A System of Singularly Perturbed Parabolic Equations with a Power Boundary Layer

Abstract

The work is devoted to the construction of the asymptotics of the solution of a singularly perturbed system of equations of parabolic type, in the case when the limit equation has a regular singularity as the small parameter tends to zero. The developed algorithm allows construction of the asymptotics of solutions containing power, parabolic, and angular boundary layer functions.

1. Introduction

1.1 A singularly perturbed system of parabolic equations

This article studies the problem

$$L_{\epsilon }u\left(x,t,\epsilon \right)\equiv \left(\epsilon +t\right){\partial }_{t}u-{\epsilon }^{2}a\left(x\right){\partial }_x^{2}u-B\left(t\right)u=f\left(x,t\right),\left(x,t\right)\in \mathrm{\Omega },$$

$${\left.u\right|}_{t=0}=u{\left|{}_{x=0}=u\right|}_{x=1}=0,$$E1

where $\epsilon >0$ is a small parameter, $\mathrm{\Omega }=\{\left(x,t\right):x\in \left(0,1\right),t\in (0,T\left]\right\},u=\left(u_1,u_2,\dots ,u_n\right)$. Suppose the following conditions:

$$0<a\left(x\right)\in C^{\infty }\left[0,1\right],B\left(t\right)\in C^{\infty }\left(\left[0,T\right],C^{n^2}\right),f\left(x,t\right)\in C^{\infty }\left(\overline{\mathrm{\Omega }},C^n\right);$$

The eigenvalues ${\lambda }_i\left(t\right)$ of the matrix $B\left(t\right)$ satisfy the conditions: $B\left(t\right){\psi }_i\left(t\right)={\lambda }_i\left(t\right){\psi }_i\left(t\right),i=1,2,\dots ,n,\Re {\lambda }_i\left(t\right)<0,{\lambda }_j\left(t\right)\ne {\lambda }_i\left(t\right),\forall i\ne j,t\in \left[0,T\right]$ ($\Re$ is the real part of the complex number).

Singularly perturbed parabolic problems in various statements are devoted to Ref. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. In Ref. [6, 9, 10, 11, 12, 13, 14, 15, 16, 17], by the method of boundary layer functions, various boundary value problems for parabolic equations with a small parameter are studied. The regularization asymptotic for solving parabolic problems [7, 8, 18, 19, 20, 21, 22, 23] with different occurrences of a small parameter in the equations and limit operators has different structures using the regularization method for singularly perturbed problems.

The papers [16, 23] are devoted to systems of singularly perturbed parabolic equations. In [16], the method of boundary functions, and in Ref. [23], the method of regularization. For singularly perturbed problems, equations are studied when the spatial derivative is preceded by a scalar function, and [24] is a matrix. In the latter case, the structure of the solution asymptotic is greatly multiplied. The order of the equation does not go down.

Problems with power boundary layers, that is, problems where, when a small parameter tends to zero, it acquires a regular feature were studied in Ref. [9, 10, 25, 26]. Ordinary differential equations with a power boundary layer are studied in Ref. [8, 9, 10, 26]. For equations of parabolic type, when a small parameter does not enter the factors of the spatial derivative, the asymptotic of the power boundary layer is constructed. In contrast to Ref. [9], in our equation, there is a small parameter in front of the spatial derivative and we will improve the algorithm for constructing the asymptotic.

1.2 Regularization of problems

We introduce regularizing variables

$${\xi }_l=\frac{{\phi }_l\left(x\right)}{\sqrt{{\epsilon }^3}},{\phi }_l\left(x\right)={\left(-1\right)}^{l-1}{\int }_{l-1}^x\frac{\mathit{ds}}{\sqrt{a\left(s\right)}},{\eta }_i={\lambda }_i\left(0\right)ln\left(1+t/\epsilon \right),\tau ={\epsilon }^{-1}ln\left(1+t/\epsilon \right)$$E2

along with the independent variables $\left(x,t\right)$ of the function $\tilde{u}\left(M,\epsilon \right)$, $M=\left(x,t,\xi ,\tau ,\eta \right),$ $\xi =\left({\xi }_1,{\xi }_2\right),$ $\eta =\left({\eta }_1,{\eta }_2,\dots ,{\eta }_n\right)$ such that

$${\left.\stackrel{\sim }{u}\left(M,\epsilon \right)\right|}_{\zeta =\gamma \left(x,t,\epsilon \right)}\equiv u\left(x,t,\epsilon \right),$$

$$\begin{array}{l}\zeta =\left(\xi ,\tau ,\eta \right),\gamma \left(x,t,\epsilon \right)\\ =\left(\frac{{\phi }_1\left(x\right)}{\sqrt{{\epsilon }^3}},\frac{{\phi }_2\left(x\right)}{\sqrt{{\epsilon }^3}},{\lambda }_1\left(0\right)ln\left(1+t/\epsilon \right),\dots ,{\lambda }_n\left(0\right)ln\left(1+t/\epsilon \right),{\epsilon }^{-1}ln\left(1+t/\epsilon \right)\right).\end{array}$$

Hence, on the basis of (2), we find

$$\begin{array}{l}{\partial }_{t}u\equiv {\left({\partial }_t\stackrel{\sim }{u}+\sum _{i=1}^n\frac{{\lambda }_i\left(0\right)}{t+\epsilon }{\partial }_{{\eta }_i}\stackrel{\sim }{u}+\frac{1}{\epsilon }\frac{1}{t+\epsilon }{\partial }_{\tau }\stackrel{\sim }{u}\right)}_{\zeta =\gamma \left(x,t,\epsilon \right)},{\partial }_{x}u\equiv {\left({\partial }_x\stackrel{\sim }{u}+\sum _{l=1}^2\frac{{\phi }_l^{\prime }\left(x\right)}{\sqrt{{\epsilon }^3}}{\partial }_{{\xi }_l}\stackrel{\sim }{u}\right)}_{\zeta =\gamma \left(x,t,\epsilon \right)},\end{array}$$

$${\partial }_x^{2}u\equiv {\left({\partial }_x^2\stackrel{\sim }{u}+\sum _{l=1}^2\left[{\left(\frac{{\phi }_l^{\prime }\left(x\right)}{\sqrt{{\epsilon }^3}}\right)}^2{\partial }_{{\xi }_l}^2\stackrel{\sim }{u}+\frac{1}{\sqrt{{\epsilon }^3}}{D}_{\xi ,l}\stackrel{\sim }{u}\right]\right)}_{\zeta =\gamma \left(x,t,\epsilon \right)},D_{\xi ,l}\equiv 2{\phi }_l^{\prime }\left(x\right){\partial }_{{\xi }_{l}x}^2+{\phi }_l^{″}\left(x\right){\partial }_{{\xi }_l},$$

then, according to (1), we set the extended problem

$$L_{\xi }\equiv {\epsilon }^{-1}{T}_0\stackrel{\sim }{u}+T_1\stackrel{\sim }{u}-\sqrt{\epsilon }{L}_{\xi }\stackrel{\sim }{u}+\epsilon {\partial }_t\stackrel{\sim }{u}-{\epsilon }^2{L}_x\stackrel{\sim }{u}=f\left(x,t\right),M\in Q,$$

$${\left.\stackrel{\sim }{u}\right|}_{t=\tau ={\eta }_i=0}=\stackrel{\sim }{u}{\left|{}_{x=0,{\xi }_1=0}=\stackrel{\sim }{u}\right|}_{x=1,{\xi }_2=0}=0,Q=\mathrm{\Omega }\times {\left(0,\infty \right)}^3.$$E3

The notation is entered here

$$T_0\equiv {\partial }_{\tau }-{\mathrm{\Delta }}_{\xi },T_1\equiv \sum _{i=1}^n{\lambda }_i\left(0\right){\partial }_{{\eta }_i}+t{\partial }_t-B\left(t\right),{\mathrm{\Delta }}_{\xi }\equiv \sum _{l=1}^2{\partial }_{{\xi }_l}^2,L_{\xi }\equiv \sum _{l=1}^{2}a\left(x\right)D_{\xi ,l},L_x\equiv a\left(x\right){\partial }_x^2.$$

Note that the identity holds

$${\left({\stackrel{\sim }{L}}_{\epsilon }\stackrel{\sim }{u}\right)}_{\zeta =\gamma \left(x,t,\epsilon \right)}\equiv L_{\epsilon }u,$$E4

solutions of problem (3) will be defined as

$$\stackrel{\sim }{u}\left(M,\epsilon \right)=\sum _{k=0}^{\infty }{\epsilon }^{\frac{k}{2}}{u}_k\left(M\right),$$E5

then for the coefficients of this series we obtain the following iterative problems:

$$T_0{u}_{\nu }=0,\nu =0,1,T_0{u}_2=f\left(x,t\right)-T_1{u}_0,T_0{u}_k=-T_1{u}_{k-2}+L_{\xi }{u}_{k-3}-{\partial }_t{u}_{k-4}+L_x{u}_{k-6},$$

$${\left.u_k\right|}_{t=\tau ={\eta }_i=0}=0,u_k{\left|{}_{x=0,{\xi }_1=0}=u_k\right|}_{x=1,{\xi }_2=0}=0,k\ge 3.$$E6

1.3 Solvability of iterative problems

We introduce the space of functions in which the iterative problems will be solved:

$$U_1=\{u_1\left(N_1\right):u_1\left(N_1\right)=<v\left(x,t\right)+\left[c\left(x,t\right)+\mathrm{\Lambda }\left(P\left(x\right)\right)\right]e^{\eta },\psi \left(t\right)>,$$

$$v\left(x,t\right)\in C^{\infty }\left(\overline{\mathrm{\Omega }},C^n\right),P\left(x\right)\in C^{\infty }\left(\left[0,1\right],C^n\right),c\left(x,t\right)\in C^{\infty }\left(\overline{\mathrm{\Omega }},C^{n^2}\right),\psi \left(t\right)\in C^{\infty }\left(\left[0,T\right],C^n\right)\},$$

$$U_2=\begin{array}{l}\{u_2\left(N^l\right):u_2\left(N^l\right)=\sum _{l=1}^2<Y\left(N^l\right)+z\left(N^l\right)e^{\eta },\psi \left(t\right)>,‖Y\left(N^l\right)‖<{\mathit{ce}}^{-\frac{{\xi }_l^2}{8\tau }},\\ ‖z\left(N^l\right)‖<{\mathit{ce}}^{-\frac{{\xi }_l^2}{8\tau }}\},\end{array}$$

$$Y\left(N^l\right)=\mathit{col}\left(Y_1\left(N^l\right),Y_2\left(N^l\right),\dots ,Y_n\left(N^l\right)\right),z\left(N^l\right)=\left(z_{\mathit{ij}}\left(N^l\right)\right),i,j=\overline{1,n},$$

$$N^l=\left(x,t,{\xi }^l,\tau \right),\eta =\mathit{col}\left({\eta }_1,{\eta }_2,\dots ,{\eta }_n\right),\mathrm{\Lambda }\left(P\left(x\right)\right)=\mathit{diag}\left(P_1\left(x\right),P_2\left(x\right),\dots ,P_n\left(x\right)\right),$$

$$<\left[c\left(x,t\right)+\mathrm{\Lambda }\left(P\left(x\right)\right)\right]e^{\eta },\psi \left(t\right)>=\sum _{i,j=1}^n{c}_{\mathit{ij}}\left(x,t\right)e^{{\eta }_j}{\psi }_i\left(t\right)+\sum _{i=1}^n{P}_i\left(x\right)e^{{\eta }_i}{\psi }_i\left(t\right).$$

$$<Y\left(N^l\right)+z\left(N^l\right)e^{\eta },\psi \left(t\right)>=\sum _{i=1}^n\left[Y_i\left(N^l\right)+\sum _{j=1}^n{z}_{\mathit{ij}}\left(N^l\right)e^{{\eta }_j}\right]{\psi }_i\left(t\right).$$

