Remote-Sensing Technologies and Data Processing Algorithms - New Ecoinformatics Tools in Environmental Science: Applications and Decision-making - page 216

Z

x

G 0 x ð x g 0 ð x Þ

; y Þ

uð x Þþ

x Þ uð y Þ dy ¼ G ð x ; k G ð x ; x Þ

a

Therefore, the argument used to find an approximate solution of ( 2.75 ) is valid

for Eq. ( 2.74 ) as well.
In traditional way the Eq. ( 2.75 ) is solved by replacing the integral of the
equation by a finite sum of some quadratic formula. Applying this approach let us

divide the interval [a,b] by a sequence of points x 0 ¼ a x 1 \ x 2 \ \ x m ¼ b

\

into elementary intervals ʔ j =[x j ,x j+1 ], and instead of ( 2.75 ) let us write the

equation:

Z

x i þ 1

X

j 1

uð x j Þk K ð x j ; y Þuð y Þ dy ¼ f ð x j Þ; ð j ¼ 0 ; 1 ; ...; m Þ ð 2 : 76 Þ

i¼0

x i

Further, because of the assumptions made on ˆ (x) and K(x,y) we can write

2

ð x i Þþ ð x x i Þ

0

00 ðn i Þ;

uð x Þ ¼ uð x i Þþð x x i Þu u

2 !

ð x i n i x x i þ 1 Þ ð 2 : 77 Þ

and assuming the existence and differentiability of K y ð x

; y Þ we have

2

x i Þþ ð y x i Þ x i Þþð y x i Þ K y ð x j ; K 0 yy ð x j ; g i Þ;

K ð x j ; y Þ ¼K ð x j ;

2

ð x i g i y x i þ 1 Þ ð 2 : 78 Þ

substituting ( 2.77 ) and ( 2.78 )in( 2.76 ) we get:

X

j 1
i¼0 f K ð x j ;

h i
2 ½K ð x j ;

0

ð x i Þþ K y ð x j ;

uð x j Þk x i Þuð x i Þþ x i Þu x i Þuð x i Þ

h i

3 K y ð x j ;

0

þ x i Þu ð x i Þg h i þ R j ¼ f ð x j Þ

where

X

j 1
i¼0 f

h i
2 u

1
3 K y ð x j ;

h i

00 ðn i Þ ½

4 K y ð x j ;

R j ¼ k x i Þþ x i Þ

h i h i

20 u

1
3 uð x i Þþ

h i
4 u 2 K 0 yy ð x j ; g i Þ ½

0 ð x i Þ þ 00 ðn i Þ K 0 yy ð x j ; g i Þg

þ