Singularly Perturbed Parabolic Problems

The aim of this work is to construct regularized asymptotic of the solution of a singularly perturbed parabolic problems. Namely, in the first paragraph, we consider the case when the scalar equation contains a free term consisting of a finite sum of the rapidly oscillating functions. In the first paragraph, it is shown that the asymptotic solution of the problem contains parabolic, power, rapidly oscillating, and angular boundary layer functions. Angular boundary layer functions have two components: the first one is described by the product of a parabolic boundary layer function and a boundary layer function, which has a rapidly oscillating change. The second section is devoted to a two-dimensional equation of parabolic type. Asymptotic of the scalar equation contains a rapidly oscillating power, parabolic boundary layer functions, and their product; then, the multidimensional equation additionally contains a multidimensional composite layer function.


Introduction
Singularly perturbed problems with rapidly oscillating free terms were studied in [1][2][3]. Ordinary differential equations with a rapidly oscillating free term whose phase does not have stationary points are studied in [1]. Using the regularization method for singularly perturbed problems [4], differential equations of parabolic type with a small parameter were studied in [2,3] when fast-oscillating functions as a free member. The asymptotic solutions constructed in [1][2][3] contain a boundary layer function having a rapidly oscillating character of change. In addition to such a boundary layer function, ordinary differential equations contain an exponential [1], and parabolic equations -parabolic [2,3] and angular boundary layer [2,5] functions. If the phase of the free term has stationary points, then boundary layers arise additionally, having a power character of change. In this case, the asymptotic solution consists of regular and boundary layer terms. The boundary layer members are parabolic, power, rapidly oscillating boundary layer functions, and their products, which are called angular boundary layer functions [4]. In this chapter we used the methods of [4,5].

Statement of the problem
In this chapter we study the following problem: 0 is a small parameter and Ω = {(x, t): x ∈ 0; 1 ð Þ, t ∈ 0; T ð g. The problem is solved under the following assumptions: 2. ∀x ∈ 0; 1 ½ function a x ð Þ . 0.

Regularization of the problem
For the regularization of problem (Eq. (1)), we introduce regularizing independent variables using methods [5,6]: Instead of the desired function u x; t; ε ð Þ, we will study the extended function such that its restriction by regularizing variables coincides with the desired solution: Taking into account (Eqs. (21)) and ((3)), we find the derivatives On the basis of (Eqs. (1)-(4)) for the extended function u M; ε ð Þ, we set the problem: The problem (Eq. (5)) is regular in ε as ε ! 0:

Solution of iterative problems
The solution of problem (Eq. (5)) will be determined in the form of a series: For the coefficients of this series, we obtain the following iterative problems: The solution of this problem contains parabolic boundary layer functions; internal power boundary layer functions which are connected with a rapidly oscillating free term in a phase which are vanished at t ¼ t l , l ¼ 0, 1, ……, n in addition; and the asymptotic also contain angular boundary layer functions. We introduce a class of functions in which the iterative problems will be solved: From these spaces we construct a new space: The element u M ð ÞϵG has the form: (9)

Solvability of intermediate tasks
The iterative problems (Eq. (9)) in general form will be written: Then, by directly substituting function u M ð Þϵ G from (Eq. (9)) in (Eq. (10)), we see that this function is a solution if and only if the function Y l k N l ð Þ will be a solution of equation: With the corresponding boundary conditions, this equation has a solution which have the estimate: The theorem is proven.
Theorem 2. Suppose that the conditions of Theorem 1 are satisfied. Then, under additional conditions: Eq. (10) is uniquely solvable.
Proof. By Theorem 1 equation (Eq. (10)) has a solution that is representable in the form (Eq. (9)). With satisfying condition (1) Due to the fact that the function erfc θ 2 ffi ffi t p is zero at θ ¼ 0, the values for w l x; t ð Þ t¼0 ; q l k x; t ð Þ t¼0 are chosen arbitrarily. We calculate Condition (3) of the theorem will be ensured, if we choose arbitrarily (Eq. (9)) as the solutions of the following equations: After this choice of arbitrariness, expression (Eq. (13)) is rewritten: 30)) under the boundary conditions from (Eq. (12)) in the form: We substitute this function in the corresponding equation from (Eq. (14)); then with respect to d l k x; t ð Þ, we obtain a differential equation, which is solving under the where P l k x; t ð Þ is known as the function.
By substituting the obtained function into condition for d l k x; t ð Þ x¼lÀ1 from (Eq. (12)), we define the value of d l k x ð Þ x¼lÀ1 . The obtained value is used as an initial condition for a differential equation with respect to d l k x ð Þ, which is obtained after substitution d l k x; t ð Þ into the first condition of (2). With that we ensure fulfillment of this condition and uniqueness of the function Y l k N l ð Þ: The last equation from (Eq. (14)) due to the fact that θ 0 The obtained ratio is used as the initial condition for the differential equation with respect to z l k, vÀ1 x; t ð Þ from (Eq. (14)). The equation with respect to v vÀ1 x; t ð Þ under the initial condition from (12) determines this function uniquely. Equations with respect to w l k, vÀ1 x; t ð Þ, q l k, vÀ1 x; t ð Þ under the corresponding condition from (Eq. (12)) have solutions representable in the form: where H l 1, vÀ1 x; t ð Þ, H l 2, vÀ1 x; t ð Þare known functions. With substituting (Eq. (16)) into the conditions under x ¼ l À 1 from (Eq. (12)), . These conditions are used in solving differential equations which are obtained from the second condition of (Eq. (21)): Thus, function u M ð Þ is determined uniquely. The theorem is proven.