Here $\mathit{col}\left({\eta }_1,{\eta }_2,\dots ,{\eta }_n\right)$ is the vector, $\mathit{diag}\left(P_1\left(x\right),P_2\left(x\right),\dots ,P_n\left(x\right)\right)$ is a diagonal matrix. Element $u_k\left(M\right)\in U=U_1\oplus U_2$ has an idea

$$\begin{array}{ll}{u}_k\left(M\right)& =<v_k\left(x,t\right)+\left[c^k\left(x,t\right)+\mathrm{\Lambda }\left(P^k\left(x\right)\right)\right]e^{\eta },\psi \left(t\right)>\\ & +\sum _{l=1}^2<Y^k\left(N^l\right)+z^k\left(N^l\right)e^{\eta },\psi \left(t\right)>.\end{array}$$E7

We calculate the action of the operators included in the extended equation on the function $u_k\left(M\right)\in M$, for which we first decompose ${\psi }_{i^{\prime }}\left(t\right)={\sum }_{j=1}^n{\alpha }_{\mathit{ij}}\left(t\right){\psi }_j\left(t\right)$, or we write by entering the notation

$$T_0{u}_k\left(M\right)=\sum _{l=1}^2<{\partial }_{\tau }{Y}^k\left(N^l\right)-{\partial }_{{\xi }_l}^2{Y}^k\left(N^l\right)+\left[{\partial }_{\tau }{z}^k\left(N^l\right)-{\partial }_{{\xi }_l}^2{z}^k\left(N^l\right)\right]e^{\eta },\psi \left(t\right)>,$$

$$T_1{u}_k\left(M\right)=<t{\partial }_t{v}_k\left(x,t\right)+{\mathit{tA}}^T\left(t\right)v_k\left(x,t\right)-B\left(t\right)v_k\left(x,t\right)+t{\partial }_t{c}^k\left(x,t\right)$$E8

$$+{\mathit{tA}}^T\left(t\right)c^k\left(x,t\right)-B\left(t\right)c^k\left(x,t\right)+c^k\left(x,t\right)\mathrm{\Lambda }\left(0\right)-B\left(t\right)\mathrm{\Lambda }\left(P^k\left(x\right)\right)+{\mathit{tA}}^T\left(t\right)\mathrm{\Lambda }\left(P^k\left(x\right)\right)$$

$$+\mathrm{\Lambda }\left(P^k\left(x\right)\right)\mathrm{\Lambda }\left(0\right),\psi \left(t\right)>+\sum _{l=1}^2<[t{\partial }_t{Y}^k\left(N^l\right)+{\mathit{tA}}^T\left(t\right)Y^k\left(N^l\right)-B\left(t\right)Y^k\left(N^l\right)$$

$$+t{\partial }_t{z}^k\left(N^l\right)+{\mathit{tA}}^T\left(t\right)z^k\left(N^l\right)-B\left(t\right)z^k\left(N^l\right)+z^k\left(N^l\right)\mathrm{\Lambda }\left(0\right)]e^{\eta },\psi \left(t\right)>$$

$$=<D_1{v}_k\left(x,t\right)+D_2\left[c^k\left(x,t\right)+\mathrm{\Lambda }\left(P^k\left(x\right)\right)\right]e^{\eta }+\sum _{l=1}^2\left[<D_1{Y}^k\left(N^l\right)+D_2{z}^k\left(N^l\right)e^{\eta }\right],\psi \left(t\right)>,$$

$$D_1\equiv t{\partial }_t+{\mathit{tA}}^T\left(t\right)-B\left(t\right),D_2\equiv D_1+D_{\lambda },D_{\lambda }=\sum _{i=1}^n{\lambda }_i\left(0\right){\partial }_{{\eta }_i},$$

$$\mathrm{\Lambda }\left(0\right)=\mathit{diag}\left({\lambda }_1\left(0\right),{\lambda }_2\left(0\right),\dots ,{\lambda }_n\left(0\right)\right),$$

$$A\left(t\right)=\left({\alpha }_{\mathit{ij}}\right),{\alpha }_{\mathit{ij}}=\left({\psi }_i^{\prime }\left(t\right),{\psi }_j\left(t\right)\right);$$

$$\begin{array}{r}{L}_{\xi }{u}_k=<a\left(x\right)\sum _{l=1}^2\{\left[2{\phi }_l^{\prime }\left(x\right){\partial }_{x{\xi }_l}^2+{\phi }_l^{″}\left(x\right){\partial }_{{\xi }_l}\right]Y^k\left(N^l\right)\\ +\left[2{\phi }_l^{\prime }\left(x\right){\partial }_{x{\xi }_l}^2+{\phi }_l^{″}\left(x\right){\partial }_{{\xi }_l}\right]z^k\left(N^l\right)e^{\eta },\psi \left(t\right)\}>\\ ,\end{array}$$

$$\begin{array}{ll}{L}_x{u}_k\left(M\right)& =<L_x{v}_k\left(x,t\right)+\sum _{l=1}^2{L}_x{Y}^k\left(N^l\right)\\ & +\left[L_x{c}^k\left(x,t\right)+L_x\mathrm{\Lambda }\left(P^k\left(x\right)\right)+\sum _{l=1}^2{L}_x{z}^k\left(N^l\right)\right]e^{\eta },\psi \left(t\right)>.\end{array}$$

Iterative equations are written in the form.

$$T_0{u}_k\left(M\right)=h^k\left(M\right).$$E9

Theorem 1: Let conditions (1), (2), and $h^k\left(M\right)\in U_2$ hold. Then Eq. (9) is solvable in $U$.

Proof: Let $h^k\left(M\right)\in U_2$:

$$h^k\left(M\right)=\sum _{l=1}^2<\left[h^{k,1}\left(N^l\right)+h^{k,2}\left(N^l\right)e^{\eta }\right],\psi \left(t\right)>.$$E10

Function (7) will be the solution of Eq. (9) in $U$. If the functions $Y^k\left(N^l\right),z^k\left(N^l\right)$ are the solution of the equations

$${\partial }_{\tau }{Y}^k\left(N^l\right)-{\partial }_{{\xi }_l}^2{Y}^k\left(N^l\right)=h^{k,1}\left(N^l\right),{\partial }_{\tau }{z}^k\left(N^l\right)-{\partial }_{{\xi }_l}^2{z}^k\left(N^l\right)=h^{k,2}\left(N^l\right).$$E11

These equations are obtained by substituting (7) into Eq. (9), taking into account calculations (8) and representation (10). Eqs. (11), with appropriate boundary conditions, have solutions that satisfy the estimates [25]:

$$‖Y^k\left(N^l\right)‖<{\mathit{ce}}^{-\frac{{\xi }_l^2}{8\tau }},‖z^k\left(N^l\right)‖<{\mathit{ce}}^{-\frac{{\xi }_l^2}{8\tau }}.$$

Theorem 2: Let conditions (1), (2), and $\overline{h^{k,3}\left(x,0\right)}=0$ hold. Then problem

$$<D_2\left[c^k\left(x,t\right)+\mathrm{\Lambda }\left(P^k\left(x\right)\right)\right],\psi \left(t\right)>=<h^{k,3}\left(x,t\right),\psi \left(t\right)>$$E12

$$\overline{c^k\left(x,0\right)}=-\mathrm{\Lambda }\left(v_k\left(x,0\right)\right)-\mathrm{\Lambda }\left(P^k\left(x\right)\right)-\mathrm{\Lambda }\left(\overline{\overline{c^k\left(x,0\right)}}\cdot \mathbf{1}\right),\mathbf{1}=\mathit{col}\left(1,1,\dots ,1\right),$$E13

$$\left({\left.c_{\mathit{ii}}^k\left(x,t\right)\right|}_{t=0}=-v_{k,i}\left(x,0\right)-P_i^k\left(x\right)-\sum _{j=1\left(i\ne j\right)}^n{c}_{\mathit{ij}}^k(x,0)\right),$$

has a unique solution. Hereinafter, $\overline{c}$ denotes a matrix with nonzero diagonal, and $\overline{\overline{c}}$ with zero diagonal elements, that is, $c=\overline{c}+\overline{\overline{c}}$.

Proof: We write the relation (12) in the coordinate form

$$\sum _{i,j=1}^n\left\{t\left[{\partial }_t{c}_{\mathit{ij}}^k\left(x,t\right)+\sum _{\mu =1}^n{\alpha }_{\mu ,i}\left(t\right)\left(c_{\mu ,i}^k\right(x,t)+P_{\mu }^k\left(x\right))\right]+\left({\lambda }_j\left(0\right)-{\lambda }_i\left(t\right)\right)c_{\mathit{ij}}^k(x,t)\right\}{\psi }_i\left(t\right)e^{{\eta }_j}$$

$$+\sum _{i=1}^n\left({\lambda }_i\left(0\right)-{\lambda }_i\left(t\right)\right)P_i^k\left(x\right)e^{{\eta }_i}{\psi }_i\left(t\right)=\sum _{i,j=1}^n{h}_{\mathit{ij}}^{k,3}\left(x,t\right)e^{{\eta }_j}{\psi }_i\left(t\right).$$

Removing the degeneracy of this equation, assuming that

$${\left.\left({\lambda }_j\left(0\right)-{\lambda }_i\left(t\right)\right)c_{\mathit{ij}}^k(x,t)\right|}_{t=0}=h_{\mathit{ij}}^{k,3}\left(x,0\right),\forall i\ne j.$$E14

Equating the coefficients $\psi$, we get

$$t\left[{\partial }_t{c}_{\mathit{ii}}^k\left(x,t\right)+{\alpha }_{\mathit{ii}}\left(t\right)c_{\mathit{ii}}^k(x,t)\right]=\left({\lambda }_i\left(t\right)-{\lambda }_i\left(0\right)\right)P_i^k\left(x\right)+h_{\mathit{ii}}^{k,3}\left(x,t\right)-t\sum _{\mu =1}^n{\alpha }_{\mu ,i}\left(t\right)P_{\mu }^k\left(x\right),i=\overline{1,n},$$E15

$$t\left[{\partial }_t{c}_{\mathit{ij}}^k\left(x,t\right)+\sum _{\mu =1\left(i\ne \mu \right)}^n{\alpha }_{\mu ,i}\left(t\right)c_{\mu ,j}^k(x,t)\right]+\left({\lambda }_j\left(0\right)-{\lambda }_i\left(t\right)\right)c_{\mathit{ij}}^k\left(x,t\right)=h_{\mathit{ij}}^k\left(x,t\right),i\ne j,i,j=\overline{1,n}.$$E16

Eq. (15), by virtue of condition $h_{\mathit{ii}}^k\left(x,0\right)=0$, under the initial condition (13), and Eq. (16) with condition (14) have one-to-one solutions.

Remarks. In iterative problems, the condition $h_{\mathit{ii}}^k\left(x,0\right)=0$ will be provided by the choice of the function $P_i^k\left(x\right)$.

Theorem 3: Let conditions (1) and (2) be fulfilled. Then Eq. (9) has a unique solution satisfying the conditions: (a) $u_k\left(M\right){\left|{}_{t=\tau =\eta =0}=0,u_k\left(M\right)\right|}_{x=l-1,{\xi }_l=0}=0$; (b) $T_1{u}_k\left(M\right)+h^k\left(M\right)\in U_2$; (c) $L_{\xi }{u}_k\left(M\right)=0$.

