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6.995, 2024 7.75
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273
THE RELEVANCE OF HYPERBOLIC EQUATIONS AND AN EFFECTIVE METHOD
OF STUDYING THEM
Basheeva Aynur Orinbasarovna
Kazakhstan In the Republic located L.N.Gumilyov
named after Eurasia National University
Algebra and geometry department Associate Professor , (PhD)
Kholmanova Klara Yangiboy kizi
Jizzakh branch of the National University of Uzbekistan
Assistant of the Department of "Applied Mathematics"
Annotatsiya:
Mazkur maqolada giperbolik tenglamalarning ilmiy-nazariy asoslari, amaliy
qo‘llanilishi va o‘qitish jarayonida qo‘llaniladigan samarali metodik yondashuvlar keng
yoritilgan. Unda ta’limning zamonaviy talablari, raqamli vositalar, fanlararo integratsiya hamda
loyihaviy faoliyat asosida o‘qitish texnologiyalari tahlil etilgan. Shuningdek, mavzuni
o‘zlashtirishda o‘quvchilarning ijodiy va analitik fikrlash ko‘nikmalarini rivojlantirishga
qaratilgan amaliy metodlar tavsiya etilgan. Maqola metodik jihatdan chuqur, zamonaviy
yondashuvlar asosida tayyorlangan bo‘lib, amaliyotchi o‘qituvchilar va metodistlar uchun
qo‘llanma vazifasini o‘tashi mumkin.
Kalit so‘zlar:
giperbolik tenglamalar, differensial tenglamalar, modellashtirish, to‘lqin
tenglamasi, metodika, STEM, interaktiv ta’lim, matematik tafakkur, raqamli texnologiyalar
Аннотация
: В статье подробно рассматриваются научно-теоретические основы
гиперболических уравнений, их практическое применение, а также эффективные
методические подходы, используемые в процессе обучения. Анализируются
современные образовательные требования, цифровые инструменты, междисциплинарная
интеграция и технологии обучения на основе проектной деятельности. Рекомендуются
также практические методы, направленные на развитие у студентов навыков творческого
и аналитического мышления при освоении предмета. Статья методически глубока,
основана на современных подходах и может служить руководством для практикующих
педагогов и методистов.
Ключевые слова:
гиперболические уравнения, дифференциальные уравнения,
моделирование, волновое уравнение, методология, STEM, интерактивное образование,
Volume 15 Issue 04, April 2025
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математическое мышление, цифровые технологии
Abstract :
This in the article hyperbolic of equations scientific-theoretical basics , practical
application
and teaching in the process applicable effective methodical approaches wide
illuminated . In it of education modern requirements , digital tools , interdisciplinary integration
and project activity based on education technologies analysis Also , the topic in mastering
students creative and analytical thinking skills to develop aimed at practical methods
recommendation Article
methodical in terms of deep , modern approaches based on
prepared to be a practitioner teachers and Methodists for manual task transition possible .
Key words :
hyperbolic
equations , differential equations , modeling , wave equation,
methodology , STEM, interactive education , mathematics thinking , digital technologies
Today on the day in our lives mathematics place incomparable that to everyone known .
So that's it every part of mathematics to the front attention our focus and our study necessary .
Including hyperbolic
equations are also important . Special productive differential of
equations three main type available : elliptical , parabolic and hyperbolic equations . Of them
hyperbolic equations in physics important processes - especially waves , acoustics ,
electromagnetic fields , earthquake waves , light and signal propagation in modeling wide is
applied .
Hyperbolic of equations general appearance :
this on the ground u( x,t ) – wave height , c – spread speed . This equation wave equation is
called and him/her D'Alembert method through solution find possible .
This real -life simulation using a mathematical model many physicist events clear analysis to
do , to calculate and in advance prophecy to do It is possible . This is hyperbolic equations to
study not only theoretically , maybe practical also relevant
that shows . This is the topic
deeper study necessity shows . Therefore this the topic education to the methodology our
attention Let's see .
Hyperbolic equations of teaching methodical basics
1.
First
lesson
effective to be for the lesson planned let 's take
Lesson topic content following in sequence organization is done :
1.
Vital problem with introduction : sound in the air spread , telephone signal arrived
Volume 15 Issue 04, April 2025
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progress .
2.
Mathematician to the model transition : hyperbolic equation to compile .
3.
Solution methods explanation : analytical , graphic and digital .
4.
Practical tasks : real reality to formulate a problem based on and solution
5.
Analysis and generalization : model and real process compatibility assessment .
2. Hyperbolic equation the topic following sciences with integration as education education
quality to increase big help gives .
Physics
:
wave , sound , light events .
Computer science : programming via model simulation (Python, MATLAB).
Technology : Arduino or sensors using practical models preparation
3. Educational technologies and from methods use the lesson further interesting and
understandable to be service does . So so , the following methods the topic lighting for suitable
that was because of this to methods attention Let's see .
STEM Approach
: Students
physicist the event selects → builds a mathematical
model → solution finds → in the program the result simulation does .
Flipped classroom class )
: subject video about home given , in class and discussion
and practice will be done .
Gamification :
“
Tol'tul ”
" Lifeguard " game - wave to spread correct answer
giving , the area from danger save .
Fishbone analysis scheme
: hyperbolic of the equation reasons and consequences
graphic in a way analysis will be done .
Technological tools
:
GeoGebra, Desmos — graph drawing
MATLAB, Python — building a numerical model .
Padlet, Jamboard — idea maps and cluster schemes create
4. Study activity assessment and monitoring
Evaluation three in stages take goes to :
Stage
Evaluation shape
Methodical recommendation
Lesson took
Diagnostic test
Previous knowledge level determination
Lesson during
Formative ( tables , reflection )
" What? " Do you understand ?", " What?
interesting it has been ? "
Volume 15 Issue 04, April 2025
Impact factor: 2019: 4.679 2020: 5.015 2021: 5.436, 2022: 5.242, 2023:
6.995, 2024 7.75
http://www.internationaljournal.co.in/index.php/jasass
276
Stage
Evaluation shape
Methodical recommendation
Lesson after
Summative test, project protection
Independent analysis to do , to think to conduct
Conclusion as telling if so , hyperbolic equations to teach student's logical his/her thinking
developmental , real -life complicated processes mathematician express to take ability
increasing strong is a tool . Education methodology modern technologies , interdisciplinary
approach and interactive methods with enrichment , education efficiency noticeable at the level
increases . Especially , the student ' s own on performance , projects through creative approach
them modern educated expert as shapes .
Used literature
1.
Karimova DX “ Mathematics in education methodical approaches ”. Tashkent, 2021.
2.
Rakhimov AA “ Partially productive differential equations ”. Tashkent, 2020.
3.
Evans, LC
Partial Differential Equations
. AMS, 2010.
4.
Strauss, WA
Partial Differential Equations: An Introduction
. Wiley, 2007.
5.
“ Innovative education technologies ”, Higher and middle special education Ministry ,
2023.
6.
GeoGebra, MATLAB, Python — official documents and open education platforms .
7.
LeVeque, R. J.
Finite Difference Methods for Ordinary and Partial Differential
Equations
, SIAM, 2007.
