Qarshi davlat universiteti international scientific and practical conference on algorithms and current problems of programming - bet 58

dx
,
1
x
R
,
0,1, 2,
n
, (5)
Следуя [7, 10], слабо аппроксимируем задачу Коши (1), (2), (3) задачей
73
1
3
xx
u
u
t
,
2
3
xx
v
v
t
,
1
3
n
t
n
, (6)
3
0
x
u
u u
t
,
3
0
x
v
v v
t
,
1
2
3
3
n
t
n
, (7)
3 (
)
u
b u
v
t
,
3 (
)
u
b u
v
t
,
2
1
3
n
t
n
, (8)
0
(0, )
( )
u
x
u x
,
0
(0, )
( )
v
x
v x
, (9)
где
2
1
b
b
,
*
N
t
,
1
N
- целое,
0,1,
,
1
n
N
, и постоянная
*
t
удовлетворяет
неравенству (10) . Решения которого имеет вид
0
0
0
3
( )
(
( )
( )) 1
bt
b
u
u
x
v x
u
x
e
b
,
0
0
0
3
3
( )
( ) 1
( ) 1
bt
bt
bt
b
b
v
v x e
v x
e
u x
e
b
b
, где
3(
)
b
b
b
.
Пусть выполнены соотношения (5) и постоянные
1
c
,
1
c
и
*
t
удовлетворяют
условиям
*
1
1
0
c t
,
*
1
1
0
c t
(10)
тогда решение
u
и
v
в полосе
*
0,
t
существует и ограничено вместе со
всеми своими производными по переменным
t
,
x
.
Очевидно, что при любом фиксированном
решение
u
и
v
задачи (6)-(9)
ограничены независимо от величины
0
( , )
u t x
c
,
0
( , )
v t x
c
. (11)
Также можно показать ограниченность частных производных решений
u
и
v
по
x
любого порядка равномерно по
:
( , )
k
k
k
u
t x
C
x
,
( , )
k
k
k
v
t x
C
x
,
*
0,
( , )
t
t x
,
0,1,
k
(12)
где
k
C
,
k
C
- некоторые положительные постоянные, такие что
k
k
C
c
,
k
k
C
c
.
Из неравенств (11), (12) и уравнений (6)-(9) следуют равномерные по
оценки:
1
( , )
k
k
k
u
t x
s
t x
,
1
( , )
k
k
k
v
t x
s
t x
,
*
0,
( , )
t
t x
,
0,1,
k
. (13)
Из этих оценок следует, что
u
,
v
и их производные по
x
любого порядка
равномерно ограничены и равностепенно непрерывны в
*
0,
t
. На основании
теоремы Арцела диагональным способом можно выбрать подпоследовательность
k
u
,
k
v
последовательностей
u
,
v
, сходящуюся в
*
0,
t
к функциям
u
и
v
соответственно вместе со всеми производными по
x
, равномерно в каждой
74
ограниченной области полосы
*
0,
t
, вследствие чего функции
u
и
v
имеют
производные любого порядка по
x
и выполняются соотношения
0
0
( )
t
u
u x
,
0
0
( )
t
v
v x
, (14)
( , )
k
k
k
u t x
C
x
,
( , )
k
k
k
v t x
C
x
,
*
0,
( , )
t
t x
,
0,1,
k
. (15)
Единственность решения доказывается стандартным способом. Следовательно, и
сами последовательности функций
u
,
v
при
0
, сходятся равномерно в
*
0,
t
к
u
и
v
соответственно, вместе со всеми производными.Случай, когда
2
1
0
0
( ),
( )
u x v x
C
R
рассматривается с помощью средних функций [6].
Список использованной литературы
1. Куликовский А.Г., Свешников Е.И., Чугайнова А.П. Математические методы
изучения разрывных решений нелинейных гиперболических систем уравнении,
Москва -2010, 122с.
2. Соболев С.Л. Некоторые применения функционального анализа в
математической физике Новосибирск: СО АН СССР, 1962, 255 с.
3. Доровский В Н Перепечко Ю.В. Феноменологическое описание двух скоростных
сред с касательными напряжениями //ПМТФ.1992. No.3. С.94-105.
