Remote-Sensing Technologies and Data Processing Algorithms - New Ecoinformatics Tools in Environmental Science: Applications and Decision-making - page 209

The general solution of Eq. ( 2.53 ) is known:

c ð k Þ
1

c ð k Þ
2

c ð k Þ
n

~

y ¼ ~

y ð x ; ; ; ...; Þ; ð 2 : 57 Þ

where the constants are determined from the initial conditions at the beginning of
each interval [x k , x k+1 ]. The calculations are carried out successively beginning with
the interval (k = 0).
Let us estimate the error in solution of Eq. ( 2.49 ). Let y(x) and

~

y ð x Þ by the exact

and the approximate solutions respectively. Let us denote

y ð m i Þ
k

f k ¼ f ð x k ; ~

P ik ¼ P i ð x k ; ~

y 0 k ; ...; ~ y k ; ~ Þ; y k Þ;

ð 2 : 58 Þ

e k ¼ y ð x k Þ~ y ð x k Þ

Let us integrate times Eqs. ( 2.49 ) and ( 2.53 ) from x k to x, and consider the final

results for x = x k+1 . For the sake of convenience and brevity let us denote

Z Z Z Z

x k þ 1 x x x k þ 1

uð x Þ dx dx
n

¼ uð x Þ dx ;

x k x k x k x k

n n

we have

Z

x k þ 1

y k þ 1 ¼ y k þ y k h k þ h k X X

n 1 n

h s 2
k
s

y ð s Þ
k

P i y ð n i Þ dx

!

s¼2 i¼1

x k

n

ð 2 : 59 Þ

Z

x k þ 1

Z

x k þ 1

f ð x ; y Þ dx þ k W½y dx

x k x k

n n

Z

x k þ 1

y k þ 1 ¼y k þ y 0 k h k þ h k X X

n 1
s¼2 y ð s Þ

n

h s 2
k
s

P ik y ð n i Þ dx

k

!

i¼1

x k

n

ð 2 : 60 Þ

Z Z

x k þ 1 x k þ 1

f k dx þ k

W k ½y dx

þ

x k
n

x k

n