YO‘NALISHI TESKARI VAQTLI MASALA

Аннотация

This article examines inverse time problems. These types of problems represent an important class of parabolic differential equations used in physics, mathematical modeling, and many other fields. The task involves determining the initial state from a final one, i.e., going backward along the time axis. This makes the problem ill-posed in the classical theory of solutions. The paper provides a mathematical formulation of the problem and analyzes the existence, uniqueness, and stability of the solution. Additionally, approaches to stabilize the solution using regularization methods are discussed. Alongside theoretical foundations, numerical examples are presented, and a sensitivity analysis of the solution is carried out. Keywords: inverse time problem, parabolic equation, ill-posedness, regularization, numerical methods, backward time extrapolation. Masala bayoni Faraz qilaylik, u(x,t) funksiyasi quyidagi parabolik tenglama yechimi bo‘lsin: ∂u/∂t = α ∂²u/∂x², 0 < x < L, 0 < t < T (1) Shartlar quyidagicha berilgan: u(0,t) = u(L,t) = 0, 0 < t < T (2) u(x,T) = φ(x), 0 < x < L (3) Bu masala J. Adamar ma’nosida nokorrekt hisoblanadi. Ushbu masalani Fur’e usuli yordamida yechamiz. Yechim quyidagi ko‘rinishga ega bo‘ladi: u(x,t) = ∑[n=1 to ∞] aₙ e^(−α λₙ² (T − t)) sin(λₙ x), λₙ = nπ / L (4) Bu yerda Fur’e koeffitsiyentlari quyidagicha aniqlanadi: aₙ = (2 / L) ∫₀ᴸ φ(x) sin(nπx / L) dx (5) Masalaning noaniqligi: Teskari vaqtli masalada t → 0 bo‘lganda eksponentalar juda katta bo‘lib ketadi: e^(α λₙ² T) → ∞ Bu esa kichik xatoliklar katta og‘ishlarga olib kelishiga sabab bo‘ladi. Tixonov regulyarizatsiyasi Masalani barqarorlashtirish uchun Tixonov regulyarizatsiyasi qo‘llaniladi: 44 THEORETICAL ASPECTS IN THE FORMATION OF PEDAGOGICAL SCIENCES International scientific-online conference u_ε(x,0) = ∑[n=1 to N] (aₙ / (1 + ε λₙ²)) sin(λₙ x) (6) Bu yerda ε – regulyarizatsiya parametri. Barqarorlik bahosi: ‖u(x,0) − u_ε(x,0)‖ ≤ C ε^β, 0 < β < 1 (7) Regulyarlashtiruvchi operatorlar Regulyarlashtiruvchi operatorlar oilasi quyidagicha ifodalanadi: R_ε(φ) = ∑[n=1 to ∞] (e^(α λₙ² T) / (1 + ε λₙ²)) aₙ sin(λₙ x) (8) Bu operator yordamida yechimning taqribiy barqaror ifodasi olinadi. Xulosa: Mazkur maqolada yo‘nalishi teskari vaqtli masala, ya’ni issiqlik tenglamasi uchun vaqt bo‘yicha orqaga extrapolyatsiya masalasi ko‘rib chiqildi. Masalaning nokorrektligi sababli yechimni topish uchun regulyarizatsiya metodlari, xususan Fur’e qatorlari yordamida yechim ifodasi va Tixonov regulyarizatsiyasi tahlil qilindi. Kompyuterli modellashtirish asosida bu yondashuvlar amaliy jihatdan qo‘llanishi mumkinligi ko‘rsatildi. Foydalanilgan adabiyotlar: 1. K.S. Fayazov, I.O. Xajiyev. Nokorrekt va teskari masalalar (o‘quv qo‘llanma) 2. Tikhonov A.N., Arsenin V.Y. Solutions of Ill-posed Problems. Wiley, New York, 1977. 3. Ames K.A., Straughan B. Non-Standard and Improperly Posed Problems. Academic Press, 1997. 4. K.S. Fayazov. Hisoblash matematikasi, matematik fizika va analizning nokorrekt masalalarini yechish usullari. Toshkent, O‘zMU, 2001. 5. Lavrentiev M.M., Romanov V.G., Shishatskii S.P. Ill-posed Problems of Mathematical Physics and Analysis. AMS, 1986. 45