CHIZIQSIZ CHEGARAVIY SHARTLAR BILAN PARABOLIK TENGLAMALAR SISTEMASI UCHUN AVTOMODEL YECHIMLAR

THE VI INTERNATIONAL SCIENTIFIC CONFERENCE “SCIENTIFIC FOUNDATIONS FOR THE USE OF INFORMATION TECHNOLOGIES OF A NEW LEVEL AND MODERN PROBLEMS OF AUTOMATION”, NOVEMBER 20, 2025 216 CHIZIQSIZ CHEGARAVIY SHARTLAR BILAN PARABOLIK TENGLAMALAR SISTEMASI UCHUN AVTOMODEL YECHIMLAR Zaripova A. R.1, Palimbetova N. M.2 1Qarshi davlat universiteti, tayanch doktorant, 2Qoraqalpoq davlat universiteti, magistr https://doi.org/10.5281/zenodo.17767774 Annotatsiya. Ushbu maqolada chiziqsiz chegaraviy shartlar bilan bog’langan parabolik tenglamalar sistemasini hisoblashlarni amalga oshirib, avtomodel tenglamalar sistemasiga keltirilgan. Masalaning kompakt yurituvchili yechimi uchun asimptotika topilgan. Topilgan asimptotik formulalardan berilgan masalani sonli yechishda boshlang‘ich yaqinlashish sifatida foydalanilgan. Kalit so‘zlar: Nochiziqli tenglamalar sistemasi, reaksiya-diffuziya, global yechim. Annotation. In this article, a system of parabolic equations with nonlinear boundary conditions is investigated, and it is reduced to a system of self-similar (automodel) equations. The asymptotic behavior of the solution with a compact support is obtained. The derived asymptotic formulas are used as an initial approximation for the numerical solution of the given problem. Key words: System of nonlinear equations, reaksiya-diffuziya, global solution KIRISH. Maqolada quyidagi nochiziqli parabolik tipdagi tenglamalar sistemasi: ( ) ( ) , 1,2, , , 0, 0 , iii mm ix p it ix x u u i k tu xT= =    ∣∣ (1) nochiziqli chegaraviy shart ( ) 1 ( , ) (0, ), 1,2, , , 0 , ij iii mm i kq p ix j j x u x t u t i ku tT = − = =     ∣∣ (2) hamda boshlang‘ich shartlar bilan berilgan 0 ( ,0) ( ), 1,2, , , 0, ii u x u x i k x= =   (3) (1)-(3) masalaning avtomodel yechimlari asimptotikasi va sonli yechimlarini tadqiq etamiz. Bu yerda ( ) 1, 1 1 , 1, 0 1,2,..., i i i ij k p m m q i k  +   = va 0( 1,2, , ) i u i k= funksiyalar + R da manfiy bo‘lmagan uzluksiz funksiyalar. (1) parabolik tipdagi nochiziqli tenglamalar sistemasi chiziqli bo‘lmagan issiqlik o‘tkazish, diffuziya, filtratsiya, kimyoviy reaksiya va portlovchi jarayonlarni tavsiflovchi umumiy matematik modeldir hamda tabiatda uchraydigan qator jarayonlarni matematik modellashtirishda keng tatbiq etiladi [1-10]. TADQIQOT METODOLOGIYASI Avtomodel tenglamalar sistemasini qurishda (1) -(3) tenglamalar sistemasi yechimini quyidagicha qidiramiz: ( , ) ( ) ( ), ( ) , ii kl i i i i u x t x tFt     −− = + = + (4) THE VI INTERNATIONAL SCIENTIFIC CONFERENCE “SCIENTIFIC FOUNDATIONS FOR THE USE OF INFORMATION TECHNOLOGIES OF A NEW LEVEL AND MODERN PROBLEMS OF AUTOMATION”, NOVEMBER 20, 2025 217 bu yerda 1 1 ( )( 1) , ( )( 1) ( 1,2,..., .1 ) k i i i i i i i i i j ij j k k m l p l k m p k q i k = + = + + + = =++  Hisoblashlar olib borish natijasida (1)-(3) masala quyidagi ko‘rinishga keltiramiz: ( ) ( ) ( ) ( ) ( ) ( ) ( ) `` ` `` 1 0 (0) (0), 1,2,..., . i ii i ij ii p mm i i i i i i i i i i k pq mm i i j j F F k F l F F F F i k     =   + + =    − = =   (5) (5) masalani avtomodel yechimni quyidagicha ( ) 2 1 () ii iii F a b   + =− qidiramiz va uni (5) masalaga qo`yib hisoblashlarni amalga oshirib, avtomodel yechimga kelamiz: 2 1 1 ( 1) 1 ( ) , i i i ii p i mp i i i p p i F a b  ++ +− + +  =−    bu yerda 0 i a , 1 1 ( 1) 1 0, max(0, ). ( 2) i p ii ii ii mp b k y y pm + + +− =  = + NATIJALAR. Teorema. (5) masalaning kompakt yurituvchili yechimi uchun 1 2 i i p p i i i a b  + +  →  da quyidagi asimptotika o‘rinli bo‘ladi ( ) ( ) ( ) 1 1 iii F F o  =+ . (6) Xulosa Chiziqsiz differensial tenglamalar sistemalarini sonli yechishda eng asosiy muammolardan biri iteratsiya jarayonining barqarorligini va tez yaqinlashuvini ta’minlay oladigan boshlang‘ich yaqinlashishni to‘g‘ri tanlashdir. Tadqiqot davomida ushbu masalani hal etish uchun qurilgan asimptotik ifodalar boshlang‘ich yaqinlashish sifatida qo‘llanildi va ular sonli parametrlar bilan yaxshi moslashishi isbotlandi. FOYDALANILGAN ADABIYOTLAR RO‘YXATI 1. G. Astrita and G. Marrucci, Principles of non-Newtonian f luid mechanics 1. (McGraw-Hill, New York, 1974). 2. B. T. Chen, Y. S.Mi and C. L. Mu, Critical exponents for a doubly degenerate parabolic system coupled via nonlinear boundary flux, Acta Mathematica Scientia 31B (2011), 681–693. 3. Z. J. 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