Remote-Sensing Technologies and Data Processing Algorithms - New Ecoinformatics Tools in Environmental Science: Applications and Decision-making - page 213

The system ( 2.66 ) can be replaced by an equivalent system of m ¼ m 1 þþ
m n equations of the first order, relative to the derivatives for all m unknown

functions. Then one of the standard software can be used to solve the last system.
Let the functions fi, i , i =1,

, n be continuous and differentiable with respect to
all arguments. Let us suppose that the solution of system ( 2.66 ) with the initial
conditions

…

y i ð t 0 Þ ¼ ð y i Þ 0 ; ...; y ð m i 1 Þ ð t 0 Þ ¼ ð y ð m i 1 Þ y i ð t 0 Þ ¼ ð y i Þ 0 ; Þ 0

i i

exists and is unique in t 0 ≤

≤

T.
Divide the interval [t0,T] 0 ,T] into elementary intervals

t

k =[t k ,t k+1 ] by a sequence of
points t 0 \ t 1 \ \ t l ¼ T. On each such interval, let us search the solution of
system ( 2.66 ) in the form of a series:

ʔ

y i ð t k Þþ X

m i 1 j m i

ð t t k Þ

y ð j Þ ð t k Þþ ð t t k Þ ð f i Þ k ;

~

y i ð t Þ ¼ ~

~

j ! ð m i Þ!

j¼1

ð t k 1 Þþ X

m i j 1 s

ð t k t k 1 Þ

y ð j Þ
i

ð t k Þ ¼y ð j Þ
i

y ð s þ j Þ
i

ð t k 1 Þ

s !

s¼1

m i j

þ ð t k t k 1 Þ ð f i Þ k ; ð j ¼ 1 ; ...; m i 1 Þ

ð m i j Þ!

The error of such a solution can be easily estimated, considering the exact
expansion of the functions y i (t) and y ð j Þ ð t Þ in a Taylor series:

i

þ M i

j þ X

m i 1

h k
j !

h m i þ 1
k
ð m i þ 1 Þ!

ð j Þ
i

j e i ð t k þ 1 Þ j e i ð t k Þ j e ð t k Þ

j¼1

X

m j 1
s¼0 e

X

h m i
k
ð m i Þ!

n

ð s Þ
j

þ M i ð t k Þ ð 2 : 67 Þ

j¼1

e þ X

m i j 1

h k 1
s

ð j Þ
i

ð j Þ
i

ð s þ j Þ
i

e ð t k Þ ð t k 1 Þ e ð t k 1 Þ

!

s¼1

!

X

m j 1
s¼0 e

X

h m i j þ 1
k
ð m i j þ 1 Þ!

h m i j
k
ð m i j Þ!

n

ð s Þ
j

þ M i þ ð t k 1 Þ ð 2 : 68 Þ

j¼1

where

( )

; ; ...;

@ f i

@ t

f i
@ y 1

@ @ f i

M i ¼ max
½t 0 ; T

y ð m n 1 Þ

@

n