ОПИСАНИЕ ПРОЦЕССОВ С КОНВЕКТИВНОГО ПЕРЕНОСА

Аннотация

The construction of self–similar solutions of differential equations is the process of creating a system or method that automatically generates solutions for given differential equations. This can be useful in various fields, such as physics, engineering, economics, etc., where it is necessary to quickly and efficiently find solutions to differential equations. Self-similar solutions, also known as approximate solutions, are a special class of solutions to differential equations that are automatically configured to fulfill the equation during development. They are used to describe complex dynamical systems or to approximate experimental data. Keywords: self-model, differential equation, dynamical systems, approximation. Avtomodel echimlarni qurish odatda eng kichik kvadratlar usuli yoki boshqa optimallashtirish usullaridan foydalanishga asoslangan. Avtomodel echimlarni yaratishda quyidagi umumiy yondashuvlar mavjud: Funktsional shaklni aniqlash: avval siz qaralayotgan tizim yoki ma'lumotlarga eng mos keladigan avtomodel echimning funktsional shaklini aniqlashingiz kerak. Bu sozlanishi kerak bo'lgan parametrlarni o'z ichiga olgan analitik formula bo'lishi mumkin. 9 50 International scientific journal ISSN: 2181-3906 «MODERN SCIENCE АND RESEARCH» 2023 VOLUME 2 / ISSUE 5 / UIF:8.2 / MODERNSCIENCE.UZ Yo'qotish funktsiyasini aniqlash: keyinchalik, modeldan olingan qiymatlar va kuzatilgan ma'lumotlar o'rtasidagi farqni o'lchaydigan yo'qotish funktsiyasini aniqlash kerak. Odatda ayirma kvadratlari yig'indisi (qoldiq kvadratlari yig'indisi) yoki boshqa mos metrika ishlatiladi. Parametrlarni optimallashtirish: keyin model parametrlarini yo'qotish funktsiyasini minimallashtirish uchun sozlash uchun eng kichik kvadratlar usuli yoki gradient tushish algoritmi kabi optimallashtirish usuli qo'llaniladi. Bu optimallashtirish muammosini hal qilishni yoki iterativ usullardan foydalanishni talab qilishi mumkin. Natijani tekshirish va tahlil qilish: avtomodel echim parametrlarini o'rnatgandan so'ng, olingan natijalarni tekshirish va tahlil qilish amalga oshiriladi. Bunga model qiymatlarini kuzatilgan ma'lumotlar bilan taqqoslash, muvofiqlik sifatini baholash va olingan model parametrlarini tahlil qilish kiradi. Avtomatik modellashtirish echimlarini yaratish - bu turli xil funktsional shakllar, dastlabki shartlar va optimallashtirish usullari bilan tajriba o'tkazishni talab qiladigan iterativ jarayondir. Jismoniy cheklovlarni va olingan parametrlarning talqin qilinishini hisobga olish ham muhimdir. Yuqoridagilarga asoslanib quyidagi masalani avtomodel echimlarini qurishni ko’rib o’tamiz: 𝛼 𝜕𝑢 𝜕 𝜕𝑢 𝜕𝑢 = (𝑢𝜎 )−𝑘(𝑡)| | 𝑢𝛽 (1) 𝜕𝑡 𝜕𝑥 𝜕𝑥 𝜕𝑥 𝑢(0,𝑥) = 𝑢 (𝑥) ≥ 0 (2) 0 𝑢(𝑡,0) = 𝑢 (𝑡) ≥ 0; 𝑢(𝑡,𝑀) = 𝑢 (𝑡) ≥ 0. (3) 1 2 bu yerda 𝑘(𝑡) = (𝑇−𝑡)𝛿. Yechimni quyidagi ko‘rinishda qidiramiz: 𝑢 = (𝑇−𝑡)𝛾𝑤(𝑧); bu yerda 𝑧 = 𝑥(𝑇−𝑡)𝜌. (1) tenglamaning har bir hadini hisoblab olamiz: 𝜕𝑢 𝜕𝑤𝜕𝑧 𝜕𝑤 = −𝛾(𝑇−𝑡)𝛾−1𝑤(𝑧)+(𝑇−𝑡)𝛾 = −𝛾(𝑇−𝑡)𝛾−1𝑤(𝑧)− (𝑇−𝑡)𝛾𝑥𝜌(𝑇−𝑡)𝜌−1 𝜕𝑡 𝜕𝑧 𝜕𝑡 𝜕𝑧 𝜕𝑤 𝜕𝑤 = −𝛾(𝑇−𝑡)𝛾−1𝑤(𝑧)−𝜌(𝑇−𝑡)𝛾−1𝑧 = (𝑇−𝑡)𝛾−1(−𝛾𝑤(𝑧)−𝜌𝑧 ); 𝜕𝑧 𝜕𝑧 𝜕 𝜕𝑢 𝜕 𝜕𝑤 (𝑢𝜎 ) = ((𝑇−𝑡)𝛾𝜎𝑤𝜎 (𝑇−𝑡)𝛾(𝑇−𝑡)𝜌 )(𝑇−𝑡)𝜌 𝜕𝑥 𝜕𝑥 𝜕𝑧 𝜕𝑧 = (𝑇−𝑡)𝛾𝜎+𝛾+2𝜌 (𝑤𝜎 ); 𝜕𝑧 𝜕𝑧 𝛼 𝛼 𝜕𝑢 𝜕𝑤 𝑘(𝑡)| | 𝑢𝛽 = (𝑇−𝑡)𝛿|(𝑇−𝑡)𝛾 (𝑇−𝑡)𝜌| (𝑇−𝑡)𝛾𝛽𝑤𝛽 𝜕𝑥 𝜕𝑧 𝛼 = (𝑇−𝑡)𝛿+𝛾𝛽+(𝛾+𝜌)𝛼| | 𝑤𝛽. Tenglamani qayta yozib olamiz: (𝑇−𝑡)𝛾−1(−𝛾𝑤(𝑧)−𝜌𝑧 ) = 𝛼 𝜕 𝜕𝑤 𝜕𝑤 = (𝑇−𝑡)𝛾𝜎+𝛾+2𝜌 (𝑤𝜎 )−(𝑇−𝑡)𝛿+𝛾𝛽+(𝛾+𝜌)𝛼| | 𝑤𝛽. 