170Academic Research in Educational Sciences VOLUME 2 | ISSUE 11 | 2021
ISSN: 2181-1385
Scientific Journal Impact Factor (SJIF) 2021: 5.723 Directory Indexing of International Research Journals-CiteFactor 2020-21: 0.89
DOI: 10.24412/2181-1385-2021-11-1576-1584
MATLAB TIZIMIDA ODDIY DIFFERENSIAL TENGLAMALARNI
YECHISH
Sherali Abduqodir o'g'li Abdullayev
Toshkent viloyati Chirchiq davlat pedagogika instituti o'qituvchisi
Maftuna Abdusamad qizi Ahmadjonova
Namangan davlat universiteti talabasi
ANNOTATSIYA
Maqolada masalani qo'yilishi Matlab tizimida oddiy differensial tenglamalarni taqribiy yechish, taxlili, grafigi ko'rilgan.
Kalit so'zlar:Matlab tizimi, ODTMatlab tizimidayechkichlari.
SOLVING SIMPLE DIFFERENTIAL EQUATIONS IN THE MATLAB
SYSTEM
ABSTRACT
Problem statement in the article Approximate solution, analysis, graph of simple differential equations in Matlab's system are considered. Keywords: Matlab system, SDE Solvers in Matlab system.
KIRISH
Ko'plab tizimlar va qurilmalarning dinamikasini taxlil qilish, tebranishlar nazariyasining masalalarini yechish va boshqalar oddiy differensial tenglamalar sistemasini (ODtT) yechishga asoslangan. Odatda ular Koshi shaklidagi birinchi tartibli differensial tenglamalar sistemasi tarzida ko'rsatiladi:
ADABIYOTLAR TAHLILI VA METODOLOGIYA
= y\ y' = f(y,t) ODT uchun chegaraviy shartlar ham ko'rsatiladi:
y(t0rtnrp)=b, bu yerda t0,tn -integralning boshlang'ich va so'nggi nuqtalari. Boshlang'ich va so'nggi shartlar b vektor yordamida beriladi, t parametr albatta vaqt bo'lishi shart emas.
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Academic Research in Educational Sciences VOLUME 2 | ISSUE 11 | 2021
ISSN: 2181-1385
Scientific Journal Impact Factor (SJIF) 2021: 5.723 Directory Indexing of International Research Journals-CiteFactor 2020-21: 0.89
DOI: 10.24412/2181-1385-2021-11-1576-1584
MATLAB
Mat hWorks- R 2015 b
1- tizimni ishga tushurish 1. Oddiy differensial tenglamalarning yechgichlari.
Oddiy differensial tenglamalarni yechish uchun MATLABda turli xil usullar mavjud. Ularni amalga oshirish oddiy differensial tenglamalarning yechkichlari deb ataladi. Keyinchalik matnda keltiriladigan umumlashtirilgan solver (yechgich)nomi, oddiy differensial tenglamalarni yechishning quyidagi sonli usullaridan birini anglatadi: ode45, ode23, ode15s,ode23s, ode23t, bvp4c yoki pdepe.
Differensial tenglamaning qattiq sistemalarni yechish uchun faqat mahsus ode 15s, ode23s, ode23t, ode23tb yechgichlardan foydalanish tavsiya etiladi:
• Ode45-bir qadamli yaqqol 4-va 5-tartibli Runge-Kutta usullari. U klassik usul bo'lib ko'plab xollarda yaxshi natijalarni beradi;
• Ode23- bir qadamli yaqqol 2-va 4-tartibli Runge-Kutta usullari;
• Ode113-bir qadamli, o'zgaruvchi tartibli Adams-Bashvort-Multon usuli. Ushbu adaptiv usul yuqori aniqlikdagi yechimni berishi mumkin.
• Ode23tb- yechimning boshlanishida yaqqol bo'lmagan Runge-Kutta usulidan va keyinchalik 2-tartibli teskari differensiallash formulasidan foydalanuvchi usul. Aniqlik pastligiga qaramasdan, ushbu usul ode15s usulidan effektivroq bo'lishi mumkin;
• Ode15s- sonli differensiallash formalalaridan foydalanuvchi, o'zgarunchi tartibli (1dan 5gacha, dastlabki xolatda 5), ko'plab qadamli usul. Ushbu adaptiv usulni ode45 yechgich yechimni ta'minlay olmasa qo'llash maqsadga muvofiq;
• Ode23s-modifikatsiyalangan 2-tartibli Rozenbroka formulasidan foydalanuvchi bir qadamli usuli. Differensail tenglamalarning qattiq sistemasini yechishda pastroq aniqlikda va yuqori xisoblash tezligiga ega;
• Ode23t- interpolyatsiyali trapetsiyalar usuli. Ushbu usul chiqish signali garmonikalari yaqin bo'lgan tebranuvchi sistemalarni hisoblashda yaxshi natijalarni beradi.
