Remote-Sensing Technologies and Data Processing Algorithms - New Ecoinformatics Tools in Environmental Science: Applications and Decision-making - page 212

y ð n Þ ¼ f ð x y 0 y ð n 1 Þ

m

; ^ y m ; ^

m ; ...; ^

Þ; ð x ; ^ y m Þ2 G ; m ¼ 0 ; 1 ; ...; s 1 ;

y ð j Þ ð x m Þ ¼ ^ y ð j m ;

y ð x m Þ ¼ ^ y m ; j ¼ 1

; ...;

n 1

; ð x m ; ^

y m Þ2 G

Then, if the function f satis es condition ( 2.65 ) and max i ½x i þ 1 x i ¼h, the

solution of this problem

y ð j Þ ¼ f y ð j Þ
0

y ð j Þ
1

y ð j Þ
s

^ y ¼ f y 0 ; ^ y 1 ; ...; ^ y s g; ^ ; ^ ; ...; ^ g; j ¼ 1 ; ...; n 1

when h 0 tends to the solution of Eqs. ( 2.63 ), ( 2.64 ) and the estimation for the

rate of convergence is as follows:

→

e 0 1 þ h

h a 1

y ð n l Þ
r

y ð n l Þ
r

r r

max
l

^ ð a 0 Þ þ ½ ð 1 þ h a 0 Þ 1 ; r ¼ 1 ; 2 ; ...; s

2 a 0

where

!

a 0 ¼ X

2 X

X

n 1 n 1 n j 2

h i 1
i

h s
s

h

þ Kn þ ;

! !

i¼1 j¼0 s¼0

" #

a 1 ¼ K X þ X

n j 2

n 1

h n j 1
ð n j 1 Þ!

N s þ j þ 1 h s
s !

;

M

j¼0 s¼0

;

y 0 y ð n 1 Þ y ð s þ j þ 1 Þ

r

M ¼ max
½a ; b

f ð x ; y ; ; ...; N s þ j þ 1 ¼ max

r

^

If the initial conditions are exactly given, the error estimation has the form:

Dh 2

1 l n 1 y ð n l Þ

y ð n l Þ
r

max ^ ; r ¼ 1 ; ...; s ;

r

where

r

1 þ he h 1 þ 0 ð : 5Knh Þ þ Kn 1

N ¼ max

1 r s N j ;

D ¼ K ð M þ nD Þ

41 þ Kh 0 ½ ð : 5h þ e h Þ

2.12.5.3 Solution of a System of Ordinary Differential Equations

For the sake of simplicity let us con ne ourselves to the important case of equa-

tions, having the canonical form

y ð m i Þ
i

y ð m 1 1 Þ
1

y 0 1 ; ...; y ð m n 1 Þ

n

ð t Þ ¼ f i ð t ;

y 1 ;

; ...; Þ;

i ¼ 1

; ...; ð 2 :

66 Þ

n