𝜕𝑧 𝜕𝑧 𝜕𝑧 (𝑇−𝑡) koeffisientlar daraja ko‘rsatkichlarini 𝛾 −1 ga tenglaymiz: 9 51 International scientific journal ISSN: 2181-3906 «MODERN SCIENCE АND RESEARCH» 2023 VOLUME 2 / ISSUE 5 / UIF:8.2 / MODERNSCIENCE.UZ 𝛾𝜎+𝛾+2𝜌 = 𝛾 −1; { 𝛿 +𝛾𝛽+(𝛾+𝜌)𝛼 = 𝛾−1. Bu sistemadan 𝛾 va 𝜌 ni topib olamiz: 1+𝛾𝜎 1+𝛾𝜎 𝜌 = − , 𝛿 +𝛾𝛽+(𝛾 − )𝛼 = 𝛾 −1, 2 2 𝛼𝜎 𝛼 𝛾(𝛽 +𝛼− −1) = −1−𝛿 + . 2 2 Demak, 𝛼−2(1+𝛿) 𝛾 = ; 2(𝛼+𝛽 −1)−𝛼𝜎 1 𝜎(𝛼−2(1+𝛿)) 𝜌 = − − . { 2 4(𝛼 +𝛽 −1)−2𝛼𝜎 2(1−𝛽) Bu yerda 2(𝛼 +𝛽 −1) ≠ 𝛼𝜎 → 𝛼(2−𝜎) ≠ 2−2𝛽 → 𝛼 ≠ . Kritik xol. 2−𝜎 Tenglamani quyidagicha yozishimiz mumkin bo‘ldi: 𝛼 𝜕𝑤 𝜕 𝜕𝑤 𝜕𝑤 (𝑇−𝑡)𝛾−1(−𝛾𝑤(𝑧)−𝜌𝑧 ) = (𝑇−𝑡)𝛾−1 (𝑤𝜎 )−(𝑇−𝑡)𝛾−1| | 𝑤𝛽. 𝜕𝑧 𝜕𝑧 𝜕𝑧 𝜕𝑧 Endi tenglamani ikkiala tomonini (𝑇−𝑡)𝛾−1 ga bo‘lib, quyidagi avtomodel tenglamaga ega bo‘lamiz: 𝛼 𝑑𝑤 𝑑 𝑑𝑤 𝑑𝑤 −𝜌𝑧 −𝛾𝑤 = (𝑤𝜎 )−| | 𝑤𝛽 𝜕𝑧 𝜕𝑧 𝑑𝑧 𝑑𝑧 yoki −𝜌𝑧𝑤′ −𝛾𝑤 = (𝑤𝜎𝑤′)′ −|𝑤′|𝛼𝑤𝛽. (4) (4) avtomodel tenglamani 𝛼, 𝛽 va 𝛿 larning tanlangan qiymatlari uchun yechimini quramiz. 𝛾 = 𝜌 bo‘lsin. U xolda 1+𝛾𝜎 𝜌 = − 2 ga ko‘ra 1+𝛾𝜎 1 𝛾 = − → 𝛾 = − 2 𝜎+2 ekanligi kelib chiqadi. Va 𝛼 −2(1+𝛿) 𝛾 = 2(𝛼 +𝛽−1)−𝛼𝜎 dan 2𝛼 +𝛽 −𝜎+1 𝛿 = 𝜎 +2 ni topamiz. Endi (4) tenglamani quyidagicha yozishimiz mumkin: 𝛼 𝑑𝑤 𝑑 𝑑𝑤 𝑑𝑤 −𝛾𝑧 −𝛾𝑤 = (𝑤𝜎 )−| | 𝑤𝛽, 𝜕𝑧 𝜕𝑧 𝑑𝑧 𝑑𝑧 𝛼 𝑑𝑧𝑤 𝑑 𝑑𝑤 𝑑𝑤 −𝛾 = (𝑤𝜎 )−| | 𝑤𝛽. 𝜕𝑧 𝜕𝑧 𝑑𝑧 𝑑𝑧 9 52 International scientific journal ISSN: 2181-3906 «MODERN SCIENCE АND RESEARCH» 2023 VOLUME 2 / ISSUE 5 / UIF:8.2 / MODERNSCIENCE.UZ 𝑤(𝑧) ≥ 0 bo‘lganligi uchun quyidagi o‘rinli: 𝛼 𝑑𝑧𝑤 𝑑 𝑑𝑤 𝛽𝑑𝑤 −𝛾 = (𝑤𝜎 )−|𝑤𝛼 | 𝜕𝑧 𝜕𝑧 𝑑𝑧 𝑑𝑧 yoki 𝛼 𝛽 +1 𝑑𝑧𝑤 𝑑 𝑑𝑤 1 𝑑𝑤𝛼 −𝛾 = (𝑤𝜎 )−| | 𝜕𝑧 𝜕𝑧 𝑑𝑧 𝛽 𝑑𝑧 +1 𝛼 𝛼 = 1 deb tanlab, quyidagiga ega bo‘lamiz: 𝑑𝑧𝑤 𝑑 𝑑𝑤 1 𝑑𝑤𝛽+1 −𝛾 = (𝑤𝜎 )− | |. 𝜕𝑧 𝜕𝑧 𝑑𝑧 𝛽 +1 𝑑𝑧 Tenglamani integrallab quyidagiga ega bo‘lamiz: 𝑑𝑤 1 −𝛾𝑧𝑤 = 𝑤𝜎 − 𝑤𝛽+1 𝑑𝑧 𝛽