Two-dimensional parabolic problem with a rapidly oscillating free term

Yıl 2019, Cilt: 7 Sayı: 1, 52 - 59, 18.06.2019

Ella Abylaeva , Asan Omuraliev

Öz

The
aim of this paper is to construct regularized asymptotics of the solution of a two-dimensional partial differential equation of parabolic
type with a small parameter for all spatial derivatives and a rapidly
oscillating free term.

     The case
when the first derivative of the phase of the free term at the initial point vanishes
is considered. The two-dimensionality of the equation leads to the
emergence of a two-dimensional boundary layer. The presence in the free term of
a rapidly oscillating factor leads to the inclusion in the asymptotic of the
boundary layer with a rapidly oscillating nature of change.  The vanishing of the derived phase of the free term
introduces into the asymptotic of a new type of boundary layer function. A
complete asymptotic solution of the problem is constructed by the method of
regularization of singularly perturbed problems developed by S.А. Lomov and
adapted by one of the authors for singularly perturbed parabolic equations.

Anahtar Kelimeler

Asymptotics, singularly perturbed parabolic problem, oscillating free term

Kaynakça

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• [2] Omuraliev A.S., Sadykova D.A., Regularization of a singularly perturbed parabolic problem with a fast-oscillating right-hand side, Khabarshy –Vestnik of the Kazak National Pedagogical University, 20, (2007), 202-207.
• [3] Omuraliev A.S., Sheishenova Sh. K., Asymptotics of the solution of a parabolic problem in the absence of the spectrum of the limit operator and with a rapidly oscillating right-hand side, Investigated on the integral-differential equations, no. 42, (2010), 122-128.
• [4] Omuraliev A., Abylaeva E., Asymptotics of the solution of the parabolic problem with a stationary phase and an additive-free member, Manas Journal of Engineering, no. 6/2, (2018), 193-202.
• [5] Lomov S., Introduction to the general theory of singular perturbations, Moscow, Nauka, 1981.
• [6] Omuraliev A., Regularization of a two-dimensional singularly perturbed parabolic problem, Journal of Computational Mathematics and Mathematical Physics, vol. 8, no. 46, (2006), 1423-1432.
• [7] Omuraliev A., Imash kyzy M., Singularly perturbed parabolic problems with multidimensional boundary layers, Differential Equations, vol. 53, no. 12, (2017), 1–15.