( о г —к 2)к р 2 + к 2
со2 - к 2 р 2 + со2
tasvir fu n k siy a ja d v a lid a n foydalanib, m asalaning quyidagi
x ( t ) = — ■
Л ,— ( - со sin kt + к sincot)
(к — со )k
’
yech im ig a ega boMamiz.
Bu yech im d an k o ‘rinib turibdiki, m asalan in g yechim i ikkita garm onik
tebranish, y a ’ni chastotasi к boMgan xususiy tebranish
, ,
Aco
. ,
x ( t ) = ------ :------ -— sm k t
(к -с о )k
va chastotasi CO boMgan m ajburiy tebranish
79
х
( t ) =
—
г ~ — г
si ncot
к ' - с о '
lar y ig ‘indisidan iborat b o ‘lar ekan.
T asvir funksiyalardan to ‘g ‘rid an -to ‘g ‘ri foydalanish m aq sad id a quyidagi
jadvalni keltiram iz.____________________________ ____________________________
N
F ( p )
f ( t )
1
1
P
1
2
a
p ' + a 1
sin at
3
P
p ' + a 2
cos at
4
1
p + a
e~at
5
a
p 2 -СС'
s h a t
6
Р
p 2 - a 2
chat
7
a
{p + a ) 2 + a 2
e * s i n at
8
p + a
( p + a ) 2 + a 2
e~M cosat
9
n\
p -
t"
10
2 p a
( p 2 + a 2 ) 2
t s i n at
11
p 2 - a 2
( p 2 + a 2) 2
t cosat
12
1
( p + a ) 2
te
13
2 p a
( p 2 + a 2) 2
1 ,
.
—
7
(sin a t - at c o s a t J
2 a ’
14
r-1
r ~ F ( p )
d p
t " f ( 0
15
Ft( p ) F, ( p )
) f ( T ) f 2( t - f ) d T
80
*
6.2. F u n k siy a n i sonli d iffe re n s ia lla s h
A g ar x - [a, b] oraliqqa tegishli ixtiyoriy x t tugun nuqtaning qiym atlarini
qabul qiluvchi erkli o ‘zgaruvchi boMsa, u h o ld a ixtiyoriy tugun nuqta qiym atini
x + kh,
k =
0
, ±
1
, ±
2
, ...
k o 'rin ish d a yozish m um kin.
5 > 1
dxb
hosila qiym atini, funksiyaning tugun nuqtalardagi qiym atlari, y a ’ni y ( x + kh) lar
orqali ifodalash, y ( x ) funksiya hosilasini ta q rib iy hisoblash yoki taqribiy
differensiallash deb ataladi.
F a ra z qilaylik
y ( x ) e C 2[ a , b ] , x < b boMsin.
y ( x + h ) nin g ikkinchi
tartibli an iq lik d a T ey lo r qatoriga yoyilm asi
y ( x + h ) = y ( x ) + ^
X ^ h + 0 ( h 2 )
dx
dan fo y d alan sak
d y ( x ) _ y ( x + h ) - y ( x ) |
dx
h
ten g lik g a e g a b o ‘lam iz. A gar bu y erd a x = x f d eb olsak, birinchi tartibli hosila
uchun ikki nuqtalik o ld in g a taqribiy hisoblash fo rm u lasi hosil b o ‘lad i:
M b l = y (* M ) - y < * , l + 0 ( h ) ' о < / < m-\
(6.3)
dx
h
O d atd a sonli differensiallash form ulasi d eg an d a quyidagi
d)’( x ,)
У(
) - У (x , )
dx
h
taqribiy fo rm u la tushuniladi.
p
d y ( x ,)
y ( x l+]) - y ( x , )
dx
h
ayirm aga sonli differensiallash xatosi deb ataladi. (6 .3 ) form ula uchun /г —>0 d a
R = 0 ( h ) .