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Academic Research, Uzbekistan 1577 www.ares.uz
Academic Research in Educational Sciences VOLUME 2 | ISSUE 11 | 2021
ISSN: 2181-1385
Scientific Journal Impact Factor (SJIF) 2021: 5.723 Directory Indexing of International Research Journals-CiteFactor 2020-21: 0.89
DOI: 10.24412/2181-1385-2021-11-1576-1584
Hamma yechgichlar y,=F(x,y) ko'rinishdagi tenglamalar sistemasini, ode15s, ode23s, ode23t va ode23t yechkichlar esa yaqqol bolmagan M(t,y) y'=F(t,y) ko'rinishdagi tenglamalarni yechishi mumkin. Hamma yechgichlar (ode23s va bvp4c dan tashqari) M(t,y) y'-F(t,y) ko'rinishdagi matrisaviy tenglamalarning ildizlarini toppish mumkin.
m
2-ishga tushgan oynasi
2. Oddiy differensial tenglamalarning yechkichlardan
foydalanish.
Differensial tenglamalarni yechish funksiyalarida quyidagi belgilash va qoidalar qabul qilingan;
• Options-odeset funksiyasi hosil qiladigan argument (yana bir funksiya -odeget yoki bvpget (faqat bvp4c uchun )-sukut bo'yicha yoki odeset/bvpset funksiyalari tomonidan o'rnatilgan parametrlarini chiqarish;
• Tspan-integralash intervalini aniqlaydigan vektor(t0, tfinal). Yechishni konkret vaqt momentlarida t0,ti...tfinal topish uchun tspan=[t0,ti,^.tfinal]dan foydalanish kerak;
• Y0-boshlang'ich shartlar vektori ;
• Pi,p2,.,.,-F funksiyaga uzatiluvchi ixtiyoriy parametrlar;
• T,Y-Y har bir satri T vektor ustunda qaytarilgan vaqtga mos keladigan funksiyalarning tavsiyasiga o'tamiz;
• Differensial tenglamalar sistemasini yechish uchun ishlatiladigan funksiyalarning tavsiyasiga o'tamiz;
• [T,Y]=solver(@F,tspan,y0)-y,=F(t,y) ko'rinishdagi differensial tenglamalar sistemasini tspan intervalda y0 boshlang'ich shartlarga asosan integrallaydi. @F-Oddiy differensial tenglamalar -funksiyaning deskriptori. Y yechimlar massividagi har bir satr T vector-ustunda qaytariluvchi vaqt qiymatlariga mos keladi;
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Academic Research in Educational Sciences VOLUME 2 | ISSUE 11 | 2021
ISSN: 2181-1385
Scientific Journal Impact Factor (SJIF) 2021: 5.723 Directory Indexing of International Research Journals-CiteFactor 2020-21: 0.89
DOI: 10.24412/2181-1385-2021-11-1576-1584
• [T,Y]=solver(@F,tspan,y0, options)-yuqoridagiga o'xshash, lekin qo'shimcha odeset funksiyasi xosil qiladigan options argumentning qiymatlari orqali aniqlovchi parametrlar bilan. Odatda bunday parametrlarga nisbiy xatolikning yol qo'yiladigan qiymati RelTol va ruxsat etiladigan absolyut xatoliklarning vektori AbsTol kiradi;
• [T,Y]=solver(@F,tspan,y0,options,p1,p2,...)-yuqoridagiga o'xshash , lekin qo'shimcha p1,p2,... parametrlarni har bir chaqirilganida m-fayl Fga uzatadi. Agar option parametrlar berilmaydigan bo'lsa ularningo,rniga[ ]deb yoziladi;
• [T,X,Y]=sim(@model,tspan.-y0options,ut.p1,p2.. ,,)-SIMULINK modelini ishlatiladi. Misol uchun:
• [T,X,Y]-sim(@model....).
Integrallash parametrlari (options) m-faylda yoki odeset komandasi yordamida komandalar satrida aniqlanishi mumkin.
Yechkichlarning parametrlari ro'yxatida quyidagi parametrlar bolishi mumkin:
• NormControl-yechim vektori normasi [on| {off}]ga bog'liq xolda xatolikni boshqaradi, norm(e)<=max(RelTol*norm(y), AbsTol) boTishi uchun 'on' o'rnatiladi;
• RelTol-nisbiy tanlash chegarasi [musbat skalyar]. Hamma yechkichlarning aniqligi sukut holatida 1e-3(0.1%)gat eng;
• AbsTol-absolyut aniqlik [musbat skalyar yoki vektor{1e-6}];
• OutputFcn-chiqarish funksiyasi[function]ning deskriptori;
Birinchi tartibli sodda differensial tenglamalar.
1.^ = fix) ko'rinishdagi tenglamalar.
Bu tenglamaning yechimi quyidagicha topiladi.
dy = f{x)dx tenglikning ikkala tomonini integrallaymiz va yechimni topamiz.
Misol.
Bu tenglamaning yechimi.