4. Демидов Г.В., Новиков В.А. О сходимости метода слабой аппроксимации в
рефлексивном банаховом пространстве // Функциональный анализ и его
приложения, 1975, т. . 9, No. 1, с.25-30.
5. Гегечкори З.Г., Демидов Г.В. О сходимости метода слабой аппроксимации // ДАН
России, 1973, т. . 213, No. 2, с. 264-266.
6. Яненко H.H. Метод дробных шагов решения многомерных задач
математической физики - Новосибирск: Наука. Сиб. отд-ние, 1967, - 197 с.
Shavkatov Sh.Sh. Actual problems of programming language c: a comprehensive analysis
ACTUAL PROBLEMS OF PROGRAMMING LANGUAGE C: A COMPREHENSIVE
ANALYSIS
Shavkatov Shoxabbos Shaxzodovich
Teacher at Kimyo International University in Tashkent
shavkatovshoxabbos88@gmail.com
Abstract:
C is one of the oldest and most popular programming languages, known for its
speed, efficiency, and versatility. Despite its long-standing popularity, C is not without its
challenges and problems. In this article, we will explore the actual problems of
programming language C, including its limitations, vulnerabilities, and potential solutions.
Key words:
vulnerability, unauthorized access, security, control systems, software
systems.
One of the primary challenges of C is its limitations. While C is known for its speed and
efficiency, it lacks some of the features and capabilities of more modern programming
languages. For example, C does not have built-in support for object-oriented programming,
which can make it more difficult to write complex, modular code.
Another limitation of C is its lack of support for dynamic memory management. In C,
programmers must manually allocate and deallocate memory, which can be a time-
75
consuming and error-prone process. Additionally, C does not have built-in support for
garbage collection, which can lead to memory leaks and other memory-related errors[1].
C is also known for its lack of support for modern programming paradigms, such as
functional programming. While C does support some functional programming concepts,
such as higher-order functions and lambda expressions, it does not provide the same level
of support as more modern languages like Haskell or Scala.
Another challenge of programming language C is its vulnerabilities. C is known for its
susceptibility to buffer overflows, a type of security vulnerability that occurs when a
program attempts to write data outside the bounds of a buffer. Buffer overflows can be
exploited by attackers to execute arbitrary code or crash the program[2].
C is also susceptible to other types of security vulnerabilities, such as integer overflows,
format string vulnerabilities, and heap-based buffer overflows. These vulnerabilities can be
difficult to detect and can be exploited by attackers to gain unauthorized access to systems
or data.
In addition to security vulnerabilities, C is also prone to other types of programming
errors, such as null pointer dereferences and memory leaks. These errors can lead to
crashes, data corruption, and other serious problems[3].
Despite its challenges, C remains a popular and widely used programming language,
particularly in systems programming and embedded systems. There are several potential
solutions to the challenges of programming language C, including the use of modern
programming practices and techniques.
One potential solution is the use of modern C libraries, such as the GNU C Library, which
provide support for dynamic memory management and other modern programming
paradigms. Additionally, the use of automated tools, such as memory analyzers and static
code analyzers, can help to detect and prevent security vulnerabilities in C code.
Another potential solution is the use of programming language extensions, such as the
Microsoft Secure Coding Extensions for C++, which provide additional security features and
capabilities for C programming. Additionally, the use of secure coding standards, such as
the CERT C Coding Standard, can help to ensure that C code is written in a secure and
robust manner.
Despite its challenges, C remains a powerful and important programming language. In
recent years, there has been a renewed interest in C programming, particularly in the fields
of systems programming and embedded systems[4].
One of the reasons for this renewed interest is the growing importance of the Internet of
Things (IoT), which requires low-level programming languages like C to interface with
hardware and control systems. Additionally, C is still widely used in the development of
operating systems, compilers, and other low-level software systems[5].
As C programming continues to evolve, it is likely that new tools and techniques will be
developed to address its limitations and vulnerabilities. One promising area of research is
the development of new programming paradigms and techniques for writing secure and
efficient C code, such as static analysis and formal verification.
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