In this paper, it is aimed to construct regularized asymptotics of the solution of a twodimensional 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 existence of a two-dimensional boundary layer. The presence in the free term as a rapidly oscillating factor leads to the inclusion in the asymptotic of the boundary layer with a rapidly oscillating nature of change. Vanishing of the derived phase of the free term leads to 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 the authors for singularly perturbed parabolic equations.

[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, 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, 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, 8, 46, (2006), 1423-1432.

[7]. Omuraliev A., Imash kyzy M. “Singularly perturbed parabolic problems with multidimensional boundary layers”, Differential Equations, 53, 12, (2017), 1616–1630.

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Two-dimensional parabolic problem with a rapidly oscillating free term

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