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

*Asan Omuraliev and Peil Esengul Kyzy*

#### **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.

**Keywords:** singularly perturbed problems, power boundary layer, parabolic boundary layer, angular boundary-layer functions, regularized asymptotics

#### **1. Introduction**

#### **1.1 A singularly perturbed system of parabolic equations**

This article studies the problem

$$L\_t u(\mathbf{x}, t, \varepsilon) \equiv (\varepsilon + t)\partial\_t u - \varepsilon^2 a(\mathbf{x}) \partial\_\mathbf{x}^2 u - B(t)u = f(\mathbf{x}, t), \ (\mathbf{x}, t) \in \Omega,$$

$$u|\_{t=0} = u|\_{\mathbf{x}=0} = u|\_{\mathbf{x}=1} = \mathbf{0},\tag{1}$$

where *ε*> 0 is a small parameter, Ω ¼ f g ð Þ *x*, *t* : *x*∈ð Þ 0, 1 ,*t*∈ ð � 0,*T* , *u* ¼ ð Þ *u*1, *u*2, … , *un* . Suppose the following conditions:

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

The eigenvalues *λi*ð Þ*t* of the matrix *B t*ð Þ satisfy the conditions: *B t*ð Þ*ψi*ðÞ¼ *t λi*ð Þ*t ψi*ð Þ*t* ,*i* ¼ 1,2, … ,*n*,ℜ*λi*ð Þ*t* < 0,*λj*ð Þ*t* 6¼ *λi*ð Þ*t* ,∀*i* 6¼ *j*,*t* ∈½ � 0, *T* (ℜ is the real part of the complex number).

Singularly perturbed parabolic problems in various statements are devoted to Ref. [1–23]. In Ref. [6, 9–17], 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, 18–23] 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. [8–10, 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

$$\xi\_l = \frac{\varrho \rho\_l(\mathbf{x})}{\sqrt{\varepsilon^3}}, \quad \varrho \rho\_l(\mathbf{x}) = (-1)^{l-1} \int\_{l-1}^{\mathbf{x}} \frac{ds}{\sqrt{a(s)}}, \quad \eta\_i = \lambda\_i(\mathbf{0}) \ln\left(1 + t/\varepsilon\right), \quad \tau = \varepsilon^{-1} \ln\left(1 + t/\varepsilon\right) \tag{2}$$

along with the independent variables ð Þ *x*, *t* of the function *u* ~ ð Þ *M*, *ε* , *M* ¼ ð Þ *x*, *t*, *ξ*, *τ*, *η* , *ξ* ¼ *ξ*1, *ξ*<sup>2</sup> ð Þ, *η* ¼ *η*1, *η*2, … , *η<sup>n</sup>* ð Þ such that

$$
\left.\tilde{\mathfrak{u}}(\mathsf{M},\,\,\varepsilon)\right|\_{\zeta=\eta(\mathsf{x},\,\,t,\,\,\varepsilon)} \equiv \mathfrak{u}(\mathsf{x},t,\,\,\varepsilon),
$$

$$\begin{aligned} \zeta\_{\varepsilon} &= (\xi, \tau, \eta), \chi(\mathbf{x}, t, \varepsilon) \\ &= \left( \frac{\varrho\_1(\mathbf{x})}{\sqrt{\varepsilon^3}}, \frac{\varrho\_2(\mathbf{x})}{\sqrt{\varepsilon^3}}, \lambda\_1(\mathbf{0}) \ln\left(1 + t/\varepsilon\right), \dots, \lambda\_n(\mathbf{0}) \ln\left(1 + t/\varepsilon\right), \varepsilon^{-1} \ln\left(1 + t/\varepsilon\right) \right). \end{aligned}$$