Proof: Let satisfy the function (7) to the boundary conditions (a):

$$<v_k\left(x,0\right)+\left[c^k\left(x,0\right)+\mathrm{\Lambda }\left(P^k\left(x\right)\right)\right]1+\sum _{l=1}^2{\left[Y^k\left(N^l\right)+z^k\left(N^l\right)1\right]}_{t=\tau =0},\psi \left(0\right)>=0,$$

$${\left.<v_k\left(l-1,t\right)+\left[c^k\right(l-1,t)+\mathrm{\Lambda }\left(P^k\left(x\right)\right)]e^{\eta }+\sum _{l=1}^2[Y^k\left(N^l\right)\right|}_{x=l-1,{\xi }_l=0}$$

$${\left.+z^k\left(N^l\right)\right|}_{x=l-1,{\xi }_l=0}{e}^{\eta }],\psi \left(t\right)>=0,$$

Hence, we define

$$\overline{c^k\left(x,0\right)}=-\mathrm{\Lambda }\left(v_k\left(x,0\right)\right)-\mathrm{\Lambda }\left(P^k\left(x\right)\right)-\mathrm{\Lambda }\left(\overline{\overline{c^k\left(x,0\right)}}1\right),$$

$${\left.Y^k\left(N^l\right)\right|}_{t=\tau =0}=0,{\left.z^k\left(N^l\right)\right|}_{t=\tau =0}=0,$$E17

$$\begin{array}{l}{z^k\left(N^l\right)|}_{{\xi }_l=0}=W^{k,l}\left(x,t\right),{W^{k,l}\left(x,t\right)|}_{x=l-1}=c^k\left(l-1,t\right)-\mathrm{\Lambda }\left(P^k\left(x\right)\right),\\ {Y^k\left(N^l\right)|}_{{\xi }_l=0}=d^{k,l}\left(x,t\right),\end{array}$$

$${\left.d^{k,l}\left(x,t\right)\right|}_{x=l-1}=-v_k\left(l-1,t\right).$$

Ensuring the solvability of Eq. (9) with the right side of

$$F_k\left(M\right)=T_1{u}_k\left(M\right)+h^k\left(M\right)\in U_2,$$

based on the calculations (8) and the representation of

$$h^k\left(M\right)=<h^{k,1}\left(x,t\right)+h^{k,2}\left(x,t\right)e^{\eta }+\sum _{l=1}^2\left[h^{k,3}\left(N^l\right)+h^{k,4}\left(N^l\right)e^{\eta }\right],\psi \left(t\right)>$$

assuming that

$$<D_2\left[c^k\left(x,t\right)+\mathrm{\Lambda }\left(P^k\left(x\right)\right)\right]e^{\eta },\psi \left(t\right)>=-<h^{k,2}\left(x,t\right)e^{\eta },\psi \left(t\right)>,$$E18

$$<D_1{v}_k\left(x,t\right)-h^{k,1}\left(x,t\right),\psi \left(t\right)>=0.$$E19

Eq. (18) under the initial condition (17), on the basis of Theorem 2, are uniquely solvable. Eqs. (19) are solved without an initial condition and have a bounded solution [27].

Eqs. (11) with boundary conditions (17) have solutions that can be represented as

$$Y_i^k\left(N^l\right)=d_i^{k,l}\left(x,t\right)\mathit{erfc}\left[{\xi }_l/\left(2{\tau }^{1/2}\right)\right]+h_i^{k,3}\left(x,t\right)I_1\left({\xi }_l,\tau \right),$$

$$z_{\mathit{ij}}^k\left(N^l\right)=W_{\mathit{ij}}^{k,l}\left(x,t\right)\mathit{erfc}\left[{\xi }_l/\left(2{\tau }^{1/2}\right)\right]+h_{\mathit{ij}}^{k,4}\left(x,t\right)I_2\left({\xi }_l,\tau \right),$$

where $\mathit{erfc}\left(x\right)=2{\pi }^{-1/2}{\int }_0^x{e}^{-t^2}\mathit{dt}$ is integral of the additional function and describes a parabolic boundary layer, $d^{k,l}\left(x,t\right),W^{k,l}\left(x,t\right)$ are arbitrary functions that are chosen, like solving the equations

$$D_x{d}_i^{k,l}\left(x,t\right)=-D_x{h}_i^{k,3}\left(x,t\right),D_x{W}_{\mathit{ij}}^{k,3}\left(x,t\right)=-D_x{h}_{\mathit{ij}}^{k,4}\left(x,t\right),D_x\equiv 2{\phi }_{l^{\prime }}\left(x\right){\partial }_x+{\phi }_{{l^{\prime }}^{\prime }}\left(x\right),$$E20

with boundary conditions from (17). Eqs. (20) are obtained by satisfying condition c) and taking into account that the functions $\mathit{erfc}\left[{\xi }_l/\left(2{\tau }^{1/2}\right)\right]$ and $I_l\left({\xi }_l,\tau \right)$ have single estimates.

Thus, a unique solution to Eq. (9) is obtained that satisfies conditions (a)–(c).

1.4 Construction of solutions of iterative equations

For $\nu =0,1$ the equations for $u_{\nu }\left(M\right)$ are homogeneous; therefore, the condition of Theorem 1 holds; therefore, the solution of these equations exists and can be represented in the form (7).

The following iterative equation, on the grounds (8), has a free term

$$F_2\left(M\right)=f\left(x,t\right)-T_1{u}_0=f\left(x,t\right)-<D_1{v}_0\left(x,t\right)+D_2\left[c^0\left(x,t\right)+\mathrm{\Lambda }\left(P^0\left(x\right)\right)\right]e^{\eta }$$

$$+\sum _{l=1}^2\left[D_1{Y}^0\left(N^l\right)+D_2{z}^0\left(N^l\right)e^{\eta }\right],\psi \left(t\right)>.$$

We decompose $f\left(x,t\right)$ by the system ${\psi }_i\left(t\right)$ and substitute it with the previous relation. Further, providing the conditions of Theorem 1, we set

$$D_1{v}_0\left(x,t\right)=f\left(x,t\right),t\left[{\partial }_t{v}_{0,i}\left(x,t\right)+\sum _{\mu =1}^n{\alpha }_{\mathit{i\mu }}\left(t\right)v_{0,\mu }(x,t)\right]-{\lambda }_i\left(t\right)v_{0i}\left(x,t\right)=\left(f\left(x,t\right),{\psi }_i\right),$$

$$<D_2\left[c^0\left(x,t\right)+\mathrm{\Lambda }\left(P^0\left(x\right)\right)\right],\psi \left(t\right)>=0.$$

The first system has a smooth solution, and the second system by Theorem 2 is solvable if

$$t\left[{\partial }_t{c}_{\mathit{ii}}^0+{\alpha }_{\mathit{ii}}\left(t\right)c_{\mathit{ii}}^0(x,t)\right]=\left({\lambda }_i\left(t\right)-{\lambda }_i\left(0\right)\right)P_i^0\left(x\right)-t\sum _{\mu =1}^n{\alpha }_{\mu ,i}\left(t\right)P_{\mu }^0\left(x\right),$$

$${\left.c_{\mathit{ii}}^0\left(x,t\right)\right|}_{t=0}=-v_{i0}\left(x,0\right)-P_i^0\left(x\right),$$E21

$$t\left[{\partial }_t{c}_{\mathit{ij}}^0+\sum _{\mu =1\left(\mu \ne i\right)}^n{\alpha }_{\mu ,i}\left(t\right)c_{\mathit{\mu i}}^0(x,t)\right]+\left({\lambda }_j\left(0\right)-{\lambda }_i\left(t\right)\right)c_{\mathit{ij}}^0\left(x,t\right)=0,$$

$$\left({\lambda }_j\left(0\right)-{\lambda }_i\left(t\right)\right)c_{\mathit{ij}}^0\left(x,0\right)=0,i\ne j,i,j=\overline{1,n}.$$E22

Problem (21) is uniquely solvable, and problem (22) has a trivial solution.

For $k=3$, by Theorem 3, ensuring condition (c) for $d^{0,l}\left(x,t\right)$ and $W^{0,l}\left(x,t\right)$, we obtain problem

$$D_x{d}_i^{0,l}\left(x,t\right)=0,{\left.d_i^{0,l}\left(x,t\right)\right|}_{x=l-1}=-v_{0,i}\left(l-1,t\right),$$

$$D_x{W}_{\mathit{ij}}^{0,l}\left(x,t\right)=0,{\left.W_{\mathit{ij}}^{0,l}\left(x,t\right)\right|}_{x=l-1}=-c_{\mathit{ij}}^0\left(l-1,t\right).$$

By this, we have determined the main term of the asymptotic. In addition, conditions (b) of Theorem 3 gives

$$D_1{v}_{1,i}=0,t\left[{\partial }_t{c}_{\mathit{ii}}^1+{\alpha }_{\mathit{ii}}\left(t\right)c_{\mathit{ii}}^1\right]=\left({\lambda }_i\left(t\right)-{\lambda }_i\left(0\right)\right)P_i^1\left(x\right)-t\sum _{\mu =1}^n{\alpha }_{\mu ,i}\left(t\right)P_{\mu }^1\left(x\right),$$

$${\left.c_{\mathit{ii}}^1\left(x,t\right)\right|}_{t=0}=-v_{1i}\left(x,0\right)-P_i^1\left(x\right),$$

$$t\left[{\partial }_t{c}_{\mathit{ij}}^1+\sum _{\mu =1\left(\mu \ne i\right)}^n{\alpha }_{\mu ,i}\left(t\right)c_{\mathit{\mu j}}^1(x,t)\right]+({\lambda }_j\left(0\right)-{\lambda }_i\left(t\right)c_{\mathit{ij}}^1\left(x,t\right)=0,{\left.c_{\mathit{ij}}^1\left(x,t\right)\right|}_{t=0}.$$

From here we define $v_{1i}\left(x,t\right)=0$, below it will be shown that $P_i^1\left(x\right)=0$, therefore $c_{\mathit{ii}}^1\left(x,t\right)=0$, and from the last equation we find $c_{\mathit{ij}}^1\left(x,t\right)=0,i\ne j$.

