Remote-Sensing Technologies and Data Processing Algorithms - New Ecoinformatics Tools in Environmental Science: Applications and Decision-making - page 207

" # 2

E ¼ X

X

P i t ðÞþ X

M N N c

C k H ð k Þ ð t s Þn i ð t s Þ

¼ min
C fg

ð 2 : 46 Þ

i

s¼1 i¼1 k¼1

Let demand:

@ E =@ C k ¼ 0 for k ¼ 1 ; ...; N c ð 2 : 47 Þ

From Eqs. ( 2.46 ) and ( 2.47 ) it is followed that

X

N c

A km C k þ B m ¼ 0 ; m ¼ 1 ; ...; N c ; ð 2 : 48 Þ

k¼1

where

A km ¼ X

X

B ¼ X

X

M N M N

H ð k Þ
i

ð t s Þ H ð m Þ x i ð t s Þ H ð m Þ

ð t s Þ; ½P i ð t s Þ^ ð t s Þ

i i

s¼1 i¼1 s¼1 i¼1

Finally, every iteration of ( 2.42 ) needs to solve the system of equations ( 2.48 ).
Unfortunately, the convergence of this procedure depends on the successful
selection of initial conditions.

2.12.5 Quasi-Linearization Method

The number of problems that crop up in ecoinformatics leads to the necessity of
integrating generally non-linear integro-differential equations; but, in a majority of
cases, these equations are not integrable by elementary or special functions. To
solve them, as a rule, it is necessary to make use of the latest achievements of
calculating methods and technics. In many problems the use of well-known
numerical methods to solve initial value problems — even by means of modern high-

speed electronic computers does not come up with desired results. The existing

approximate methods of solving integro-differential equations are based as rule on
replacing derivatives by

—

finite differences and represent a complicated multi-step
process, which in practical problems cannot be solved on computers reasonably
quickly. Therefore in solving practical problems we have to search other means of
approximate solutions for integro-differential equations, without using the

finite-

difference methods.
In the method considered here, the integro-differential equation is substituted in
each subinterval of the independent variable by an easily integrable ordinary dif-
ferential equation with constant coef

cients; this method is not a new theoretical
idea for it was known to Euler. However, here the error estimations are obtained for
the first time, and methods applicable to various problems are developed in detail.