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

$$\begin{split} \partial\_t \boldsymbol{u} &\equiv \left( \partial\_t \boldsymbol{\tilde{u}} + \sum\_{i=1}^n \frac{\lambda\_i(0)}{t+\varepsilon} \partial\_{\boldsymbol{u}} \boldsymbol{\tilde{u}} + \frac{1}{\varepsilon} \frac{1}{t+\varepsilon} \partial\_{\boldsymbol{t}} \boldsymbol{\tilde{u}} \right)\_{\boldsymbol{\xi}=\boldsymbol{\gamma}(\boldsymbol{x},\ t\_{\cdot},\epsilon)}, \partial\_{\boldsymbol{t}} \boldsymbol{u} \equiv \left( \partial\_{\boldsymbol{x}} \boldsymbol{\tilde{u}} + \sum\_{l=1}^2 \frac{\varrho\_l'(\boldsymbol{x})}{\sqrt{\varepsilon^3}} \partial\_{\boldsymbol{\xi}} \boldsymbol{\tilde{u}} \right)\_{\boldsymbol{\xi}=\boldsymbol{\gamma}(\boldsymbol{x},\ t\_{\cdot},\epsilon)}, \\ \partial\_{\boldsymbol{x}}^2 \boldsymbol{u} &\equiv \left( \partial\_{\boldsymbol{x}}^2 \boldsymbol{\tilde{u}} + \sum\_{l=1}^2 \left[ \left( \frac{\varrho\_l'(\boldsymbol{x})}{\sqrt{\varepsilon^3}} \right)^2 \partial\_{\boldsymbol{\xi}}^2 \boldsymbol{\tilde{u}} + \frac{1}{\sqrt{\varepsilon^3}} D\_{\boldsymbol{\xi}\boldsymbol{\tilde{u}}} \boldsymbol{\tilde{u}} \boldsymbol{\tilde{u}} \right] \right)\_{\boldsymbol{\xi}=\boldsymbol{\gamma}(\boldsymbol{x},\ t\_{\cdot},\epsilon)}, \quad D\_{\boldsymbol{\xi},\boldsymbol{\tilde{t}}} \equiv 2\varrho\_l'(\boldsymbol{x}) \partial\_{\boldsymbol{\xi}\boldsymbol{x}}^2 + \varrho\_l''(\boldsymbol{x}) \partial\_{\boldsymbol{\xi}\boldsymbol{\tilde{u}}}, \end{split}$$

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

$$L\_{\xi} \equiv e^{-1} T\_0 \ddot{u} + T\_1 \ddot{u} - \sqrt{\epsilon} L\_{\xi} \ddot{u} + \epsilon b\_l \ddot{u} - e^2 L\_{\ge} \ddot{u} = f(\varkappa, t), \ \mathcal{M} \in \mathcal{Q},$$

$$\left. \ddot{u} \right|\_{t=\tau=\eta\_1=0} = \left. \ddot{u} \right|\_{x=0, \xi\_1=0} = \left. \ddot{u} \right|\_{x=1, \xi\_2=0} = \mathbf{0}, \ \mathcal{Q} = \boldsymbol{\Omega} \times \left( \mathbf{0}, \ \boldsymbol{\infty} \right)^3. \tag{3}$$

*A System of Singularly Perturbed Parabolic Equations with a Power Boundary Layer DOI: http://dx.doi.org/10.5772/intechopen.106239*

The notation is entered here

$$T\_0 \equiv \partial\_{\mathbf{r}} - \Delta\_{\xi},\ T\_1 \equiv \sum\_{i=1}^n \lambda\_i(\mathbf{0})\partial\_{\eta\_i} + t\partial\_t - B(t),\ \Delta\_{\xi} \equiv \sum\_{l=1}^2 \partial\_{\xi\_l}^2 L\_{\xi} \equiv \sum\_{l=1}^2 a(\mathbf{x})D\_{\xi,l},\ L\_{\mathbf{x}} \equiv a(\mathbf{x})\partial\_{\mathbf{x}}^2.$$

Note that the identity holds

$$\left(\tilde{L}\_{\varepsilon}\tilde{u}\right)\_{\zeta=\gamma(\mathbf{x},\ t,\ \varepsilon)} \equiv L\_{\varepsilon}u,\tag{4}$$

solutions of problem (3) will be defined as

$$
\tilde{u}(M,\varepsilon) = \sum\_{k=0}^{\infty} \varepsilon^k u\_k(M),
\tag{5}
$$

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

$$T\_0 u\_\nu = 0, \ \nu = 0, 1, \ T\_0 u\_2 = f(\mathbf{x}, t) - T\_1 u\_0, \ T\_0 u\_k = -T\_1 u\_{k-2} + L\_\xi u\_{k-3} - \partial\_l u\_{k-4} + L\_\mathbf{x} u\_{k-6},$$

$$u\_k|\_{t=\tau=\eta\_i=0} = 0, \ \left. u\_k \right|\_{\mathbf{x}=0, \xi\_i=0} = u\_k \Big|\_{\mathbf{x}=1, \xi\_i=0} = 0, \ k \ge 3. \tag{6}$$

#### **1.3 Solvability of iterative problems**

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

$$U\_1 = \{u\_1(N\_1) : u\_1(N\_1) = < \nu(\mathbf{x}, t) + [c(\mathbf{x}, t) + \Lambda(P(\mathbf{x}))]e^{\eta}, \nu(t) > \delta\},$$

$$\nu(\mathbf{x}, t) \in C^\infty(\overline{\Omega}, C^n), P(\mathbf{x}) \in C^\infty([\overline{\Omega}, \mathbf{1}], C^n), c(\mathbf{x}, t) \in C^\infty(\overline{\Omega}, C^{n^2}), \nu(t) \in C^\infty([\overline{\Omega}, \mathbf{1}], C^n)\},$$

$$\bigcup\_{\mathbf{x} \in \mathbb{R}^n} \left\{ \bigcup\_{\mathbf{x} \in \mathbb{R}^n, \mathbf{1} \le \mathbf{1} \le \mathbf{1}} \sum\_{\mathbf{x} \in \mathbb{R}^n} \mathbf{x}(\mathbf{x}) \right\}, \quad (\mathbf{x}^\dagger \in \mathbb{R}^n, \mathbf{1}^\dagger) \in C^\infty(\mathbf{x}, t) \subset C^\infty(\overline{\Omega}, \mathbf{1}), \quad t \in [\overline{\Omega}, \mathbf{1}],$$

$$U\_2 = \left\{ u\_2(N^l) : u\_2(N^l) = \sum\_{l=1} < Y(N^l) + z(N^l)e^{\eta}, \psi(t) > , \left| \left| Y(N^l) \right| \right| < ce^{\frac{\gamma}{4\kappa}}, \right. $$

$$\left| \left| z\left(N^l\right) \right| \right| < ce^{-\frac{\gamma^2}{4\kappa}} \right\},$$

$$Y(\mathbf{N}^l) = \text{col}\left(Y\_1(\mathbf{N}^l), Y\_2(\mathbf{N}^l), \dots, Y\_n(\mathbf{N}^l)\right), \ z\left(\mathbf{N}^l\right) = \left(z\_{\vec{\imath}\boldsymbol{\jmath}}(\mathbf{N}^l)\right), \ i\_{\vec{\jmath}} = \overline{1,n}, \quad \mathbf{N}^l = \text{col}\left(Y\_1(\mathbf{N}^l), Y\_2(\mathbf{N}^l), \dots, Y\_n(\mathbf{N}^l)\right)$$

$$N^l = \left(\mathbf{x}, t, \boldsymbol{\xi}^l, \tau\right), \quad \eta = \operatorname{col}(\eta\_1, \eta\_2, \dots, \eta\_n), \ \Lambda(\mathbf{P}(\mathbf{x})) = \operatorname{diag}(P\_1(\mathbf{x}), P\_2(\mathbf{x}), \dots, P\_n(\mathbf{x})),$$

