

0 ð x Þ ¼f 0 ð x Þþk

K x ð x

u K ð x ; x Þuð x Þþk ; y Þ dy

a

From this equation, we have at x=x i

Z

x s þ 1

X

i 1

0

ð x i Þ ¼f 0 ð x i Þþk K x ð x i ; u K ð x i ; x i Þuð x i Þþk y Þuð y Þ dy ð 2 : 79 Þ

s¼0

x s

where

0 x ðÞ ¼f 0 ðÞþk

u x ðÞ ¼f ðÞ; u Ka ð u ; ðÞ

Neglecting the quantity

X

i 1
s¼0 f K 0 xy ð x i ; Q s Þ

h s h s

2
3 u 0 ð x s Þ h s þ 4 u 00 ðn s Þ

r i ¼ k

2 uð x s Þþ

h s

þ K x ð x i ;

00

x s Þ

6 u

ðn s Þg; ð x s n s ; Q s y x s þ 1 Þ

Equation ( 2.79 ) can be written as:

X

i 1

h s
2 u

0

ð x i Þ ¼f 0 ð x i Þþk h s K x ð x i ;

0

u K ð x i ; x i Þuð x i Þþk x s Þ uð x s Þþ ð x s Þ

s¼0

We get finally the equation for determining uð x j Þ :

X

j 1
i¼0 uð x i Þ K ð x j ;

h i ¼ 0

uð x j Þ f ð x j Þk x i Þ l ji þ h i c ji d i ; ð 2 : 80 Þ

where, for sake of brevity, the following notation is introduced: