XUSUSIY HOSILALI CHIZIQLI BO‘LMAGAN DIFFERENSIAL TENGLAMALARNING AVTOMODEL VA UMUMLASHGAN AVTOMODEL YECHIMI
Kalit so'zlar:
avtomodel yechim, nolinear tenglama, diffuziya, Burgers tenglamasi, KdV tenglamasiAnnotatsiya
Ushbu maqolada xususiy hosilali chiziqli bo'lmagan differensial tenglamalar turli tabiiy va texnik jarayonlarni matematik modellashtirishda muhim o'rin egallashi ko‘rsatib o'tilgan. Mazkur ishda avtomodel hamda umumlashgan avtomodel yondashuvlarga asoslangan holda bunday tenglamalar uchun yechimlar qurish usullari tahlil qilinadi. Avtomodel o'zgaruvchilardan foydalanish orqali ko'p o'zgaruvchili nolinear tenglamalarni oddiy differensial tenglamalar ko'rinishiga keltirish imkoniyati asoslab beriladi. Tadqiqot natijalari matematik fizikaning murakkab masalalarini tahlil qilishda samarali qo'llanishi mumkin.
Foydalanilgan adabiyotlar
Barenblatt G.I. Scaling, Self-Similarity, and Intermediate Asymptotics. Cambridge University Press, 1996. – 12–25-betlar (avtomodel va o'z-o'ziga o'xshashlik tushunchasi) – 67–82-betlar (diffuziya tenglamasi uchun avtomodel yechimlar)
Ovsyannikov L.V. Group Analysis of Differential Equations. Academic Press, 1982. – 31–44-betlar (masshtablash va invariant o'zgaruvchilar) – 98–115-betlar (xususiy hosilali tenglamalar uchun simmetriya usullari)
Polyanin A.D., Zaitsev V.F. Handbook of Nonlinear Partial Differential Equations. CRC Press, 2012. – 145–168-betlar (nolinear diffuziya tenglamalari) – 312–328-betlar (Burgers tenglamasi va aniq yechimlar)
Whitham G.B. Linear and Nonlinear Waves. Wiley, 1999. – 201–220-betlar (nolinear to'lqinlar va tarqalish) – 245–260-betlar (zarba va yumshatilgan zarba to'lqinlar)
Olver P.J. Applications of Lie Groups to Differential Equations. Springer, 2000. – 89–104-betlar (masshtablash simmetriyasi) – 181–195 betlar (avtomodel yechimlarni qurish)
Zeldovich Ya.B., Raizer Yu.P. Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena. Academic Press, 1966. – 52–70-betlar (zarba to'lqinlarining fizik modeli) – 134–150-betlar (nolinear gidrodinamik jarayonlar)
Bluman G.W., Kumei S. Symmetries and Differential Equations. Springer, 1989. – 23–39-betlar (Lie simmetriyalar asoslari) – 117–132-betlar (xususiy hosilali tenglamalar uchun invariant yechimlar)
Debnath L. Nonlinear Partial Differential Equations for Scientists and Engineers. Birkhäuser, 2012. – 56–74-betlar (nolinear tenglamalarning umumiy xususiyatlari) – 210–225-betlar (diffuziya va konveksiya modellari)
Samarskii A.A., Galaktionov V.A., Kurdyumov S.P. Blow-up in Quasilinear Parabolic Equations. Walter de Gruyter, 1995. – 14–28-betlar (parabolik tenglamalarning umumiy nazariyasi) – 95–110-betlar (avtomodel tipdagi yechimlar)
Lie S. Theory of Transformation Groups. Chelsea Publishing, 1970. – 61–78-betlar (o'lcham va masshtablash o'zgarishlari) – 140–155-betlar (invariant yechimlar nazariyasi)
