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Asymptotic study of singularly perturbed differential equations of hyperbolic type has received relatively little attention from researchers. In this paper, the asymptotic solution of the singularly perturbed Cauchy problem for a hyperbolic system is constructed. In addition, the regularization method for singularly perturbed problems of S. A. Lomov is used for the first time for the asymptotic solution of a hyperbolic system. It is shown that this approach greatly simplifies the construction of the asymptotics of the solution for singularly perturbed differential equations of hyperbolic type.
The first boundary value problem for a multidimensional parabolic differential equation with a small parameter ε multiplying all derivatives is studied. A complete (i.e., of any order with respect to the parameter) regularized asymptotics of the solution is constructed, which contains a multidimensional boundary layer function that is bounded for x = (x 1, x 2) = 0 and tends to zero as ε → +0 for x ≠ 0. In addition, it contains corner boundary layer functions described by the product of a boundary layer function of the exponential type by a multidimensional parabolic boundary layer function
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 na ...More
In this paper we construct the asymptotics of the solution of the singularly perturbed parabolic problem with the stationary phase and the additive free term.