Remote-Sensing Technologies and Data Processing Algorithms - New Ecoinformatics Tools in Environmental Science: Applications and Decision-making - page 211

In case that the Eq. ( 2.49 ) has the form:

Z

x

y 0 y ð m l Þ dx L½y ¼f ð x ; y Þþ F ð x ; y ; ; ...; ; x 2 ½a ; b ;

a

then the Eq. ( 2.62 ) will read

p ð 1 Þ h n
p ð 0 Þ

k k

E k ð 1 þ hp ð 0 Þ Þ ½ ð 1 þ hp ð 0 Þ Þ

e 0 þ 1 ;

where

" #

p ð 0 Þ ¼ X

n 1

b þ X

n
i¼1 ð p i þð m l þ 1 Þc i b n i Þþð b a Þ c ð m l þ 1 Þ

h s 1
s !

h n 1
n !

þ ;

s¼1

!

a þ b b 1 þ 2B þ X

n

þ ð b a Þ q
n

1
ð n þ 1 Þ!

p ð 1 Þ ¼

p 0i ;

!

i¼0

; ; ;

n o

; y ð i Þ

@ f
@

@ f
@

@ P i
@ y ð s Þ

b ¼ max
½a ; b

y ð i Þ

a ¼ max
½a ; b

c i ¼ max
s ; ½a ; b

b i ¼ max
½a ; b

;

x y

! !

;

5 A þ C X

m l
s¼0 b s þ 1

l i þ c i X

m l
s¼0 b s þ 1

@ F

c ¼ max
s ; ½a ; b

q ¼ 0 : ; p oi ¼ b n i þ 2 a i b n þ 1 i ;

@ y ð s Þ

;

@ F

j ; F

A ¼ max
½a

B ¼ max
½a

@ x

; b ; b

2.12.5.2 Solution of Equation y ðÞ ¼ f ð x y 0 ; ...; y n 1

ð Þ Þ

; y ;

Let us apply the approximate method of the solution presented in Sect. 2.12.5.1 to
integro-differential equations to solve the initial value problem:

y ðÞ ¼ f ð x y 0 y ð n 1 Þ Þ;

; y ; ; ...; ðÞ2 G x ; ; ð 2 : 63 Þ

y ð j Þ ð x 0 Þ ¼y ð j Þ

yx ðÞ ¼y 0 ; ; j ¼ 1 ; ...; n 1 ; ð x 0 ; y 0 Þ G ; ð 2 : 64 Þ

0

where the function f satis es the Lipschitz condition

K X

n 1
i¼0 djj ð 2

y ð n 1 Þ þ d n 1 Þ f ð x y 0 y ð n 1 Þ Þ

f ð x ; y þ d 0 ; ...; ; y ; ; ...; : 65 Þ

Let us divide the interval [a,b] by a sequence of points x 0 = a, x 1 , … , x s = b into

elementary intervals. Let E ={x 0 , … , x s }. On each interval [x ʽ ,x ʽ +1 ], let us solve the

initial value problem: