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2.3.
O’zgaruvchan koeffisiyentli bir o’lchovli ko’chirish tenglamasini
notekis to'r yordamida sonli yechish.
Masalaning qo’yilishi.
Ko’chirish tenglamasi quyidagi ko’rinishda bo’lsin:
),
,
(
)
,
(
)
,
(
)
,
(
t
x
f
u
t
x
q
x
u
t
x
p
t
u
t
x
,
0
l
x
,
0
T
t
(2.1'')
uning boshlang’ich sharti:
),
(
)
0
,
(
0
x
u
x
u
(2.2'')
va chegaraviy shartlari:
)
,
(
),
,
0
(
0
t
l
p
P
t
p
P
N
.
q
yo'
shartlar
chegaraviy
,
0
,
0
)
4
),
(
)
,
(
),
(
)
,
0
(
,
0
,
0
)
3
),
(
)
,
0
(
,
0
,
0
)
2
),
(
)
,
(
,
0
,
0
)
1
0
2
1
0
1
0
2
0
N
N
N
N
P
P
t
t
l
u
t
t
u
P
P
t
t
u
P
P
t
t
l
u
P
P
(2.3'')
Kiritiladigan ma’lumotlar:
1)
0
,
0
0
N
P
P
1
2
0
2
2
)
(
,
)
(
]
3
)
1
(
)
1
(
[
)
,
(
2
)
,
(
,
5
)
,
(
l
x
t
x
Ae
t
Ae
x
u
t
t
x
x
Ae
t
x
f
t
x
t
x
p
t
x
t
x
l
=1, T=1
Aniq yechim:
1
,
)
,
(
A
Ae
t
x
u
t
x
2)
0
,
0
0
N
P
P
39
t
x
t
x
Ae
t
Ae
x
u
t
t
x
x
Ae
t
x
f
t
x
t
x
p
t
x
t
x
)
(
,
)
(
]
7
)
1
(
)
1
(
[
)
,
(
)
2
(
)
,
(
,
5
)
,
(
1
0
2
2
Aniq yechim:
1
,
)
,
(
A
Ae
t
x
u
t
x
3)
0
,
0
0
N
P
P
t
l
t
x
t
t
x
t
Ae
t
Ae
t
Ae
x
u
e
x
t
x
Ae
t
x
f
e
x
t
x
p
t
x
t
x
)
(
,
)
(
,
)
(
]
)
1
2
(
)
10
[(
)
,
(
)
1
2
(
)
,
(
,
10
)
,
(
2
1
0
2
2
Aniq yechim: :
1
,
)
,
(
A
Ae
t
x
u
t
x
4)
0
,
0
0
N
P
P
x
t
t
x
t
Ae
x
u
e
x
t
x
Ae
t
x
f
e
x
t
x
p
t
x
t
x
)
(
]
)
1
2
(
)
10
[(
)
,
(
)
1
2
(
)
,
(
,
10
)
,
(
0
2
2
Aniq yechim:
1
,
)
,
(
A
Ae
t
x
u
t
x
(2.1'') – (2.3'') chegaraviy masalani yechish uchun ishonchli, oshkor va
oshkormas sxemalardan foydalanamiz.
Oshkor sxemalar.
p(x,t) > 0, (p
0
>0, p
N
>0) yoki p(x,t)<0, (p
0
<0, p
N
<0)
bo’lganda oshkor sxemalardan foydalaniladi. Amaliyotda ko’pincha yugiruvchi hisob
sxemasidan foydalaniladi. p(x,t) funksiyaning ishorasiga qarab chap yoki o’ng
ayirmali sxema qo’llaniladi.
Har ikkala hol uchun yugiruvchi hisob sxemasini qaraylik.
1) p(x,t)>0, (p
0
>0, p
N
>0)
O’ng ayirmali sxema:
1
1
1
1
1
1
1
1
1
j
i
i
j
i
j
i
j
i
j
j
i
j
i
j
i
f
h
y
y
p
y
y
;
(2.1''′)
)
(
0
0
i
i
x
u
y
; (2.2′'')
1
2
1
j
j
N
y
;
(2.3''′)
(2.1′'') dan kelib chiqadiki,
1
1
1
1
1
1
1
1
1
*
j
i
j
i
j
i
j
j
i
j
i
j
i
j
i
j
i
R
f
y
y
R
y
,
bu yerda
1
1
1
1
*
j
i
i
j
j
i
p
h
R
0
,
1
N
i
1
,
0
0
j
j
.
2) p(x,t)<0, (p
0
<0, p
N
<0)
Chap ayirmali sxema:
1
1
1
1
1
1
1
1
1
j
i
i
j
i
j
i
j
i
j
j
i
j
i
j
i
f
h
y
y
p
y
y
;
(2.1*)
40
)
(
0
0
i
i
x
u
y
; (2.2*)
1
1
1
0
j
j
y
;
(2.3*)
(2.1*) dan kelib chiqadiki,
1
1
1
1
1
1
1
1
1
*
j
i
j
i
j
i
j
j
i
j
i
j
i
j
i
j
i
R
f
y
y
R
y
,
bu yerda
1
1
1
1
*
j
i
i
j
j
i
p
h
R
,
N
i
,
1
,
1
,
0
0
j
j
.
1-jadval. O’zgaruvchan koeffisiyentli bir o’lchovli ko’chirish tenglamasini oshkor
(o’ng ayirmali) sxemaning yugiruvchi hisob sxemasi bo’yicha sonli hisob
natijalari
p0>0, pN>0 va 50-qatlam uchun
N Taqribiy yechim Aniq yechim
Xatolik
0
0.10039200
0.10004559
0.00034641
1
0.10731313
0.10694264
0.00037049
2
0.11471141
0.11431517
0.00039623
3
0.12261970
0.12219596
0.00042375
4
0.13107319
0.13062004
0.00045315
5
0.14010945
0.13962487
0.00048458
6
0.14976865
0.14925048
0.00051817
7
0.16009374
0.15953968
0.00055407
8
0.17113063
0.17053820
0.00059243
9
0.18292837
0.18229495
0.00063342
10 0.19553941
0.19486220
0.00067721
11 0.20901984
0.20829583
0.00072401
12 0.22342957
0.22265555
0.00077402
13 0.23883258
0.23800523
0.00082736
14 0.25528740
0.25441310
0.00087431
15 0.27195211
0.27195211
0.00000000
41
2.9-rasm. Yechim ustivir ekan.
2-jadval. O’zgaruvchan koeffisiyentli bir o’lchovli ko’chirish tenglamasini oshkor (chap
ayirmali) sxemaning yugiruvchi hisob sxemasi bo’yicha sonli hisob natijalari
p0<0, pN<0 va 50-qatlam uchun
N Taqribiy yechim Aniq yechim
Xatolik
0
0.14715178
0.14715178
0.00000000
1
0.14242453
0.14232757
0.00009697
2
0.13785337
0.13766151
0.00019185
3
0.13343317
0.13314843
0.00028474
4
0.12915902
0.12878331
0.00037571
5
0.12502613
0.12456129
0.00046484
6
0.12102988
0.12047768
0.00055219
7
0.11716580
0.11652796
0.00063785
8
0.11342959
0.11270772
0.00072187
9
0.10981705
0.10901272
0.00080434
10 0.10632415
0.10543886
0.00088530
11 0.10294698
0.10198216
0.00096483
12 0.09968176
0.09863879
0.00104298
13 0.09652483
0.09540502
0.00111981
14 0.09347266
0.09227727
0.00119539
15 0.09052183
0.08925206
0.00126976
42
2.10-rasm. Yechim ustivir ekan. Dastur matni 1-ilovada keltirilgan.
Oshkormas sxemalar.
(2.1) - (2.3) chegaraviy masalani sonli yechishda
oshkor sxemadan farqli ravishda oshkormas sxemalardan quyidagi holatlarda
foydalaniladi: 1) p
0
>0, p
N
>0; 2) p
0
<0, p
N
<0; 3) p
0
>0, p
N
<0; 4) p
0
<0, p
N
>0.
Ikki xil ayirmali sxemalarni qaraylik:
1)
Markaziy ayirmali sxema.
2)
Vaznli uchnuqtali sxema.
Bu
sxemalarning
hammasi
progonka
usuli
bilan
yechiladi
va
approksimatsiyalash sxemalaridan olingan bu barcha ayirmali tenglamalar ishonchli,
ular quyidagi ko’rinishga keltiriladi:
j
N
j
N
N
j
N
N
j
i
j
i
i
j
i
i
j
i
i
j
j
j
F
u
C
u
A
F
u
B
u
C
u
A
F
u
B
u
С
1
1
1
1
1
1
1
1
0
1
1
0
1
0
0
1
,
1
N
i
(2.4*)
Bundagi
A
i
,
B
i
,
C
i
koeffisiyentlar quyidagi shartlarni bajarishi zarur:
.
1
,
1
,
0
,
0
,
,
,
0
0
0
N
i
C
C
B
A
C
A
C
B
B
A
С
N
i
i
N
N
i
i
i
(2.5*)
B
0
,
C
0
,
F
0
,
A
N
,
C
N
,
F
N
koeffisiyentlar esa chegaraviy shartlardan topiladi. Bu
masalada p(x,t) funksiyaning ishorasiga qarab chegaraviy shartlar qo’yiladi va bunga
mos koeffisiyentlar topiladi.
Barcha 4 ta holatni qaraylik:
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