INTERNATIONAL JOURNAL OF SCIENTIFIC RESEARCHERS
ISSN: 3030-332X Impact factor: 8,293
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BA’ZI KECHIKUVCHI ARGUMENTLI XUSUSIY HOSILALI KVACHIZIQLI
TENGLAMA UCHUN ARALASH MASALA
A.Kalandarov, M.Anorbayev
Guliston davlat universiteti ‘‘Matematika’’ kafedrasi
Maqola kechikuvchi argumentli giperbolik turdagi kvachiziqli quyidagi tenglamaning
Q = {τ ≤
t ≤ T, 0 ≤ x ≤ l
1
, 0 ≤ y ≤ l
2
, 0 ≤ z ≤ l
3
}
sohada
∂
2
u
∂t
2
= a
2 ∂
2
u
∂x
2
+
∂
2
u
∂y
2
+
∂
2
u
∂z
2
+ b
2 ∂
2
u(t−τ,x,y,z)
∂x
2
+
∂
2
u(t−τ,x,y,z)
∂y
2
+
∂
2
u(t−τ,x,y,z)
∂z
2
+
f t, x, y, z, u t, x, y, z , u t − τ, x, y, z + f t, x, y, z, u t, x, y, z , u t − τ, x, y, z
(1)
t, x, y, z ∈ E = {τ ≤ t ≤ T, 0 ≤ x ≤ l
1
, 0 ≤ y ≤ l
2
, 0 ≤ z ≤ l
3
}
bo’lganda boshlang’ich
u t, x, y, z = φ(t, x, y, z)
u
t
t, x, y, z = φ
t'
(2)
berilgan boshlang’ich shart va
τ ≤ t ≤ T
bo’lganda
u t, 0, y, z = 0
u t, l
1
, y, z = 0
0 ≤ y ≤ l
2
, 0 ≤ z ≤ l
3
,
u t, x, 0, z = 0
u t, x, l
2
, z = 0
0 ≤ x ≤ l
1
, 0 ≤ z ≤ l
3
,
u t, x, y, 0 = 0
u t, x, y, l
2
= 0
0 ≤ x ≤ l
1
, 0 ≤ y ≤ l
2
(3)
bir jinsli chegaraviy shartlarni qanoatlantiruvchi yechimini chekli-ayirmali usulda toppish
masalasi o’rganilgan. Bu masalani klassik, deyarli va umumlashgan yechimlarini mavjudligi va
yagonaligi masalasi (2) da muallif tomonidan tatqiq etilgan. Shuningdek koordinata ikkita
bo’lganda (1)-(3) masalasi sonli yechimlari (3) da oshkormas sxema asosida haydash usulida
yechilgan. Q sohani to’rlaymiz:
t
n
= nτ, x
i
= i∆
i
, y
j
= j∆
2
, z
k
= k∆
3
deb, to’rlangan to’r
funksiya uchun
u t
n
, x
i
, y
j
, z
k
= u
n
ijk
belgilashlarni kiritamiz,
n = 0,1,2, …, N
1
, j =
0,1,2, …, N
2
k = 0,1,2, …, N
3
,
bunda
Mτ = T, N
1
∆
1
= l
1
, N
2
∆
2
= l
2
, N
3
∆
3
= l
3
, N
1
, N
2
, N
3
natural sonlardir. (2) boshlang’ich
shartlardan
(t, x, y, z) ∈ E
bo’lganda
(n = 0 va n = 1)
da.
= φ 0, x
i
, y
j
, z
k
,
u
1
ijk
−u
0
ijk
τ
≈ φ
'
t
(τ, x
i
, y
j
, z
k
)
(4)
INTERNATIONAL JOURNAL OF SCIENTIFIC RESEARCHERS
ISSN: 3030-332X Impact factor: 8,293
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32
u
1
ijk
≈ u
0
ijk
+ τφ
t'
τ, x
i
, y
j
, z
k
= φ 0, x
i
, y
j
, z
k
+ τφ
t'
τ, x
i
, y
j
, z
k
(5)
(3) chegaraviy shartlardan
u
n
0jk
= 0, u
n
Njk
= 0, u
n
i0k
= 0, u
n
iNk
= 0, u
n
ijN
= 0
qiymatlarga ega bo’lamiz, bunda N mos ravishda
N
1
, N
2
, N
3
ga teng. Demak Q sohaning yon
tomonlari va asosida
u t, x, y, z
ning qiymatlari berilgan. Yuqoridagilardan foydalanib Q
sohaning ichki tugun nuqtalarida
u t, x, y, z
ning sonli qiymatlarini topamiz.