In the next $k=4$ step free member

$${\partial }_t,F_4\left(M\right)=-T_1{u}_2+L_{\xi }{u}_1-{\partial }_t{u}_0.$$

Satisfying condition (c) of Theorem 3, we obtain the problems

$$D_x{d}_i^{1,l}\left(x,t\right)=0,{\left.d_i^{1,l}\left(x,t\right)\right|}_{x=l-1}=-v_{1,i}\left(l-1,t\right)=0,$$

$$D_x{W}_{\mathit{ij}}^{1,l}\left(x,t\right)=0,{\left.W_{\mathit{ij}}^{1,l}\left(x,t\right)\right|}_{x=l-1}=-c_{\mathit{ij}}^1\left(l-1,t\right)=0.$$

That is, $d_i^{1,l}\left(x,t\right)=0,$ $W_{\mathit{ij}}^1,l\left(x,t\right)=0$, and, therefore, $u_1\left(M\right)=0$. Regarding $Y_i^3\left(N^l\right),W_{\mathit{ij}}^3\left(N^l\right)$ we obtain homogeneous equations, therefore

$$Y_i^3\left(N^l\right)=d_i^{3l}\left(x,t\right)\mathit{erfc}\left[{\xi }_l/\left(2{\tau }^{1/2}\right)\right],W_{\mathit{ij}}^3\left(N^l\right)=W_{\mathit{ij}}^{3,l}\left(x,t\right)\mathit{erfc}\left[{\xi }_l/\left(2{\tau }^{1/2}\right)\right].$$

We calculate

$$F_4\left(M\right)=-<D_1{v}_2\left(x,t\right)+D_2\left[c^2\left(x,t\right)+\mathrm{\Lambda }\left(P^2\left(x\right)\right)\right]e^{\eta }$$

$$+\sum _{l=1}^2\left[Y^2\left(N^l\right)+z^2\left(N^l\right)e^{\eta }\right],\psi \left(t\right)>-<{\partial }_t{v}_0\left(x,t\right)+A^T\left(t\right)v_0\left(x,t\right)$$

$$+\left[{\partial }_t{c}^0\left(x,t\right)+A^T\left(t\right)c^0(x,t)+A\left(t\right)\mathrm{\Lambda }\left(P^0\left(x\right)\right)\right.e^{\eta }$$

$$+\sum _{l=1}^2\left[{\partial }_t{Y}^0\left(N^l\right)+A^T\left(t\right)Y^0\left(N^l\right)\right.\left.+\left({\partial }_t{z}^0\left(N^l\right)+A^T\left(t\right)z^0\left(N^l\right)\right)e^{\eta }\right],\psi \left(t\right)>$$

and ensuring the solvability in $U$ of the iteration equation for $k=4$, we assume

$$<D_t{v}_2\left(x,t\right),\psi \left(t\right)>=-<{\partial }_t{v}_0\left(x,t\right)+A\left(t\right)v_0\left(x,t\right),\psi \left(t\right)>,$$E23

$$D_2\left[c^2\left(x,t\right)+\mathrm{\Lambda }\left(P^2\left(x\right)\right)\right],\psi \left(t\right)>=-<{\partial }_t{c}^0\left(x,t\right)+A\left(t\right)c^0\left(x,t\right)+A\left(t\right)\mathrm{\Lambda }\left(P^0\left(x\right)\right),\psi \left(t\right)>.$$

The solvability of the second system is ensured by the choice of $P^0\left(x\right)$:

$$\begin{array}{l}\mathrm{\Lambda }\left(P^0\left(x\right)\right)=-\overline{A^{-1}\left(t\right)\left[{\partial }_t{c}^0\left(x,t\right)+A\left(t\right)c^0\left(x,t\right)\right]},\\ P_i^0\left(x\right)={\alpha }_{\mathit{ii}}^{-1}{\left[{\partial }_t{c}_{\mathit{ii}}^0\left(x,t\right)+\sum _{k=1}^n{\alpha }_{\mathit{ki}}\left(t\right)c_{\mathit{ki}}^0\left(x,t\right)\right]}_{t=0},\end{array}$$

as well as

$$\begin{array}{l}{<\overline{c^2\left(x,t\right)\mathrm{\Lambda }\left(0\right)-\mathrm{\Lambda }\left(t\right)c^2\left(x,t\right),\psi \left(t\right)}>|}_{t=0}\\ =-<\overline{\overline{{\partial }_t{c}^0\left(x,t\right)+A\left(t\right)c^0\left(x,t\right)+A\left(t\right)\mathrm{\Lambda }\left(P^0\left(x\right)\right),\psi \left(t\right)}}>\end{array}$$

$${\left.c_{\mathit{ij}}^2\left(x,t\right)\right|}_{t=0}=\frac{1}{{\lambda }_j\left(0\right)-{\lambda }_i\left(0\right)}{\left[{\partial }_t{c}_{\mathit{ij}}^0\left(x,t\right)+\sum _{k=1}^n{\alpha }_{\mathit{ki}}\left(t\right)c_{\mathit{kj}}^0(x,t)+{\alpha }_{\mathit{ji}}\left(t\right)P_j^0\left(x\right)\right]}_{t=0},i\ne j.$$E24

Under such assumptions, the second system of (23) is solvable. It is solved under the initial conditions (24) and the initial condition ${\left.c_{\mathit{ii}}^2\left(x,t\right)\right|}_{t=0}=-v_{2i}\left(x,0\right)-P_i^2\left(x\right),$ which is obtained from (17). This completely defines the main term of the asymptotic.

Further, repeating the process described above, we find all the coefficients of the partial sum (5), and the coefficients with odd indices are zero.

1.5 Estimates of the remainder term

Denote by $R_{\mathit{\epsilon n}}\left(M\right)=\stackrel{\sim }{u}\left(M,\epsilon \right)-u_{\mathit{\epsilon n}}\left(M\right)$, where $u_{\mathit{\epsilon n}}\left(M\right)={\sum }_{k=0}^n{\epsilon }^k{u}_{2k}\left(M\right).$ Substituting $\stackrel{\sim }{u}\left(M,\epsilon \right)=u_{\mathit{\epsilon n}}\left(M\right)+R_{\mathit{\epsilon n}}\left(M\right)$ into the extended problem (3), then, taking into account (6), with respect to $R_{\mathit{\epsilon n}}\left(M\right)$, we obtain

$${\stackrel{\sim }{L}}_{\epsilon }{R}_{\mathit{\epsilon n}\left(M\right)}={\epsilon }^{n+1}{g}_{\mathit{\epsilon n}}\left(M\right),{\left.R_{\mathit{\epsilon n}}\left(M\right)\right|}_{t=\tau =0}={\left.R_{\mathit{\epsilon n}}\left(M\right)\right|}_{x=l-1,{\xi }_l=0}=0,l=1,2,$$

where the function $g_{\mathit{\epsilon n}}\left(M\right)$ is expressed through $u_{\epsilon ,k}\left(M\right)$. Producing constriction this problem, taking into account the identity (4), with respect to ${\left.R_{\epsilon ,n}\left(x,t,\epsilon \right)\equiv \left(R_{\mathit{\epsilon n}}\left(M\right)\right)\right|}_{\zeta =\gamma \left(x,t,\epsilon \right)}$ we get the problem

$$L_{\epsilon }{R}_{\epsilon ,n}\left(x,t,\epsilon \right)={\epsilon }^{n+1}{g}_{\mathit{\epsilon n}}\left(x,t,\epsilon \right),{\left.R_{\mathit{\epsilon n}}\right|}_{t=0}={\left.R_{\mathit{\epsilon n}}\right|}_{x=l-1}=0,l=1,2.$$

From the construction of solutions to iterative problems, it can be seen that the function $g_{\mathit{\epsilon n}}\left(x,t,\epsilon \right)$ is uniformly bounded in $\overline{\mathrm{\Omega }}$. Applying the maximum principle [28], we can establish a uniform estimate in $\overline{\mathrm{\Omega }}$

$$‖R_{\mathit{\epsilon n}}\left(x,t,\epsilon \right)‖<c{\epsilon }^{n+1}.$$E25

Theorem 4: Let conditions (1) and (2) be fulfilled. Then the function $u_{\mathit{\epsilon n}}\left(x,t,\epsilon \right)$ is a uniform asymptotic solution of problem (1), that is, the estimate (25) holds.

2. A singularly perturbed parabolic equation system

We consider the first boundary-value problem for a system of singularly perturbed parabolic equations

$$L_{\epsilon }u\equiv \left(\epsilon +t\right){\partial }_{t}u-{\epsilon }^{2}A\left(x\right){\partial }_x^{2}u-D\left(t\right)u=f\left(x,t\right),\left(x,t\right)\in \mathrm{\Omega },$$E26

$$u\left(x,0,\epsilon \right)=h\left(x\right),u\left(0,t,\epsilon \right)=u\left(1,t,\epsilon \right)=0,$$

where $\mathrm{\Omega }=\left(0,1\right)\times \left(0,T\right],\epsilon >0$ is a small parameter,

$$u=u\left(x,t,\epsilon \right)=\mathit{col}\left(u_1\left(x,t,\epsilon \right),u_2\left(x,t,\epsilon \right),u_n\left(x,t,\epsilon \right)\right),A\left(x\right)\in C^{\infty }\left(\left[0,1\right],{\mathrm{ℂ}}^{n^2}\right),$$

$$D\left(t\right)\in C^{\infty }\left(\left[0,T\right],{\mathrm{ℂ}}^{n^2}\right),f\left(x,t\right)\in C^{\infty }\left(\overline{\mathrm{\Omega }},{\mathrm{ℂ}}^n\right).$$

The work is a continuation of [29], where instead of the matrix-function $A\left(x\right)$, there was a scalar function and an asymptotic of the solution was constructed containing two functions describing the boundary layers along $x=0$ and $x=1$. Lomov was the first to introduce the concept of a power-law boundary layer based on the study of the Lighthill equation and he based his method on it [29]. In this case, the asymptotics contains $2m$ parabolic boundary layer functions describing the boundary layers along $x=0$ and $x=1$.

The asymptotics of the solution to this problem, along with the parabolic boundary layer function (the parabolic boundary layer is described by the function), $\mathit{erfc}\left(\frac{\phi \left(x\right)}{2\epsilon \sqrt{t}}\right)$, also contain the power boundary layer function

$${\mathrm{\Pi }}_{\epsilon }\left(t\right)={\left(\frac{\epsilon }{\epsilon +t}\right)}^{\lambda },\lambda >0$$

as well as their products, which describe the corner boundary layer in the vicinity of $\left(0,0\right)$.

Construction of the asymptotic solution of a singularly perturbed system of parabolic equations is devoted to works [8, 9, 10] and [30]. In Ref. [8], a regularized asymptotic is constructed in the case when the matrix of coefficients for the desired function has zero multiple eigenvalues. A similar problem was studied in [9] and an asymptotic of the boundary layer type was constructed. The method of boundary functions in [10] studied the bisingular problem for systems of parabolic equations, which is characterized by the presence of non-smoothness of the asymptotic terms and a singular dependence on a small parameter. In Ref. [30] and [26], various problems for split systems of two equations of parabolic type were studied, and asymptotics of the boundary layer type were constructed. The problems of differential equations of parabolic type with a small parameter were studied in Ref. [24, 31, 32].

