ISSN:
2181-3906
2023
International scientific journal
«MODERN SCIENCE АND RESEARCH»
VOLUME 2 / ISSUE 5 / UIF:8.2 / MODERNSCIENCE.UZ
695
BAZI XUSUSIY INTEGRAL TENGLAMALAR SISTEMASINI YECHISHDA
FREDGOLM USULI
Turdiyev Islomjon Alisherovich
Denov tadbirkorlik va pedagogika instituti 2-bosqich magistranti
https://doi.org/10.5281/zenodo.7953645
Annotatsiya.
Ushbu maqolada integral tenglamalarning kelib chiqish tarixi haqida
qisqacha ma’lumotlar berilgan. Fredgolmning birinchi va ikkinchi tur integral tenglamalarining
amaliy ahamiyati, uni yechish bo‘yicha ketma-ket yaqinlashish hamda iteratsiyalangan yadrolar
usullarini qo‘llash yo‘llari bayon qilingan. Bir nechta misollar yechib ko‘rsatilgan, mustaqil
bajarish uchun misollar tuzilgan va taqdim qilingan. Maqolani tayyorlashda mavzuning
talabalarga tushunarli qilib yoritilishiga katta ahamiyat berilgan.
Kalit so‘zlar:
integral tenglama, Furye almashtirishi, ketma-ket yaqinlashishlar usuli,
funksiya argumenti, Fredgolmning birinchi tur integral tenglamasi, Fredgolmning ikkinchi tur
integral tenglamasi, iteratsiyalangan yadrolar usuli.
FREDHOLM'S METHOD FOR SOLVING A SYSTEM OF PARTIAL
INTEGRAL EQUATIONS
Abstract.
The article presents a brief history of the emergence of integral equations. The
practical significance of the Fredholm integral equations of the first and second type, methods of
successive approximation and repeated kernels for their solution are described. Several examples
are illustrated, the examples are compiled and presented for self-fulfillment. When preparing an
article, great importance is attached to ensuring that the topic is understandable to students.
Keywords:
Fourier transform, method of successive approximations, function argument,
Fredholm integral equation of the first kind, Fredholm integral equation of the second kind,
method of iterated kernels.
МЕТОД ФРЕДГОЛЬМА РЕШЕНИЯ СИСТЕМЫ ИНТЕГРАЛЬНЫХ
УРАВНЕНИЙ В ЧАСТНЫХ ПРОИЗВОДНЫХ
Аннотация.
В статье дана краткая информация об истории возникновения
интегральных уравнений. Описано практическое значение интегральных уравнений
Фредгольма первого и второго рода, методов последовательного приближения и
повторных ядер для их решения. Проиллюстрировано несколько примеров, примеры
составлены и представлены для самостоятельного выполнения. При подготовке статьи
большое значение придается тому, чтобы тема была понятна учащимся.
Ключевые слова:
интегральное уравнение, преобразование Фурье, метод
последовательных приближений, аргумент функции, интегральное уравнение Фредгольма
первого рода, интегральное уравнение Фредгольма второго рода, метод повторных ядер.
Integral tenglamalar deb, noma’lum funksiya integral ishorasi ostida bo‘lgan
tenglamalarga aytiladi.
Mexanika, matematika fizika va texnikaning juda ko‘plab masalalar ushbu
)
(
)
(
)
,
(
)
(
x
f
dy
y
y
x
K
x
b
a
(1)
ISSN:
2181-3906
2023
International scientific journal
«MODERN SCIENCE АND RESEARCH»
VOLUME 2 / ISSUE 5 / UIF:8.2 / MODERNSCIENCE.UZ
696
ko‘rinishdagi integral tenglamalarning tekshirishga olib kelinadi, bu erda
)
(
x
noma’lum
funksiya,
)
,
(
y
x
K
va
)
(
x
f
funksiyalar mos ravishda
b
y
a
b
x
a
,
va
b
x
a
(
b
a
,
-
o‘zgarmas sonlar) yopiq sohalarda berilgan uzluksiz haqiqiy funksiyalar.
)
(
x
f
funksiya (1)
integral tenglamaning ozod hadi,
)
,
(
y
x
K
tenglmaning yadrosi, sonli
ko‘paytuvchi
tenglamaning parametiri deyiladi.
Fredgol’mning birinchi tur integlamasi deb,
)
(
)
(
)
,
(
x
f
dy
y
y
x
K
b
a
(2)
ko‘rinishdagi integral tenglamaga aytiladi.
Agar (1) tenglamada
0
)
(
x
f
bo‘lsa, ya’ni
0
)
(
)
,
(
)
(
dy
y
y
x
K
x
b
a
(3)
tenglama (1) mos bo‘lgan bir jinsli integral tenglama deyiladi.
Bir jinsli
0
)
(
)
,
(
)
(
dy
y
y
x
K
x
b
a
(4)
tenglama (3) bir jinsli tenglamaga qo‘shma integral tenglama deyiladi.
a
x
x
f
dy
y
y
x
K
x
x
a
),
(
)
(
)
,
(
)
(
(5)
ko‘rinishda bo‘lsa, u Vol’terraning ikkinchi tur integral tenglamasi,
)
(
)
(
)
,
(
x
f
dy
y
y
x
K
x
a
(6)
Tekshirib ko‘rish qiyin emaski, agar ikkinchi tur Fredgolintegral tenglamasining umumiy echimi
)
(
x
mavjud bo‘lsa
)
(
)
(
)
(
0
x
x
x
(7)
ko‘rinishga ega bo‘ladi, bunda
)
(
0
x
(3) tenglamaning umumiy echimi,
esa (1) tenglamaning umumiy va xususiy echimlari bo‘lsa, bularning ayirmasi
)
(
)
(
)
(
0
x
x
x
(3) tenglamaning echimidan iborat bo‘ladi. Bundan darhol (7) tenglik kelib chiqadi.
Fredgolm ikkinchi tur integral tenglamasini parametr kichik
bo‘lganda ketma-ket yaqinlashish usuli bilan echish.
(1) tenglamani tekshiramiz,
)
,
(
y
x
K
va
)
(
x
f
funksiyalar o‘zlari aniqlangan sohalrda
uzliuksiz bo‘lgani uchun
m
x
f
b
x
a
M
y
x
K
x
a
b
x
a
)
(
max
,
,
)
,
(
(8)
bo‘ladi.
ISSN:
2181-3906
2023
International scientific journal
«MODERN SCIENCE АND RESEARCH»
VOLUME 2 / ISSUE 5 / UIF:8.2 / MODERNSCIENCE.UZ
697
Agar (1) tenglama
parametr
M
1
(9)
shartni qanoatlantirsa, u holda bu tenglamalarning yagona
)
(
x
echimi mavjud bo‘lib, uni ketma-
ket yaqinlashish usuli bilan topish mumkin.
Nolinchi yaqinlashish sifatida (1) tenglamaning ozod hadini qabul qilamiz
)
(
)
(
0
x
f
x
Binchi yaqinlashishni
dy
y
f
y
x
K
x
f
x
b
a
)
(
)
,
(
)
(
)
(
1
munosabat bilan aniqlaymiz. Bu jarayonni davom ettirib, n-yaqinlashishni
n
n
dy
y
y
x
K
x
f
x
n
b
a
n
,.....,
2
,
1
,
)
(
)
,
(
)
(
)
(
1
(10)
munosabat bilan aniqlaymiz.
Shunday qilib, (10) rekurrent munosabatlarni qanoatlantiruvchi
),......
(
,
),........
(
),
(
1
0
x
x
x
n
funksional ketma-ketligiga ega bo‘lamiz.
Matematik analizdan ma’lumki, (11) ketma-ketlikning yaqinlashishi
1
1
0
)
(
)
(
)
(
n
n
n
x
x
x
(12)
qatorning yaqinlashishiga teng kuchludir. (10) formulani
,.....
4
,
3
,
2
)
(
)
(
)
,
(
)
(
)
(
)
(
)
,
(
)
(
)
,
(
)
(
)
(
)
(
)
(
)
,
(
)
(
)
(
2
1
1
2
1
2
1
2
1
n
dy
y
y
y
x
K
x
dy
y
y
y
x
K
dy
y
y
x
K
x
f
dy
y
y
y
y
x
K
x
f
x
n
n
b
a
n
n
n
b
a
n
b
a
n
n
n
b
a
n
(13)
ko‘rinishda yozib olamiz.
(8) ga asosan (13) dan quyidagi tengsizliklar kelib chiqadi:
n
n
n
n
M
m
x
x
M
m
x
x
M
m
x
x
m
x
)
(
)
(
.........
..........
..........
..........
)
(
)
(
)
(
)
(
)
(
1
2
2
1
2
0
1
0
Shunday qilib, (12) qatorning har bir hadi musbat sonli
ISSN:
2181-3906
2023
International scientific journal
«MODERN SCIENCE АND RESEARCH»
VOLUME 2 / ISSUE 5 / UIF:8.2 / MODERNSCIENCE.UZ
698
0
n
n
n
M
m
Ta’kidlash joizki, integral tenglamalar nazariyasi matematik fizika, differensial
tenglamalar masalalarini hal qilishda keng qo’llaniladi. Jumladan, [3-15] ishlarda operatorli
matritsalar uchun Fredgolm determinantini qurish, Fadeev tenglamasini qurish jarayonida
integral tenglamalarni yechish usullaridan foydalanilgan. [16-36] ilmiy maqolalarda olib borilgan
izlanishlarda aralash tipdagi tenglamalar uchun qo‘yilgan chegaraviy masalalar Fredgol'mning
ikkinchi tur tenglamasiga keltiriladi. Fredgol'mning ikkinchi tur tenglamasi yechimining
mavjudligidan (tegishli shartlar ostida) chegaraviy masalaning ham yechimi mavjudligi haqida
xulosa qilinadi.
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