Umumiylikka halal bermasdan quyidagi oshkormas sxemalardan foydalanishimiz mumkin:
u
n+1
ijk
−2u
n
ijk
+u
n−1
ijk
τ
2
= a
2
(
u
n+1
i+1jk
−2u
n+1
ijk
+u
n
ijk
∆
1
2
+
u
n
ij+1k
−2u
n
ijk
+u
n
ij−1k
∆
2
2
+
u
n
ijk+1
−2u
n
ijk
+u
n
ijk−1
∆
3
2
) +
b
2
(
u
n
i+1jk
−2u
n
ijk
+u
n
i−1jk
∆
1
2
+
u
n
ij+1k
−2u
n
ijk
+u
n
ij−1k
∆
2
2
+
u
n
ijk+1
−2u
n
ijk
+u
n
ijk−1
∆
3
2
) + f
n
ijk
(7)
u
n+1
ijk
−2u
n
ijk
+u
n−1
ijk
τ
2
= a
2
(
u
n
i+1jk
−2u
n
ijk
+u
n
i−1jk
∆
1
2
+
u
n
ij+1k
−2u
n
ijk
+u
n
ij−1k
∆
2
2
+
u
n+1
ijk+1
−2u
n+1
ijk
+u
n+1
ijk−1
∆
3
2
) +
b
2
(
u
n
i+1jk
−2u
n
ijk
+u
n
i−1jk
∆
1
2
+
u
n
ij+1k
−2u
n
ijk
+u
n
ij−1k
∆
2
2
+
u
n
ijk+1
−2u
n
ijk
+u
n
ijk−1
∆
3
2
) + f
n
ijk
(8)
oshkormas sxemalardanfoydalanish maqsadga muvofiq. Bu yerda
f
n
ijk
= f(nτ, i∆
1
, j∆
2
, k∆
3
, u(nτ, i∆
1
, j∆
2
, k∆
3
), u((k − 1)τ, i∆
1
, j∆
2
, k∆
3
))
τ ≤ t ≤ 2τ
bo’lganda, yuqoridagi oshkormas sxemaga haydash usulini qo’llaymiz.
(7) Sxemada n=1 bo’lganda haydash koeffitsiyentlarni to’g’ri usulda o’sish tartibida
topib boriladi. Shu zaylda davom etib,
u
2
ijk
ni teskari usulda kamayish tartibida topib boriladi.
Shunday qilib ikkinchi qatlamda
u
2
ijk
lar to’rning tugun nuqtalaridagi qiymatlar topiladi.
2
τ ≤ t ≤ 3τ
bo’lganda, yuqoridagi (7) oshkormas sxemaga haydash usulini qo’llaymiz.
N=2 bo’lganda barcha yuqoridagi jarayonlar takrorlanib uchinchi qatlamda
u
3
ijk
topiladi va h.k.
u
n
ijk
hisoblab chiqiladi.
Yuqoridagi (7) va (8) - ayirmali sxemadan foydalanib ham
u
n
ijk
hisoblab chiqish mumkin.
INTERNATIONAL JOURNAL OF SCIENTIFIC RESEARCHERS
ISSN: 3030-332X Impact factor: 8,293
Volume 11, issue 2, May 2025
https://wordlyknowledge.uz/index.php/IJSR
worldly knowledge
Index:
google scholar, research gate, research bib, zenodo, open aire.
https://scholar.google.com/scholar?hl=ru&as_sdt=0%2C5&q=wosjournals.com&btnG
https://www.researchgate.net/profile/Worldly-Knowledge
https://journalseeker.researchbib.com/view/issn/3030-332X
33
Adabiyotlar:
1.
M.Isroilov, Hisoblash metodlari. 2 - qism. Toshkent, ‘‘O’zbekiston’’,2008.
2.
A.Kalandarov,
Smeshannaya
zadacha
dlya
giperbolicheskix
uravneniy s
apazdivayushimsya argumentami. Baku. Uchyoniye zapiski AGU,1975y.
3.
A.Kalandarov, M.Anorbayev,I.jangibayev. A Mixing problem for a quasi linear
equation with particular derivatives with some late argument. American journal of Business
Management, Economics and Banking. 2024