2.1 Statement of the problem

We consider the first boundary-value problem (26). The problem is solved under the following assumptions:

1. For $n$-dimensional vector functions $f\left(x,t\right)$ and $h\left(x\right)$, the inclusions

   $$f\left(x,t\right)\in C^{\infty }\left(\overline{\mathrm{\Omega }},{\mathrm{ℂ}}^n\right),h\left(x\right)\in C^{\infty }\left(\left[0,1\right],{\mathrm{ℂ}}^n\right),$$

   are fulfilled for $n\times n$-matrix-valued functions $D\left(t\right)$ and $A\left(x\right)$-inclusions

   $$D\left(t\right)\in C^{\infty }\left(\left[0,T\right],{\mathrm{ℂ}}^{n\times n}\right),A\left(x\right)\in C^{\infty }\left(\left[0,1\right],{\mathrm{ℂ}}^{n\times n}\right);$$

2. The real parts of all roots ${\lambda }_i\left(x\right),i=\overline{1,n},$ of the equation $\mathit{det}\left(A\left(x\right)-\mathit{\lambda E}\right)=0$ are positive and ${\lambda }_i\left(x\right)\ne {\lambda }_j\left(x\right)$ for all $x\in \left[0,1\right],$ $i\ne j,i,j=\overline{1,n}$;

3. The real parts of the eigenvalues ${\beta }_j\left(t\right),j=\overline{1,n}$ of the matrix $D\left(t\right)$ are negative, that is, $\mathit{Re}{\beta }_j\left(0\right)<0$ and ${\beta }_i\left(0\right)\ne {\beta }_j\left(t\right)\forall t\in \left[0,T\right],i\ne j,i,j=\overline{1,n}$;

4. Completed the conditions of approval of the initial and boundary conditions $h\left(0\right)=h\left(1\right)=0$.

2.2 Regularization of the problem

Following [29], p. 316; 30, P. 18], we introduce regularizing variables

$${\xi }_{i,l}=\frac{{\phi }_{i,l}\left(x\right)}{\sqrt{{\epsilon }^3}};{\phi }_{i,l}\left(x\right)={\left(-1\right)}^{l-1}{\int }_{l-1}^x\frac{\mathit{ds}}{\sqrt{{\lambda }_i\left(s\right)}},l=1,2,i=\overline{1,n},$$E27

$$\tau =\frac{1}{\epsilon }ln\left(\frac{t+\epsilon }{\epsilon }\right),{\mu }_j={\beta }_j\left(0\right)ln\left(\frac{t+\epsilon }{\epsilon }\right)\equiv K_j\left(t,\epsilon \right),j=\overline{1,n},$$

and an extended function such that

$${\left.\stackrel{\sim }{u}\left(M,\epsilon \right)\right|}_{\xi =\phi \left(x\right)/\epsilon }\equiv u\left(x,t,\epsilon \right),M=\left(x,t,\xi ,\tau ,\mu \right),\xi =\left({\xi }_1,{\xi }_2\right),$$

$${\xi }_l=\left({\xi }_{1,l},{\xi }_{2,l},\dots ,{\xi }_{n,l}\right),\phi \left(x\right)=\left({\phi }_1\left(x\right),{\phi }_2\left(x\right)\right),$$E28

$${\phi }_l\left(x\right)=\left({\phi }_{1,l},{\phi }_{2,l}\left(x\right),\dots ,{\phi }_{n,l}\left(x\right)\right),l=1,2,\mu =\left({\mu }_1,{\mu }_2,\dots ,{\mu }_n\right).$$

Based on (27), we find the derivatives from (28):

$${\left.{\partial }_{t}u\equiv \left({\partial }_t\stackrel{\sim }{u}+\frac{1}{\epsilon \left(t+\epsilon \right)}{\partial }_{\tau }\stackrel{\sim }{u}+\sum _{j=1}^n\frac{{\beta }_j\left(0\right)}{t+\epsilon }{\partial }_{{\mu }_j}\stackrel{\sim }{u}\right)\right|}_{\theta =\chi \left(x,t,\xi ,\tau ,\mu \right)},$$

$${\left.{\partial }_{x}u\equiv \left({\partial }_x\stackrel{\sim }{u}+\sum _{l=1}^2\sum _{i=1}^n\left[\frac{1}{\sqrt{{\epsilon }^3}}{\phi }_{i,l}^{\prime }\left(x\right){\partial }_{{\xi }_i,l}\stackrel{\sim }{u}\right]\right)\right|}_{\theta =\chi \left(x,t,\xi ,\tau ,\mu \right)},$$

$${\left.{\partial }_x^{2}u\equiv \left({\partial }_x^2\stackrel{\sim }{u}+\sum _{l=1}^2\sum _{i=1}^n\left[\frac{1}{{\epsilon }^3}{\left({\phi }_{i,l}^{\prime }\left(x\right)\right)}^2{\partial }_{{\xi }_{i,l}}^2\stackrel{\sim }{u}+\frac{1}{\sqrt{{\epsilon }^3}}{L}_{i,l}^{\xi }\stackrel{\sim }{u}\right]\right)\right|}_{\theta =\chi \left(x,t,\xi ,\tau ,\mu \right)},$$

$$L_{i,l}^{\xi }\stackrel{\sim }{u}=2{\phi }_{i,l}^{\prime }\left(x\right){\partial }_{x{\xi }_{i,l}}^2\stackrel{\sim }{u}+{{\phi }^{\prime }}_{i,l}^{\prime }\left(x\right){\partial }_{{\xi }_{i,l}}\stackrel{\sim }{u},\chi \left(x,t,\epsilon \right)=\left(\frac{\phi \left(x\right)}{\sqrt{{\epsilon }^3}},\frac{1}{\epsilon }\mathit{ln}\left(\frac{t+\epsilon }{\epsilon }\right),K(t,\epsilon )\right),$$

$$K\left(t,\epsilon \right)=\left(K_1\left(t,\epsilon \right),K_2(t,\epsilon ),\dots ,K_n(t,\epsilon )\right),\theta =\left(\tau ,\mu ,\xi \right).$$

According to these calculations, as well as (26) and (28), we pose the following extended problem

$${\stackrel{\sim }{L}}_{\epsilon }\stackrel{\sim }{u}\equiv \epsilon {\partial }_t\stackrel{\sim }{u}+\frac{1}{\epsilon }{T}_0\stackrel{\sim }{u}+T_1\stackrel{\sim }{u}-\sqrt{\epsilon }{L}_{\xi }\stackrel{\sim }{u}-t{\partial }_t\stackrel{\sim }{u}-{\epsilon }^2{L}_x\stackrel{\sim }{u}=f\left(x,t\right),$$E29

$${\left.\stackrel{\sim }{u}\right|}_{t=0}=0,\stackrel{\sim }{u}{\left|{}_{x=0,{\xi }_{i,1}=0}=\stackrel{\sim }{u}\right|}_{x=1,{\xi }_{i,2}=0}=0,i=\overline{1,n},$$

$$T_0\stackrel{\sim }{u}={\partial }_{\tau }\stackrel{\sim }{u}-\sum _{l=1}^2\sum _{i=1}^n{\left({\phi }_{i,l}^{\prime }\left(x\right)\right)}^{2}A\left(x\right){\partial }_{{\xi }_{i,l}}^2\stackrel{\sim }{u},T_1\stackrel{\sim }{u}=t{\partial }_t\stackrel{\sim }{u}+\sum _{j=1}^{\infty }{\beta }_j\left(0\right){\partial }_{{\mu }_j}\stackrel{\sim }{u}-D\left(t\right)\stackrel{\sim }{u},$$

$$L_{\xi }\stackrel{\sim }{u}=\sum _{l=1}^2\sum _{i=1}^{n}A\left(x\right)L_{i,l}^{\xi }\stackrel{\sim }{u},L_x\stackrel{\sim }{u}=A\left(x\right){\partial }_x^2\stackrel{\sim }{u}.$$

In this case, the identity holds

$${\left({\stackrel{\sim }{L}}_{\epsilon }\stackrel{\sim }{u}\left(M,\epsilon \right)\right)}_{\theta =\chi \left(x,t,\xi ,\tau ,\mu \right)}\equiv L_{\epsilon }u\left(x,t,\epsilon \right).$$E30

The solution to the extended problem (29) will be defined as a series

$$\stackrel{\sim }{u}\left(M,\epsilon \right)=\sum _{k=0}^{\infty }{\epsilon }^{k/2}{u}_k\left(M\right).$$E31

Substituting (31) into problem (29) and equating the coefficients for the same powers of P, we obtain the following equations:

$$T_0{u}_0=0,T_0{u}_1=0,T_0{u}_2=f\left(x,t\right)-T_1{u}_0,T_0{u}_3=L_{\xi }{u}_0-T_1{u}_1,$$E32

$$T_0{u}_k=L_{\xi }{u}_{k-3}+L_x{u}_{k-6}-{\partial }_t{u}_{k-4}-T_1{u}_{k-2}.$$

The initial and boundary conditions for them are set in the form

$${\left.u_k\left(M\right)\right|}_{t=\mu =\tau =0}=0,$$

$${\left.u_k\left(M\right)\right|}_{x=l-1,{\xi }_{i,l}=0}=0,k\ge 0,i=\overline{1,n},l=1,2.$$

2.3 Solvability of iterative problems

Each of the problems (32) has innumerable solutions, therefore, we single out a class of functions in which these problems were uniquely solvable. We introduce the following function classes:

$$U_1=\left\{V\left(x,t\right):V(x,t)=\sum _{i=1}^n{v}_i(x,t){\psi }_i\left(t\right),v_i(x,t)\in C^{\infty }\left(\overline{\mathrm{\Omega }}\right)\right\},$$

$$U_2=\left\{Y\left(N\right):Y\left(N\right)=\sum _{l=1}^2\sum _{i=1}^n{y}_i^l\left(N_i^l\right)b_i\left(x\right),\left|y_i^l\left(N_i^l\right)\right|<cexp\left(-\frac{{\xi }_{i,l}}{8\tau }\right)\right\},$$

$$U_3=\left\{C\left(x,t\right):C(x,t)=\sum _{i=1}^n\left[\sum _{j=1}^n{c}_{\mathit{ij}}\left(x,t\right)exp\left({\mu }_j\right)+p_i\left(x\right)\right]{\psi }_i\left(t\right),c_{\mathit{ij}}(x,t)\in C^{\infty }\left(\overline{\mathrm{\Omega }}\right)\right\},$$

$$U_4=\left\{Z\left(N\right):Z\left(N\right)=\sum _{l=1}^2\sum _{i,j=1}^n{z}_{i,j}^l\left(N_i^l\right)b_i\left(x\right)exp\left({\mu }_j\right),\left|z_{i,j}^l\left(N_i^l\right)\right|<cexp\left(-\frac{{\xi }_{i,l}^2}{8\tau }\right)\right\},$$

where $N_i^l=\left(x,t,{\xi }_{i,l},\tau ,{\mu }_i\right),i=1,2,\dots ,n,l=1,2.$ From these classes of functions we construct a new class as a direct sum: $U=U_1\oplus U_2\oplus U_3\oplus U_4.$ The function $u_k\left(M\right)\in U$ is representable in vector form

$$u_k\left(M\right)=\mathrm{\Psi }\left(t\right)\left[V_k\left(x,t\right)+C^k(x,t)exp\left(\mu \right)\right]$$

$$+\sum _{l=1}^{2}B\left(x\right)\left[Y^{k,l}\left(N^l\right)+Z^{k,l}\left(N^l\right)exp\left(\mu \right)\right],C^k\left(x,t\right)=C_1^k\left(x,t\right)+\mathrm{\Lambda }\left(P\left(x\right)\right),$$

$$C_1^k\left(x,t\right)=\left(c_{\mathit{ij}}\left(x,t\right)\right),V_k\left(x,t\right)=\mathit{col}\left(v_{k1},v_{k2},\dots ,v_{\mathit{kn}}\right),$$

$$Y^{k,l}\left(N^l\right)=\mathit{col}\left(y_1^{k,l}\left(N_1^l\right),y_2^{k,l}\left(N_2^l\right),\dots ,y_n^{k,l}\left(N_n^l\right)\right),Z^{k,l}\left(N^l\right)=\left(z_{\mathit{ij}}^{k,l}\left(N_i^l\right)\right),$$

$$\mathrm{\Psi }\left(t\right)=\left({\psi }_1\left(t\right),{\psi }_2\left(t\right),\dots ,{\psi }_n\left(t\right)\right),B\left(x\right)=\left(b_1\left(x\right),b_2\left(x\right),\dots ,b_n\left(x\right)\right),$$

$$exp\left(\mu \right)=\mathit{col}\left(exp\left({\mu }_1\right),exp\left({\mu }_2\right),\dots ,exp\left({\mu }_n\right)\right)$$

or in coordinate form

$$u_k\left(M\right)=\sum _{i=1}^n{v}_{k,i}\left(x,t\right){\psi }_i\left(t\right)+\sum _{l=1}^2\sum _{i=1}^n{y}_i^{k,l}\left(N_i^l\right)b_i\left(x\right)$$E33

$$+\sum _{i,j=1}^n\left\{c_{i,j}^k\left(x,t\right){\psi }_i\left(t\right)+\sum _{l=1}^2{z}_{i,j}^{k,l}\left(N_i^l\right)b_i\left(x\right)\right\}exp\left({\mu }_j\right)+\sum _{i=1}^n{p}_i^k\left(x\right){\psi }_i\left(t\right)exp\left({\mu }_i\right).$$

The vector-functions $b_i\left(x\right),{\psi }_i\left(t\right)$ included in these classes are eigenfunctions of the matrices $A\left(x\right)$ and $D\left(t\right)$, respectively

$$A\left(x\right)b_i\left(x\right)={\lambda }_i\left(x\right)b_i\left(x\right),D\left(t\right){\psi }_i\left(t\right)={\beta }_i\left(t\right){\psi }_i\left(t\right),i=\overline{1,n}.$$E34

Moreover, according to condition (1), they are smooth in their arguments.