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Academic Research in Educational Sciences VOLUME 2 | ISSUE 11 | 2021
ISSN: 2181-1385
Scientific Journal Impact Factor (SJIF) 2021: 5.723 Directory Indexing of International Research Journals-CiteFactor 2020-21: 0.89
DOI: 10.24412/2181-1385-2021-11-1576-1584
y = sinOO -I- c
Tenglamaning Matlabda yechimi. Tenglamaning analitik yechimi.
>> y=dsolve('Dy-cos(t)=0')
y =
sin(t)+C1
Tenglamaning MATLAB tizimidagi sonli aniq yechimi.
M.file da funksiyani e'lon qilamiz.
>> [T,Y]=ode45(@f1,[0,1],1)
T = Y =
0 1.0000
0.0250 1.0250
0.0500 1.0500
0.0750 1.0749
0.1000 1.0998
0.1250 1.1247
0.1500 1.1494
0.1750 1.1741
0.2000 1.1987
0.2250 1.2231
0.2500 1.2474
0.2750 1.2715
0.3000 1.2955
0.3250 1.3193
0.3500 1.3429
0.3750 1.3663
0.4000 1.3894
0.4250 1.4123
2. = 9(y) ko'rinishdagi tenglamalarning yechimi quyidagicha topiladi.
dy = ¿¡•'yy)dx tenglikning ikkala tomonini integrallaymiz va quyidagicha yechim hosil bo'ladi.
x =
f—
J div)
drj + c
Misol.
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Academic Research in Educational Sciences VOLUME 2 | ISSUE 11 | 2021
ISSN: 2181-1385
Scientific Journal Impact Factor (SJIF) 2021: 5.723 Directory Indexing of International Research Journals-CiteFactor 2020-21: 0.89
DOI: 10.24412/2181-1385-2021-11-1576-1584
dy dx
Tenglamani yechamiz.
dy y2 +1
= y2 + l
= dx
Tenglikning ikkala tarafini integrallaymiz va quyidagicha yechimni olamiz.
dy
Endi MATLAB paketida yechimi. >> y=dsolve('Dy-yA2-1=0') y =tan(t+C1)
Tenglamaning sonli yechimi va grafigi.
M.file da funksiyani e'lon qilamiz.
MATLAB R2015b
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1 0.5
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1 H function odt2=f2(x,y)
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» [I,Y]=ode23(@f2,[0 0.5],0.5)
.0320 . 0820 .1320 .1820 .2320 .2820 .3320 .3820 .4320 .4820 .5000
®
0.5000 0.5407 fxw 0.6071
>> [T,Y]=ode23(@f2,[0 0.5],0.5)
T = Y =
0 0.5000
0.0320 0.5407
0.0820 0.6071
0.1320 0.6778
0.1820 0.7534
0.2320 0.8349
0.2820 0.9235
0.3320 1.0207
0.3820 1.1284
0.4320 1.2489
Grafigi.
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Academic Research in Educational Sciences VOLUME 2 | ISSUE 11 | 2021
ISSN: 2181-1385
Scientific Journal Impact Factor (SJIF) 2021: 5.723 Directory Indexing of International Research Journals-CiteFactor 2020-21: 0.89
DOI: 10.24412/2181-1385-2021-11-1576-1584
0.4820 1.3856
0.5000 1.43
>> plot(T,Y)
File Edit View Insert Tools Desktop Window Help
DE on
1.4 1.3 1.2 1.1 1 0.S 0.8 0.7 0.6
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 5
3. O'zgaruvchilari ajraladigan differensial tenglamalar.
ko'rinishdagi tenglamalar o 'zgaruvchilari ajraladigan differensial tenglamalar deyiladi. Misol uchun:
Tenglamani yechamiz va quyidagicha natijaga ega bo'lamiz.
MATLAB paketida yechimi quyidagicha. >> y=dsolve(,Dy-eAy*sin(t)=0')
У =
log(-1 /log(e)/(-cos(t)+C1 ))/log(e)
Tenglamaning sonli yechimi va grafigi.
M.file da funksiyani e'lon qilamiz.
>> [T,Y]=ode113(@f3,[-1 1],1)
T = Y =
-1.0000 1.0000
-0.9965 0.9921
-0.9896 0.9766
-0.9756 0.9465
-0.9478 0.8897
-0.8921 0.7879
-0.7808 0.6217
-0.5808 0.4125
-0.4808 0.3389
-0.3808 0.2821
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Academic Research in Educational Sciences VOLUME 2 | ISSUE 11 | 2021
ISSN: 2181-1385
Scientific Journal Impact Factor (SJIF) 2021: 5.723 Directory Indexing of International Research Journals-CiteFactor 2020-21: 0.89
DOI: 10.24412/2181-1385-2021-11-1576-1584
>>plot(T,Y) Grafigi.
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ISSN: 2181-1385
Scientific Journal Impact Factor (SJIF) 2021: 5.723 Directory Indexing of International Research Journals-CiteFactor 2020-21: 0.89
DOI: 10.24412/2181-1385-2021-11-1576-1584
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