Superdiffuziya tenglamalari uchun boshlang‘ich-chegaraviy masalalarning mavjudlik va yagonalik shartlari

Jumanazarova Hafiza G'ayrat qizi

Oxu universiteti matematika mutaxasisligi magistranti

Keywords: superdiffuziya tenglamasi, kasr tartibli hosila, boshlang‘ich-chegaraviy masala, mavjudlik va yagonalik teoremasi, Caputo hosilasi, Riemann-Liouville hosilasi, spektral metod, zaif yechim, anomal diffuziya, Sobolev fazolari.

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Abstract

Ushbu maqolada superdiffuziya tenglamalari, ya’ni vaqt yoki fazo bo‘yicha kasr tartibli hosilalar bilan ifodalangan anomal diffuziya jarayonlarini tavsiflovchi differensial tenglamalar uchun boshlang‘ich-chegaraviy masalalarning mavjudligi va yagonaligi masalalari batafsil tadqiq etiladi. Superdiffuziya (tezroq tarqalish) ko‘pincha fazo bo‘yicha kasr tartibli Laplas operatori yoki vaqt bo‘yicha kasr hosilasi (0 < α < 1 yoki α > 1 holatlari) orqali modellashadi. Maqolada Caputo va Riemann-Liouville operatorlaridan foydalangan holda muammolarning to‘g‘ri qo‘yilishi, zaif va kuchli yechimlarning mavjudligi, yagonaligi hamda barqarorligi uchun yetarli shartlar ko‘rib chiqiladi. Spektral metodlar (Fourier kengayishlari, eigenfunksiyalar), energiya integral usullari, integral tenglamalarga qaytarish va maksimum printsipi kabi yondashuvlar tahlil qilinadi. O‘zbekistonlik va xalqaro tadqiqotchilarning ishlariga asoslanib, numerik va analitik yechimlarning aniqligi, ilovaviy ahamiyati (poroz materiallar, biologik transport, neft-gaz konlari) muhokama qilinadi. Natijalar anomal diffuziya modellarini amaliyotda qo‘llash uchun nazariy asos yaratadi.

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