Remote-Sensing Technologies and Data Processing Algorithms - New Ecoinformatics Tools in Environmental Science: Applications and Decision-making - page 214

Formulas ( 2.67 ) and ( 2.68 ) give a recurrent estimation for error. From them it is
possible to obtain an error estimation applicable to the entire interval [t0,T]: 0 ,T]:

h i

hP 1
p 0

k k

E k e 0 ð 1 þ hp 0 Þ þ ð 1 þ hp 0 Þ 1 ;

where the following notations are introduced

;

ð j Þ
i

h ¼ max
k

h k ; E k ¼ max

i ; j

e ð t k Þ M ¼ max

i

M i ; m ¼ min

s

m s ;

þ l 1
s¼1

h m 1
ðm þ 1 Þ! ;

p 0 ¼ mM h m 1
m!

h s 1
s

P 1 ¼ M ; l ¼ max

s

m s

!

2.12.5.4 Solutions of Equations with Known Moments on the Right Side

Let us consider a particular case of Eq. ( 2.49 ):
L½y ¼y ð n Þ þ X

n

p i y ð n i Þ ¼ f ð x Þ; x 0 ð 2 : 69 Þ

i¼1

where p i is the constant coef cients, f(x) is a single-valued and differentiable

function, f(x) 0asx → ∞ and its moments are known:

→

Z 1

x m f ð x Þ dx M m f ð x Þ ¼

\ 1;

m ¼ 0 ; 1 ; ...; m ð 2 : 70 Þ

0

It is necessary to solve the Eq. ( 2.69 ) with the following initial conditions:

y ð s Þ ð x 0 Þ ¼y ð s Þ ð s ¼ 0 ; 1 ; ...; n 1 Þ

0

We shall approximate f(x) in the following manner:

f ð x Þ exp ð kx Þ X

m

a i x i ¼ P m ð x Þ exp ð kx Þ;

ð 2 :

71 Þ

i¼0

where m > 0 is an integer, k > 0 and a i are constants to be determined. Then, from
( 2.70 ) and ( 2.71 ) we have

" # dx ¼ X

Z 1

x m exp ð kx Þ X

m m

a i ðm þ i Þ!
k mþ i þ 1

M m f ð x Þ ¼

a i x i

i¼0 i¼0

0