Two-dimensional parabolic problem with a rapidly oscillating free term

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.

Kaynakça

[1]. Feschenko S., Shkil N., Nikolaenko L. “Asymptotic methods in the theory of linear differential equations,” Kiev, Naukova Dumka, 1966.

[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.

Kaynak Göster