$$< [\boldsymbol{\varepsilon}(\mathbf{x}, t) + \Lambda(\mathbf{P}(\mathbf{x}))] \boldsymbol{\varepsilon}^{\eta}, \boldsymbol{\psi}(t) > \ = \sum\_{i,j=1}^n \boldsymbol{c}\_{i\overline{\boldsymbol{\varepsilon}}}(\mathbf{x}, t) \boldsymbol{\varepsilon}^{\eta} \boldsymbol{\psi}\_i(t) + \sum\_{i=1}^n P\_i(\mathbf{x}) \boldsymbol{\varepsilon}^{\eta\_i} \boldsymbol{\psi}\_i(t).$$

$$< \boldsymbol{Y}(\boldsymbol{N}^l) + \boldsymbol{z}(\boldsymbol{N}^l) \boldsymbol{\varepsilon}^{\eta}, \boldsymbol{\psi}(t) > \ = \sum\_{i=1}^n \left[ \boldsymbol{Y}\_i(\boldsymbol{N}^l) + \sum\_{j=1}^n \boldsymbol{z}\_{i\overline{\boldsymbol{\varepsilon}}}(\mathbf{N}^l) \boldsymbol{\varepsilon}^{\eta} \right] \boldsymbol{\psi}\_i(t).$$

Here *col η*1, *η*2, … , *η<sup>n</sup>* ð Þ is the vector, *diag P*ð Þ <sup>1</sup>ð Þ *x* , *P*2ð Þ *x* , … , *Pn*ð Þ *x* is a diagonal matrix. Element *uk*ð Þ *M* ∈ *U* ¼ *U*<sup>1</sup> ⊕ *U*<sup>2</sup> has an idea

*Boundary Layer Flows - Modelling, Computation, and Applications of Laminar,Turbulent…*

$$\begin{split} \mu\_{k}(\mathbf{M}) &= \, \, \mathrm{ev}\_{k}(\mathbf{x}, t) + \left[ \boldsymbol{c}^{k}(\mathbf{x}, \, t) + \Lambda \left( \boldsymbol{P}^{k}(\mathbf{x}) \right) \right] \boldsymbol{e}^{\eta}, \boldsymbol{\mu}(t) > \\ &+ \sum\_{l=1}^{2} \, \mathrm{c} \, \mathrm{Y}^{k} \left( \mathrm{N}^{l} \right) + \mathrm{z}^{k} \left( \mathrm{N}^{l} \right) \boldsymbol{e}^{\eta}, \boldsymbol{\mu}(t) > . \end{split} \tag{7}$$

We calculate the action of the operators included in the extended equation on the function *uk*ð Þ *M* ∈ *M*, for which we first decompose *ψ<sup>i</sup>* 0ðÞ¼ *<sup>t</sup>* <sup>P</sup>*<sup>n</sup> <sup>j</sup>*¼1*αij*ð Þ*<sup>t</sup> <sup>ψ</sup><sup>j</sup>* ð Þ*t* , or we write by entering the notation