Along with the eigenvectors $b_i\left(x\right)$ and ${\psi }_i\left(t\right)$, the eigenvectors $b_i^{\ast }\left(x\right),{\psi }_i^{\ast }\left(t\right),i=\overline{1,n}$ of the conjugate matrices $A^{\ast }\left(x\right),D^{\ast }\left(t\right)$ will be used

$$A^{\ast }\left(x\right)b_i^{\ast }\left(x\right)={\overline{\lambda }}_i\left(x\right)b_i^{\ast }\left(x\right),D^{\ast }\left(t\right){\psi }_i^{\ast }\left(t\right)={\overline{\beta }}_i\left(t\right){\psi }_i^{\ast }\left(t\right)$$

and they are selected biorthogonal

$$\left(b_i\left(x\right),b_j^{\ast }\left(x\right)\right)={\delta }_{i,j},\left({\psi }_i\left(t\right),{\psi }_j^{\ast }\left(t\right)\right)={\delta }_{i,j},i,j=\overline{1,n},$$

where ${\delta }_{i,j}$ is the Kronecker symbol.

We calculate the action of the operators $T_0,T_1,L_{\xi },L_x$ on the function $u_k\left(M,\epsilon \right)$ from (34), taking into account relations (35) and

$${\phi }_{i,l}^{\text{'}2}\left(x\right)=\frac{1}{{\lambda }_i\left(x\right)},i=\overline{1,n}$$

we have

$$T_0{u}_k\left(M\right)\equiv \sum _{i=1}^n\sum _{l=1}^2\left\{{\partial }_{\tau }{y}_i^{k,l}\left(N_i^l\right)-{\partial }_{{\xi }_{i,l}}^2{y}_i^{k,l}\left(N_i^l\right)\right.$$E35

$$+\sum _{j=1}^n\left.\left[{\partial }_{\tau }{z}_{i,j}^{k,l}\left(N_i^l\right)-{\partial }_{{\xi }_{i,l}}^2{z}_{i,j}^{k,l}\left(N_i^l\right)\right]exp\left({\mu }_j\right)\right\}{b}_i\left(x\right);$$

or vector form

$$T_0{u}_k\left(M\right)\equiv \sum _{l=1}^{2}B\left(x\right)\left\{{\partial }_{\tau }{Y}^{k,l}\left(N^l\right)-{\partial }_{{\xi }_l}^2{Y}^{k,l}\left(N^l\right)\right.\left.+\left[{\partial }_{\tau }{Z}^{k,l}\left(N^l\right)-{\partial }_{{\xi }_l}^2{Z}^{k,l}\left(N^l\right)\right]\mathit{exp}\left(\mu \right)\right\},$$

$Y^{k,l}\left(N^l\right)$ is $n$-vector, $Z^{k,l}\left(N^l\right)$ is $n\times n$-matrix. Here $B\left(x\right)$ is a matrix function $\left(n\times n\right)$ whose columns are the eigenvectors $b_i\left(x\right)$ of the matrix $A\left(x\right)$. We calculate

$$T_1{u}_k=t{\partial }_t{u}_k+\sum _{j=1}^n{\beta }_j\left(0\right){\partial }_{{\mu }_j}{u}_k-D\left(t\right)u_k$$

$$=t\sum _{i=1}^n\left\{\left[{\partial }_t{v}_{k,i}+\sum _{r=1}^n{\alpha }_{r,i}\left(t\right)v_{k,r}(x,t)\right]{\psi }_i\left(t\right)+\sum _{l=1}^2{\partial }_t{y}_i^{k,l}\left(N_i^l\right)b_i\left(x\right)\right.$$

$$+\sum _{j=1}^n\left[\left({\partial }_t{c}_{\mathit{ij}}^k\left(x,t\right)+\sum _{r=1}^n{\alpha }_{\mathit{ri}}\left(t\right)c_{r,j}^k(x,t)+{\alpha }_{\mathit{ji}}\left(t\right)p_j^k\left(x\right)\right){\psi }_i\left(t\right)\right.$$E36

$$\left.\left.+\sum _{l=1}^2{z}_{\mathit{ij}}^{k,l}\left(N_i^l\right)b_i\left(x\right)\right]exp\left({\mu }_j\right)\right\}$$

$$+\sum _{i,j=1}^n{\beta }_j\left(0\right)c_{\mathit{ij}}^k\left(x,t\right){\psi }_i\left(t\right)exp\left({\mu }_j\right)+\sum _{i=1}^n{\beta }_i\left(0\right)p_i^k\left(x\right){\psi }_i\left(t\right)exp\left({\mu }_i\right)$$

$$+\sum _{l=1}^2\sum _{i,j=1}^n{\beta }_j\left(0\right)z_{i,j}^{k,l}\left(x,t\right)b_i\left(x\right)exp\left({\mu }_j\right)$$

$$-\sum _{i=1}^n\left\{{\beta }_i\left(t\right)v_{\mathit{ki}}(x,t){\psi }_i\left(t\right)+\sum _{l=1}^2\sum _{r=1}^n{\gamma }_{r,i}(x,t)y_i^{k,l}\left(N_i^l\right)b_i\left(x\right)\right\}$$

$$-\sum _{i,j=1}^n{\beta }_j\left(t\right)c_{i,j}^k\left(x,t\right){\psi }_i\left(t\right)exp\left({\mu }_j\right)-\sum _{i=1}^n{\beta }_i\left(t\right)p_i^k\left(x\right){\psi }_i\left(t\right)exp\left({\mu }_i\right)$$

$$-\sum _{l=1}^2\sum _{i,j=1}^n\sum _{r=1}^n{\gamma }_{r,i}\left(x,t\right)z_{r,j}^{k,l}\left(N_i^l\right)b_i\left(x\right)exp\left({\mu }_j\right).$$

Here ${\alpha }_{i,r}=\left({\psi }_i^{\prime }\left(t\right),{\psi }_r^{\ast }\left(t\right)\right),{\gamma }_{i,r}\left(x,t\right)=\left(D\left(t\right)b_i\left(x\right),b_r^{\ast }\left(x\right)\right).$

It will be shown below that the scalar functions $y_i^{k,l}\left(N^l\right)$ and $z_{i,j}^{k,l}\left(N^l\right)$ are representable in the form

$$y_i^{k,l}\left(N_i^l\right)=d_i^{k,l}\left(x,t\right)y_i^{k,l}\left({\xi }_l,\tau \right),z_{i,j}^{k,l}\left(N_i^l\right)={\omega }_{i,j}^{k,l}\left(x,t\right)z_{i,j}^{k,l}\left({\xi }_l,\tau \right).$$

Given these representations, we calculate

$$L_{\xi }{u}_k\left(M\right)=\sum _{l=1}^{2}A\left(x\right)\sum _{i=1}^n\left\{\left[2{\phi }_i^{\prime }\left(x\right){\left(b_i\left(x\right)d_i^{k,l}(x,t)\right)}_x\right.\right.$$E37

$$\left.+{{\phi }^{\prime }}_i^{\prime }\left(x\right)\left(b_i\left(x\right)d_i^{k,l}(x,t)\right)\right]{\partial }_{{\xi }_{i,l}}{y}_i^{k,l}\left({\xi }_{i,l},\tau \right)$$

$$+\sum _{j=1}^n\left[2{\phi }_i^{\prime }\left(x\right){\left(b_i\left(x\right){\omega }_{i,j}^{k,l}(x,t)\right)}_x\right.\left.\left.+{{\phi }^{\prime }}_i^{\prime }\left(x\right)\left(b_i\left(x\right){\omega }_{i,j}^{k,l}(x,t)\right)\right]{\partial }_{{\xi }_{i,l}}{z}_{i,j}^{k,l}\left({\xi }_{i,l},\tau \right)exp\left({\mu }_j\right)\right\},$$

$$L_x{u}_k\left(M\right)=A\left(x\right)\left[{\partial }_x^2\left(V_k\left(x,t\right)\right)+\sum _{l=1}^2{\partial }_x^2\left(B\left(x\right)Y^{l,k}\left(N^l\right)\right)\right.$$

$$\left.+{\partial }_x^2\left(\mathrm{\Psi }\left(t\right)C^{k,0}(x,t)\right)exp\left(\mu \right)+\sum _{l=1}^2{\partial }_x^2\left(B\left(x\right)Z^{l,k}\left(N^l\right)\right)exp\left(\mu \right)\right].$$

Satisfy the function (34) of the boundary conditions from (29)

$$y_i^{k,l}\left(N_i^l\right){\left|{}_{t=\tau =\mu =0}=0,z_{i,j}^{k,l}\left(N_i^l\right)\right|}_{t=\tau =\mu =0}=0,$$E38

$$c_{\mathit{ii}}^k\left(x,0\right)=-v_{k,i}\left(x,0\right)-p_i^k\left(x\right)-\sum _{i\ne j}{c}_{\mathit{ij}}^k\left(x,0\right),$$

$$y_i^{k,l}\left(N_i^l\right){\left|{}_{{\xi }_{i,l}=0}=d_i^{k,l}(x,t),d_i^{k,l}(x,t)b_i\left(x\right)\right|}_{x=l-1}=-v_i\left(l-1,t\right){\psi }_i\left(t\right),$$

$$z_{i,j}^{k,l}\left(N_i^l\right){\left|{}_{{\xi }_{i,l}=0}={\omega }_{i,j}^{k,l}(x,t),{\omega }_{i,j}^{k,l}(x,t)b_i\left(x\right)\right|}_{x=l-1}=-\left[c_{i,j}\left(l-1,t\right)+p_i\left(l-1\right)\right]{\psi }_i\left(t\right).$$

In general form, the iterative Eqs. (32) are written as

$$T_0{u}_k\left(M\right)=h_k\left(M\right).$$E39

Theorem 1: Let $h_k\left(M\right)\in U$ and conditions (2) and (3) on hold. Then Eq. (40) has a solution $u_k\left(M\right)\in U$, if the equations are solvable

$${\partial }_{\tau }{y}_i^{k,l}\left(N_i^l\right)-{\partial }_{{\xi }_{i,l}}^2{y}_i^{k,l}\left(N_i^l\right)=h_i^{k,1}\left(N_i^l\right)\equiv {{\overline{h}}_i}^{k,1}\left(x,t\right){{\overline{\overline{h}}}_i}^{k,1}\left({\xi }_{i,1},\tau \right),$$E40

$${\partial }_{\tau }{z}_{i,j}^{k,l}\left(N_i^l\right)-{\partial }_{{\xi }_{i,l}}^2{z}_{i,j}^{k,l}\left(N_i^l\right)=h_{\mathit{ij}}^{k,2}\left(N_i^l\right)\equiv {\overline{h}}_{\mathit{ij}}^{k,2}\left(x,t\right){{\overline{\overline{h}}}_i}^{k,2}\left({\xi }_{i,2},\tau \right).$$

Proof: Let $h_k\left(M\right)={\sum }_{l=1}^2{\sum }_{i=1}^n\left[h_i^{k,1}\left(N_i^l\right)+{\sum }_{j=1}^n{h}_{\mathit{ij}}^{k,2}\left(N_i^l\right)exp\left({\mu }_j\right)\right]b_i\left(x\right)\in U$. Pose (34) into Eq. (40), then, on the basis of calculations (38), with respect $y_i^{k,l}\left(N_i^l\right),z_{\mathit{ij}}^{k,l}\left(N_i^l\right)$, we obtain Eqs. (41). These equations, under appropriate boundary value conditions

$$y_i^{k,l}\left(N_i^l\right){\left|{}_{t=\tau =\mu =0}=0,y_i^{k,l}\left(N_i^l\right)\right|}_{{\xi }_{i,l}=0}=d_i^{k,l}\left(x,t\right),$$

$$z_{i,j}^{k,l}\left(N_i^l\right){\left|{}_{t=\tau =\mu =0}=0,z_{i,j}^{k,l}\left(N_i^l\right)\right|}_{{\xi }_{i,l}=0}={\omega }_{i,j}^{k,l}\left(x,t\right).$$

Solutions are represented in the form

$$y_i^{k,l}\left(N_i^l\right)=d_i^{k,l}\left(x,t\right)\mathit{erfc}\left(\frac{{\xi }_{i,l}}{2\sqrt{\tau }}\right)+{{\overline{h}}_i}^{k,1}\left(x,t\right)I_1\left({\xi }_{i,l},\tau \right),$$

$$z_{i,j}^{k,l}\left(N_i^l\right)={\omega }_{i,j}^{k,l}\left(x,t\right)\mathit{erfc}\left(\frac{{\xi }_{i,l}}{2\sqrt{\tau }}\right)+{{\overline{h}}_{\mathit{ij}}}^{k,2}\left(x,t\right)I_2\left({\xi }_{i,l},\tau \right),$$

$$I_r\left({\xi }_{i,l},\tau \right)=\frac{1}{2\sqrt{\pi }}{\int }_0^{\tau }{\int }_0^{\infty }\frac{{\overline{\overline{h}}}_i^{k,r}\left(\eta ,s\right)}{\sqrt{\tau -s}}\left[exp\left(-\frac{{\left({\xi }_{i,l}-\eta \right)}^2}{4\left(\tau -s\right)}\right)\right.$$

$$\left.-exp\left(-\frac{{\left({\xi }_{i,l}+\eta \right)}^2}{4\left(\tau -s\right)}\right)\right]\mathit{\text{dηds}},r=1,2,$$

where ${\overline{h}}_i^{k,r}\left(x,t\right),{\overline{h}}_i^{k,r}\left(\eta ,s\right)$ are known functions. Evaluation of the integral

$$\left|I_r\left({\xi }_{i,l},\tau \right)\right|\le cexp\left(-\frac{{\xi }_{i,l}^2}{8\tau }\right).$$

Theorem 2: Let conditions (1)–(4) be satisfied, then Eq. (32) under additional conditions

1. $u_k{\left|{}_{t=\mu =\tau =0}=0,u_k\right|}_{x=l-1,{\xi }_{i,l}=0}=0,l=1,2;$

2. $-T_1{u}_k-{\partial }_t{u}_{k-2}+L_x{u}_{k-4}\in U_2\oplus U_4;$

3. $L_{\xi }{u}_k=0$,

has a unique solutionin $U$.

Proof: Satisfying the function $u_k\left(M\right)\in U$ with the boundary conditions from (26) we obtain (39). Based on calculations (36–38) we have

$$-T_1{u}_k-{\partial }_t{u}_{k-2}+L_x{u}_{k-4}=-t\sum _{i=1}^n\left\{{\partial }_t{v}_{k,i}\left(x,t\right){\psi }_i\left(t\right)\right.$$

$$+\sum _{r=1}^n{\alpha }_{\mathit{ri}}\left(t\right)v_{k,r}\left(x,t\right){\psi }_i\left(t\right)+\sum _{r=1}^n{\alpha }_{\mathit{ri}}\left(t\right)p_r^k\left(x\right){\psi }_i\left(t\right)exp\left({\mu }_r\right)$$

$$+\sum _{j=1}^n\left[{\partial }_t{c}_{\mathit{ij}}^k+\sum _{r=1}^n{\alpha }_{r,i}\left(t\right)c_{r,j}^k(x,t)\right]{\psi }_i\left(t\right)exp\left({\mu }_j\right)$$

$$\left.+\sum _{l=1}^2\left[{\partial }_t{y}_i^{k,l}\left(N^l\right)+\sum _{j=1}^n{\partial }_t{z}_{\mathit{ij}}^k\left(N^l\right)exp\left({\mu }_j\right)\right]b_i\left(x\right)\right\}$$

$$-\sum _{i,j=1}^n{\psi }_i\left(t\right)c_{i,j}^k\left(x,t\right)\left({\beta }_j\left(0\right)-{\beta }_j\left(t\right)\right)exp\left({\mu }_j\right)$$

$$-\sum _{i=1}^n{\psi }_i\left(t\right)p_i^k\left(x\right)\left({\beta }_i\left(0\right)-{\beta }_i\left(t\right)\right)exp\left({\mu }_i\right)$$

$$+\sum _{l=1}^2\sum _{i=1}^n\left\{\sum _{r=1}^n{\gamma }_{\mathit{ir}}\left(x,t\right)b_r\left(x\right)y_i^{k,l}\left(N^l\right)+\sum _{j=1}^n\sum _{r=1}^n{\gamma }_{i,r}(x,t)b_r\left(x\right)z_{i,j}^{k,l}\left(N^l\right)exp\left({\mu }_j\right)\right\}$$

$$-\sum _{i=1}^n\sum _{r=1}^n{\alpha }_{r,i}\left(t\right)p_r^{k-2}\left(x\right)exp\left({\mu }_r\right){\psi }_i\left(t\right)$$

$$-\sum _{i=1}^n\left\{{\partial }_t{v}_{k-2,i}+\sum _{r=1}^n{\alpha }_{r,i}\left(t\right)v_{k-2,r}+\sum _{j=1}^n\left[{\partial }_t{c}_{i,j}^{k-2}+\sum _{r=1}^n{\alpha }_{r,i}\left(t\right)c_{r,j}^{k-2}(x,t)\right]exp\left({\mu }_j\right)\right\}{\psi }_i\left(t\right)$$

$$-\sum _{l=1}^2\sum _{i=1}^n\left\{{\partial }_t{y}_i^{k,l}\left(N^l\right)+\sum _{j=1}^n{\partial }_t{z}_{i,j}^{k,l}\left(N^l\right)exp\left({\mu }_j\right)\right\}{b}_i\left(x\right)$$

$$+A\left(x\right)\sum _{i=1}^n\left\{{\partial }_x^2{v}_{k-4,i}+\sum _{j=1}^n{\partial }_x^2\left[c_{i,j}^{k-4}\left(x,t\right)+p_i^{k-4}\left(x\right)\right]exp\left({\mu }_j\right)\right\}{\psi }_i\left(t\right)$$

$$+A\left(x\right)\sum _{l=1}^2\sum _{i=1}^n{\partial }_x^2\left(\left[y_i^{k,l}\left(N^l\right)+\sum _{j=1}^n{z}_{i,j}^{k,l}\left(N^l\right)exp\left({\mu }_j\right)\right]b_i\left(x\right)\right).$$

Providing the condition of Theorem 1, we set

$$t\left[{\partial }_t{V}_k+A^T\left(t\right)V_k\right]=-{\partial }_t{V}_{k-2}-L_x{V}_{k-4}\left(x,t\right),$$E41

$$t\left[{\partial }_t{C}^k+A^T\left(t\right)\left(C^k\left(x,t\right)+\mathrm{\Lambda }\left(P^k\left(x\right)\right)\right)\right]+\left[C^k\left(x,t\right)+\mathrm{\Lambda }\left(P^k\left(x\right)\right)\right]\mathrm{\Lambda }\left(\beta \left(0\right)\right)$$

$$-\mathrm{\Lambda }\left(\beta \left(t\right)\right)\left[C^k\left(x,t\right)+\mathrm{\Lambda }\left(P^k\left(x\right)\right)\right]$$E42

$$=-{\partial }_t{C}^{k-2}-A^T\left(t\right)\left[C^{k-2}+\mathrm{\Lambda }\left(P^{k-2}\left(x\right)\right)\right]+L_x\left[C^{k-4}+\mathrm{\Lambda }\left(P^{k-4}\left(x\right)\right)\right],$$

Eq. (43) is solved without an initial condition and uniquely determines the function $V_k\left(x,t\right)$ [27].

Providing the solvability of Eq. (44) we set

$${\left.\overline{\overline{C^k\left(x,t\right)\mathrm{\Lambda }\left(\beta \left(0\right)\right)-\mathrm{\Lambda }\left(\beta \left(t\right)\right)C^k\left(x,t\right)}}\right|}_{t=0}$$

$${\left.=-\overline{\overline{{\partial }_t{C}^{k-2}-A^T\left(t\right)C^{k-2}\left(x,t\right)-A^T\left(t\right)\mathrm{\Lambda }\left(P^{k-2}\right)+L_x\left(C^{k-4}\right)}}\right|}_{t=0}$$

$$\overline{A^T\left(t\right)\mathrm{\Lambda }\left(P^{k-2}\left(x\right)\right)}=\overline{\left[-{\partial }_t{C}^{k-2}-A^T\left(t\right)C^{k-2}-L_x\left(C^{k-4}\left(x,t\right)+\mathrm{\Lambda }\left(P^{k-4}\left(x\right)\right)\right)\right]}$$

or in coordinate form

$${\left.c_{\mathit{ij}}^k\left(x,t\right)\right|}_{t=0}=-\frac{1}{{\beta }_j\left(0\right)-{\beta }_i\left(t\right)}{\left[{\partial }_t{c}_{\mathit{ij}}^{k-2}+\sum _{r\ne j}{\alpha }_{\mathit{ri}}\left(t\right)c_{\mathit{rj}}^{k-2}(x,t)-q_{\mathit{ij}}(x,t)\right]}_{t=0},$$

$$p_i^{k-2}\left(x\right)=-\frac{1}{{\alpha }_{\mathit{ii}}\left(t\right)}{\left[{\partial }_t{c}_{\mathit{ii}}^{k-2}+{\alpha }_{\mathit{ii}}\left(t\right)c_{\mathit{ii}}^{k-2}-q_{\mathit{ii}}(x,t)\right]}_{t=0},$$

where $q_{\mathit{ij}}\left(x,t\right)$ is known function included in $A\left(x\right){\partial }_x^2\left[C^{k-4}\left(x,t\right)+\mathrm{\Lambda }\left(P^{k-4}\left(x\right)\right)\right]$.

On the basis of (38), condition (3), Theorem 2 is ensured if arbitrary functions $d_i^{l,k}\left(x,t\right)b_i\left(x\right),$.

${\omega }_{\mathit{ij}}^{k,l}\left(x,t\right)b_i\left(x\right)$ are solutions to the problems

$$2{\phi }_{i,l}^{\prime }\left(x\right){\left(d_i^{l,k}\left(x,t\right)b_i\left(x\right)\right)}_{x^{\prime }}+{\phi }_{i,l}^{″}\left(x\right)\left(d_i^{l,k}\left(x,t\right)b_i\left(x\right)\right)=0,$$

$${\left.d_i^{l,k}\left(x,t\right)b_i\left(x\right)\right|}_{x=l-1}=-v_k\left(l-1,t\right){\psi }_i\left(t\right),i=\overline{1,n},$$

$$2{\phi }_{i,l}^{\prime }\left(x\right){\left({\omega }_{i,j}^{l,k}\left(x,t\right)b_i\left(x\right)\right)}_{x^{\prime }}+{\phi }_{i,l}^{″}\left(x\right)\left({\omega }_{i,j}^{l,k}\left(x,t\right)b_i\left(x\right)\right)=0,$$

$${\left.{\omega }_{i,j}^{l,k}\left(x,t\right)b_i\left(x\right)\right|}_{x=l-1}=-\left[c_{\mathit{ij}}^k\left(l-1,t\right)+p_i\left(l-1\right)\right]{\psi }_j\left(t\right).$$

Thus, arbitrary functions $d_i^{l,k}\left(x,t\right),{\omega }_{\mathit{ij}}^{k,l}\left(x,t\right),v_{\mathit{ki}}\left(x,t\right),c_{\mathit{ij}}^{k,l}\left(x,t\right)$ included in (34) are uniquely determined.

Iterative Eq. (32) for $k=0,1$ is homogeneous; therefore, by Theorem 1, it has a solution $u_k\left(M\right)\in U$ if the functions $y_i^{l,k}\left(N_i^l\right),z_{i,j}^{l,k}\left(N_i^l\right)$ are solutions of the equations

$${\partial }_{\tau }{y}_i^{k,l}\left(N_i^l\right)={\partial }_{{\xi }_{i,l}^2}{y}_i^{k,l}\left(N_i^l\right),{\partial }_{\tau }{z}_{i,j}^{k,l}\left(N_i^l\right)={\partial }_{{\xi }_{i,l}^2}{z}_{i,j}^{k,l}\left(N_i^l\right)$$

for boundary value conditions

$$y_i^{k,l}\left(N_i^l\right){\left|{}_{t=\tau =0}=0,y_i^{k,l}\left(N_i^l\right)\right|}_{{\xi }_{i,l}=0}=d_i^{k,l}\left(x,t\right),$$

$$z_{i,j}^{k,l}\left(N_i^l\right){\left|{}_{t=\tau =0}=0,z_{i,j}^{k,l}\left(N_i^l\right)\right|}_{{\xi }_{i,l}=0}={\omega }_{i,j}^{k,l}\left(x,t\right).$$

From this problem, we find

$$y_i^{0,l}\left(N_i^l\right)=d_i^{0,l}\left(x,t\right)\mathit{erfc}\left(\frac{{\xi }_{i,l}}{2\sqrt{\tau }}\right),z_{i,j}^{0,l}\left(N_i^l\right)={\omega }_{i,j}^{0,l}\left(x,t\right)\mathit{erfc}\left(\frac{{\xi }_{i,l}}{2\sqrt{\tau }}\right).$$

The functions $d_i^{0,l}\left(x,t\right),{\omega }_{i,j}^{0,l}\left(x,t\right)$ are determined from problems (44) which ensure that condition $L^{\xi }{u}_0=0$ is satisfied. Using calculations (37), the free term of iterative Eq. (32) at $k=2$ is written as $F_2\left(M\right)=-T_1{u}_0\left(M\right)+f\left(x,t\right)$ by Theorem 1, an equation with such a free term is solvable in $U$, if

$$t\sum _{i=1}^n\left\{\left[{\partial }_t{v}_{0,i}\left(x,t\right)+\sum _{r=1}^n{\alpha }_{\mathit{ri}}\left(x\right)v_{0,r}(x,t)\right]-{\beta }_i\left(t\right)v_{0,i}(x,t)\right\}=f\left(x,t\right)$$

$$t\sum _{i,j=1}^n\left\{\left[{\partial }_t{c}_{\mathit{ij}}^0\left(x,t\right)+\sum _{r=1}^n{\alpha }_{\mathit{ri}}\left(t\right)c_{\mathit{rj}}^0(x,t)\right]+{\alpha }_{\mathit{ji}}\left(t\right)p_j^0\left(x\right)\right\}$$

$$+\sum _{i,j=1}^n\left[{\beta }_j\left(0\right)-{\beta }_i\left(t\right)\right]c_{\mathit{ij}}^0\left(x,t\right)+\sum _{i=1}^n\left[{\beta }_i\left(0\right)-{\beta }_i\left(t\right)\right]p_i^0\left(x\right)=0.$$E43

From (47) we uniquely determine $c_{i,j}^0\left(x,t\right)=0,\forall i\ne j$ and the function $c_{i,i}^0\left(x,t\right)$ is determined from the equation

$$t\left[{\partial }_t{c}_{\mathit{ii}}^0\left(x,t\right)+{\alpha }_{\mathit{ii}}\left(t\right)c_{\mathit{ii}}^0(x,t)\right]+\left({\beta }_i\left(0\right)-{\beta }_i\left(t\right)\right)c_{\mathit{ii}}^0\left(x,t\right)+\left[{\beta }_i\left(0\right)-{\beta }_i\left(t\right)\right]p_i^0\left(x\right)=0$$

under the initial condition

$$c_{\mathit{ii}}^0\left(x,0\right)=-v_{0,i}\left(x,0\right)-p_i^0\left(x\right).$$

The first equation from (47), by virtue of condition (2), has a solution satisfying the condition [27] $‖v^0\left(x,0\right)‖<\infty .$ We calculate the free term of Eq. (32) at $k=3$:

$$F_3\left(M\right)=-T_1{u}_1,$$

which has the same view as $T_1{u}_0$. Providing the solvability of equation $T_0{u}_3=-T_1{u}_1$ in $U$, with respect to $c_{\mathit{ij}}^1\left(x,t\right),v_{1i}\left(x,t\right)$ we obtain Eqs. (47).

In the next step, $\left(k=4\right)$ the free term has the view

$$F_4\left(M\right)=-T_1{u}_2-{\partial }_t{u}_0+L_{\xi }{u}_1.$$

The functions $d_i^{1,l}\left(x,t\right),{\omega }_{i,j}^{1,l}\left(x,t\right)$ entering the $u_1\left(M\right)$ provide the condition $L_{\xi }{u}_1=0$. Providing the solvability of the iterative equation at $k=4$, we set

$$t\left[{\partial }_t{v}_{2i}\left(x,t\right)+\sum _{k=1}^n{\alpha }_{\mathit{ki}}\left(x\right)v_{2,k}(x,t)\right]-{\beta }_i\left(t\right)v_{2,i}\left(x,t\right)=-{\partial }_t{v}_{0,i}\left(x,t\right).$$

For $c_{\mathit{ij}}^2\left(x,t\right)$ we obtain the same equation of the form (47), but with the right-hand side ${\partial }_t{c}_{i,j}^0\left(x,t\right)+{\sum }_{k=1}^n{\alpha }_{k,i}\left(x\right)\left(c_{k,j}^0\left(x,t\right)+{\alpha }_{j,i}\left(x\right)p_j^0\left(x\right)\right)$. Taking off the degeneracy of this equation as $t=0$, we set $p_i^0\left(x\right)=-\frac{1}{{\alpha }_{\mathit{ii}}\left(t\right)}{\left[{\partial }_t{c}_{\mathit{ii}}^0+{\alpha }_{\mathit{ii}}\left(x\right)c_{\mathit{ii}}^0\right]}_{t=0}$. Further repeating the described process, using Theorems 1 and 2, sequentially determining $u_k\left(M\right),k=0,1,\dots ,n$, we construct a partial sum

$$u_{\mathit{\epsilon n}}\left(M\right)=\sum _{k=0}^n{\epsilon }^{k/2}{u}_k\left(M\right).$$E44

2.4 Estimate for the remainder

For the remainder term

$$R_{\mathit{\epsilon n}}\left(M\right)=u\left(M,\epsilon \right)-u_{\mathit{\epsilon n}}\left(M\right)=u\left(M,\epsilon \right)-\sum _{k=0}^{n+2}{\epsilon }^{k/2}{u}_k\left(M\right)+\sum _{l=1}^2{\epsilon }^{\frac{n+l}{2}}{u}_{n+l}$$

we get the problem

$${\stackrel{\sim }{L}}_{\epsilon }{R}_{\mathit{\epsilon n}}\left(M\right)={\epsilon }^{\frac{n+1}{2}}{g}_{\mathit{\epsilon n}}\left(M\right),$$

$$R_{\mathit{\epsilon n}}\left(M\right){\left|{}_{t=\tau =\mu =0}=R_{\mathit{\epsilon n}}\right|}_{x=l-1,\xi =0}=0,l=1,2,$$

where

$$g_{\mathit{n\epsilon }}\left(M\right)=-\sum _{l=0}^1\left(T_1{u}_{n+1+l}\right){\epsilon }^{l/2}-\sum _{l=0}^3{\epsilon }^{l/2}{\partial }_t{u}_{n-1+l}+\sum _{l=0}^5{\epsilon }^{l/2}{L}_x{u}_{n-3+l}-\sum _{l=0}^1{\epsilon }^{l/2}{\stackrel{\sim }{L}}_{\epsilon }{u}_{n+1+l}.$$

From the form (34) of the function $U$, based on conditions (1)–(3) and the form of regularizing variables from (27), we conclude that

$$‖g_{\mathit{n\epsilon }}\left(M\right)‖<C.$$

In the equation for R, we make the restriction by means of regularizing functions, then, based on (30), we obtain the problem

$$L_{\epsilon }{R}_{\mathit{\epsilon n}}\left(x,t,\epsilon \right)={\epsilon }^{\left(n+1\right)/2}{g}_{\mathit{n\epsilon }}\left(x,t,\epsilon \right),$$

$${\left.R_{\mathit{\epsilon n}}\right|}_{t=0}=R_{\mathit{\epsilon n}}{\left|{}_{x=0}=R_{\mathit{\epsilon n}}\right|}_{x=1}=0.$$

We divide both sides of this equation by $\left(t+\epsilon \right)$, while the properties of the matrices A are preserved. Therefore, using Theorem 5.5 of [33], we obtain the estimate

$$‖R_{\mathit{\epsilon n}}\left(x,t,\epsilon \right)‖<c{\epsilon }^{\left(n+1\right)/2}.$$

Passing to Euclidean norms, like [28], or the same estimate can be obtained using the maximum principle [33], p. 23.

By narrowing this problem by means of regularizing functions. Following [33] and [28], passing to Euclidean norms, we obtain a problem that is limited by the maximum principle

$$‖R_{\mathit{\epsilon n}}\left(x,t,\epsilon \right)‖<c{\epsilon }^{\frac{n+1}{2}}.$$E45

Theorem 3: Let conditions (1)–(4) be satisfied. Then the restriction of the constructed solution (44) is an asymptotic solution to the problem (26), that is, $\forall n=0,1,\dots$ the estimate (45) holds at $\epsilon \to 0$.

Proof: Let us rewrite the problem (31)

$${\partial }_{t}u-{\epsilon }^2\frac{1}{t+\epsilon }A\left(x\right){\partial }_x^{2}u-\frac{1}{t+\epsilon }D\left(t\right)u=\frac{1}{t+\epsilon }f\left(x,t\right)$$

here, the expression $\left(t+\epsilon \right)$ does not affect sufficiently small $\epsilon$ the properties of the matrices $A\left(x\right),D\left(t\right)$ for which the conditions of the maximum principle theorem are valid [28], p. 20]. Therefore, on the basis of this theorem, it is easy to establish an estimate (45).