Solution of iterative problems
Eq. (8) is homogeneous for k = 0; therefore, by Theorem 1, it has a solution in G, representable in the form: If the function Y l k, 0 N l ð Þ is the solution of the equation ∂ η Y l k, 0 N l ð Þ ¼ ∂ 2 ς l Y l k, 0 N l ð Þ which is satisfying that Y l k, 0 N l ð Þ t¼η¼0 ¼ 0; Y l k, 0 N l ð Þ x¼lÀ1, ς l ¼0 ¼ Àc k, 0 l À 1; t ð Þ: from the last problem, we define , 0 x ð Þ is the arbitrary function. In the next step, equation (Eq. (8)) for k = 1 takes the form: According to Theorem 1, this equation is solvable in U, if c k, 0 x; t ð Þ=0; the function Y l k, 0 N l ð Þ is the solution of the differential equation ensuring the solvability of the equation with respect to c l k, 1 x; t ð Þ: Suppose that This relation is used by the initial condition for determining Z k, 0 x; t ð Þ from the equation entering into (Eq. (18)).
Further repeating this process, we can determine all the coefficients of u k m ð Þ of the partial sum: In each iteration with respect to v i x; t ð Þ, w l i x; t ð Þ, d l k, i x; t ð Þ, z k, i x; t ð Þ, q l k, i x; t ð Þ, we obtain inhomogeneous equations.
2. Two-dimensional parabolic problem with a rapidly oscillating free term

Introduction
In the case when a small parameter is also included as a multiplier with a temporal derivative, the asymptotic of the solution acquires a complex structure.
Different classes of singularly perturbed parabolic equations are studied in [2]. There, regularized asymptotics of the solution of these equations are constructed, when a small parameter is in front of the time derivative and with one spatial derivative. It is shown that the constructed asymptotic contains exponential, parabolic, and angular products of exponential and parabolic boundary layer functions. The equations are studied when the limiting equation has a regular singularity. Such equations have a power boundary layer. If a small parameter is entering as the multiplier for all spatial derivatives, then the asymptotic solution contains a multidimensional parabolic boundary layer function. When entering into the equation, as free terms of rapidly oscillating functions, then the asymptotic of the solution additionally contains fast-oscillating boundary layer functions. If it is additionally assumed that the phase of this free term has a stationary point, in addition to the rapidly oscillating boundary layer function that arises as a power boundary layer.
This section is devoted to a two-dimensional equation of parabolic type.

Statement of the problem
Consider the problem: where ε . 0 is the small parameter, The problem is solved under the following assumptions: 3. θ 0 0 ð Þ ¼ 0:

Solution of iterative problems
For the solution of the extended function (Eq. (24)), we search in the form of series Then, for the coefficients of this series, we get the following problems: T 0 u q ¼ Àiθ 0 t ð Þ∂ τ 2 u qÀ2 À T 1 u qÀ2 , q ¼ 2, 3: We introduce a class of functions: From these classes we will construct a new one, as a direct sum: Any item u M ð Þ∈ U is representable in the form: where d l x; t ð Þis arbitrary function such as G η l ; η rþ2, l ; ξ; η; τ 1 À Á Due to the fact that the function d l v x; t ð Þ при t ¼ τ 1 ¼ 0 multiplied by the function becomes as d l 0 x; t ð Þ  27), i=4, has a free term: