DOIRADA BERILGAN KASR TARTIBLI SUPERDIFFUZIYA TENGLAMASI UCHUN BOSHLANG‘ICH CHEGARAVIY MASALANI YECHISH
Keywords:
kasr tartibli superdiffuziya tenglamasi, boshlang‘ich chegaraviy masala, Caputo hosilasi, Riemann-Liouville hosilasi, numerik metodlar, finite difference, spektral metodlar, transport jarayoni, xotira ta’siri.Abstract
Doiraviy hududda berilgan kasr tartibli superdiffuziya tenglamasi uchun boshlang‘ich chegaraviy masalani yechish ilmiy tadqiqotlar sohasida muhim ahamiyatga ega. Superdiffuziya jarayonlari odatiy diffuziyaga nisbatan tezroq yoki sekinroq tarqalish hodisalarini tavsiflaydi va u ko‘plab fizik, biologik va muhandislik tizimlarida uchraydi. Kasr tartibli hosila operatorlari yordamida ifodalangan superdiffuziya tenglamalari an’anaviy differensial tenglamalarga qaraganda ko‘proq ma’lumotni o‘z ichiga oladi, chunki ular tizim xotirasini va tarqalishning noan’anaviy xususiyatlarini hisobga oladi. Dairaviy hududda boshlang‘ich va chegaraviy shartlar bilan berilgan masalani yechish numerik metodlar va analitik usullar orqali amalga oshiriladi, jumladan Grünwald-Letnikov, Caputo va Riemann-Liouville formulalari asosida. Tadqiqot natijalari kasr tartibli superdiffuziya tenglamalarini turli sharoitlarda modellash imkonini beradi va real dunyodagi transport jarayonlarini oldindan bashorat qilishga yordam beradi. Ushbu masalada raqamli metodlar, xususan finite difference usuli, spektral metodlar va variatsion yondoshuvlar tadqiq etilgan, ularning aniqligi va barqarorligi tahlil qilingan. Tadqiqotlar shuni ko‘rsatadiki, boshlang‘ich chegaraviy shartlarni aniqlik bilan qo‘llash superdiffuziya jarayonini to‘g‘ri modellashtirish uchun muhimdir. Shuningdek, kasr tartibli modellar biologik hujayra transporti, molekulyar diffuziya, neft va gaz sohalarida energiya va moddalar oqimining tavsifi kabi amaliy sohalarda qo‘llaniladi.
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