Remote-Sensing Technologies and Data Processing Algorithms - New Ecoinformatics Tools in Environmental Science: Applications and Decision-making - page 215

k is fixed from the conditions of best approximation by ( 2.71 ). Then the Eq. ( 2.69 )

is replaced by the approximate equation:

y ¼exp ð kx Þ X

m

a i x i

L½ ~

; ð 2 : 72 Þ

i¼0

which can be easily solved. For the error eð x Þ ¼y ð x Þ~ y ð x Þ , we obtain an equation

from ( 2.69 )to( 2.72 ):
L½ e ðÞ ¼f ðÞ exp kx

ð Þ P m ðÞ R m ðÞ ð 2 : 73 Þ

Solving Eq. ( 2.73 ), we have

1

n þ k

j n

ð b a Þ

M k j eð x Þ ;

! k !

k¼0

where

s Þ ¼ X

n i 1

p i ð x s Þ M ¼

a s x b K ð x

max j ; s Þ j; g ¼ max

½a

j R m ð x Þ j; K ð x ;

ð i 1 Þ! ; b

i¼1

2.12.5.5 Re nements of Approximate Solutions of Volterra Integral

Equations

Let us consider Volterra integral equation of the first and of the second kind arising

in the remote monitoring problems:

Z

x

k G ð x ; y Þuð y Þ dy ¼ g ð x Þ; ð 2 : 74 Þ

a

Z

x

uð x Þk K ð x ; y Þuð y Þ dy ¼ f ð x Þ; ð 2 : 75 Þ

a

where x ∈ [a,b], the kernel K(x,y) and its derivatives K x ð x ; y Þ are continuous in the
region R{a

b}, f(x) is a continuously differentiable function in (a,b),
the kernek G(x,y) and g(x) are twice continuously differentiable functions of x,
G(x,x)

≤ y ≤ x ≤

≠ 0. Then, as it is known, Eqs. ( 2.74 ) and ( 2.75 ) have unique solutions

ˆ 1 (x) and ˆ 2 (x) respectively which are continuous and differentiable in [a,b] for any

ʻ

value of . The case when G(x,x) = 0 for some point in the interval [a,b] or for the

entire interval needs special consideration. In our case, Eq. ( 2.74 ) is equivalent to
the equation of second kind