X uddi sh u n g a o ‘xshash birinchi tartibli h o sila uchun ikki nuqtalik taqribiy
hisoblash form ulasini hosil qilish m um kin:
dx
h
A g ar y ( x ) e C 3[ a , b j b o ‘lsa, birinchi tartib li h osila uchun y an ad a aniqroq
ikki n u q talik taqribiy hisoblash form ulasi - m ark aziy ayirm a form ulasi m avjud v a
u quyidagi k o ‘rin ish d a boMadi:
d y ( X j) _ У (x i+A) ~ y ( x , A )
dx
2 h
yoki
+ 0 ( 1 r ) ,
1 < / < m - \ .
81
ф ( х , ) у ( х м ) - у ( х , _ х)
(6.4)
dx
2
Л
Birinchi tartibli hosilani taqribiy h iso b lash d a k o ‘p nuqtali, m asalan uch nuqtali
=
h
J ■- 3
* . , )
dx
2 h
form uladan ham foydalanish m um kin.
U m um iy holda birinchi tartibli h o sila u chun k o ‘p nuqtali form ulaning
k o ‘rinishi q u y id ag ich a b o ‘ladi:
d ) ’( x i )
s r *
/
\ , / ^ / L iP
1
■
= L aky ( xJ + ° ( h )
dx
k=o K
Bu yerda a k larni shunday tanlash m um kinki, form ulaning aniqlik tartibi p ga
teng b o ia d i.
Ikkinchi tartibli hosilani hisoblash uchun ham taqribiy hisoblash form ula-
larini keltirish m um kin:
d 2y ( x ; )
y ( x i+l) - 2 y ( x l ) + y ( x j_] )
d x 1
+ 0(h2)
yoki
d 2y ( x t )
y ( x M ) - 2 y ( x i ) + y ( x l_i )
d x 2
h !
(
6
.
6
)
1-m iso l. у — X J + e funksiya uchun у ( 2 ) ni hisoblang.
Y e c h ish . h = 0,05 deb, quyidagi
X
1,9
1,95
2 , 0 0
2,05
2 ,1
У
13,5449
14,4436
15,3891
16,3830
17,4272
jad v aln i
tu zib
olam iz.
y 0 = y ( l , 9 ) = 13.5449;
y\ = y ( \.9 5 ) = 14,4436
va
y 2 = y ( 2^ = 15,3891 ^ , = ^ 2 , 0 5 ^ = 16,3830 y 4 = y ( 2,1 ^ = 1 7 ,4 2 7 2 ekanligini h i
sobga olsak, old in g a taqribiy hisoblash
dy(x, )
y ( x M ) - y ( x i )
form ulasiga asosan
у ( V
dx
h
' 1 6 ,3 8 3 0 -1 5 ,3 8 9 1
0,05
= 19,878
U
<1
•
л
*
-U- U- l i u
d y ( x i )
y ( x , ) - y ( x i - \ ) f
ga ega bo lam iz. A gar orqaga taqribiy h is o b la s h ----------= ------------ :---------- torm u -
dx
lasi dan foydalansak
82
1 5 3 8 9 1- 14 ’4 4 3 6 = 1 8 9
i
0,05
ni hosil qilam iz. A gar ikki n u q talik taq rib iy hisoblash form ulasidan foydalansak
ш
ш
-- н , 4 4 з б =
1
2 -0 ,0 5
boMadi. A g ar uch nuqtali fo rm u la (6.5) dan foydalansak
» .
- 1 7 .4 2 7 2 + 4 1 6 ,3 8 3 0 -3 -1 5 ,3 8 9 1
у ( 2 ) & --------------------------------------------------= 19,3777
2 • 0,05
ga eg a boMamiz.
A gar у ( 2 ) n in g aniq q iym ati 19,3891 ekanligini hisobga olsak, taqribiy
hisoblashdagi absolyut xato birinchi holda 0 ,4889 ga, ikkinchi h o ld a 0,4 791 ga,
uchinchi holda esa 0,0049 ga v a to ‘rtinchi h o ld a esa 0,0114 teng boMadi. Bu h o l
larda nisbiy xatolar m os rav ish d a 2 ,5 2 % , 2,47% , 0 ,025% va 0 ,059% ga teng
boMadi.
2-misol.
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