*<sup>T</sup>*0*uk*ð Þ¼ *<sup>M</sup>* <sup>X</sup> 2 *l*¼1 <sup>&</sup>lt;*∂τY<sup>k</sup> <sup>N</sup><sup>l</sup>* � � � *<sup>∂</sup>*<sup>2</sup> *ξl <sup>Y</sup><sup>k</sup> <sup>N</sup><sup>l</sup>* � � <sup>þ</sup> *<sup>∂</sup>τzk <sup>N</sup><sup>l</sup>* � � � *<sup>∂</sup>*<sup>2</sup> *ξl <sup>z</sup><sup>k</sup> <sup>N</sup><sup>l</sup>* � � h i*<sup>e</sup> η* ,*ψ*ð Þ*t* >, *<sup>T</sup>*1*uk*ð Þ¼ *<sup>M</sup>* <sup>&</sup>lt;*t∂tvk*ð Þþ *<sup>x</sup>*, *<sup>t</sup> tA<sup>T</sup>*ð Þ*<sup>t</sup> vk*ð Þ� *<sup>x</sup>*, *<sup>t</sup> B t*ð Þ*vk*ð Þþ *<sup>x</sup>*, *<sup>t</sup> <sup>t</sup>∂tc <sup>k</sup>*ð Þ *<sup>x</sup>*, *<sup>t</sup>* (8) <sup>þ</sup>*tAT*ð Þ*<sup>t</sup> <sup>c</sup> k* ð Þ� *x*, *t B t*ð Þ*c k* ð Þþ *x*, *t c k* ð Þ *<sup>x</sup>*, *<sup>t</sup>* <sup>Λ</sup>ð Þ� <sup>0</sup> *B t*ð Þ<sup>Λ</sup> *Pk* ð Þ *<sup>x</sup>* � � <sup>þ</sup> *tAT*ð Þ*<sup>t</sup>* <sup>Λ</sup> *<sup>P</sup><sup>k</sup>* ð Þ *<sup>x</sup>* � � <sup>þ</sup><sup>Λ</sup> *<sup>P</sup><sup>k</sup>* ð Þ *<sup>x</sup>* � �Λð Þ <sup>0</sup> ,*ψ*ð Þ*<sup>t</sup>* <sup>&</sup>gt; <sup>þ</sup><sup>X</sup> 2 *l*¼1 <sup>&</sup>lt; *<sup>t</sup>∂tY<sup>k</sup> <sup>N</sup><sup>l</sup>* � � <sup>þ</sup> *tA<sup>T</sup>*ð Þ*<sup>t</sup> <sup>Y</sup><sup>k</sup> <sup>N</sup><sup>l</sup>* � � � *B t*ð Þ*Y<sup>k</sup> <sup>N</sup><sup>l</sup>* � � � <sup>þ</sup>*t∂tzk <sup>N</sup><sup>l</sup>* � � <sup>þ</sup> *tA<sup>T</sup>*ð Þ*<sup>t</sup> <sup>z</sup><sup>k</sup> <sup>N</sup><sup>l</sup>* � � � *B t*ð Þ*z<sup>k</sup> <sup>N</sup><sup>l</sup>* � � <sup>þ</sup> *<sup>z</sup><sup>k</sup> <sup>N</sup><sup>l</sup>* � �Λð Þ� <sup>0</sup> *<sup>e</sup> η* ,*ψ*ð Þ*t* > ¼ < *D*1*vk*ð Þþ *x*, *t D*<sup>2</sup> *c k* ð Þþ *<sup>x</sup>*, *<sup>t</sup>* <sup>Λ</sup> *Pk* ð Þ *<sup>x</sup>* � � � � *<sup>e</sup> <sup>η</sup>* <sup>þ</sup><sup>X</sup> 2 *l*¼1 <sup>&</sup>lt; *<sup>D</sup>*1*Y<sup>k</sup> <sup>N</sup><sup>l</sup>* � � <sup>þ</sup> *<sup>D</sup>*2*z<sup>k</sup> <sup>N</sup><sup>l</sup>* � �*<sup>e</sup> <sup>η</sup>* � �,*ψ*ð Þ*<sup>t</sup>* <sup>&</sup>gt; , *<sup>D</sup>*<sup>1</sup> � *<sup>t</sup>∂<sup>t</sup>* <sup>þ</sup> *tA<sup>T</sup>*ðÞ�*<sup>t</sup> B t*ð Þ, *<sup>D</sup>*<sup>2</sup> � *<sup>D</sup>*<sup>1</sup> <sup>þ</sup> *<sup>D</sup>λ*, *<sup>D</sup><sup>λ</sup>* <sup>¼</sup> <sup>X</sup>*<sup>n</sup> i*¼1 *<sup>λ</sup>i*ð Þ <sup>0</sup> *<sup>∂</sup>η<sup>i</sup>* , Λð Þ¼ 0 *diag*ð Þ *λ*1ð Þ 0 , *λ*2ð Þ 0 , … , *λn*ð Þ 0 , *A t*ðÞ¼ *<sup>α</sup>ij* � �, *<sup>α</sup>ij* <sup>¼</sup> *<sup>ψ</sup>*<sup>0</sup> *i* ð Þ*t* , *ψ<sup>j</sup>* ð Þ*t* � �; *<sup>L</sup>ξuk* <sup>¼</sup> <sup>&</sup>lt;*a x*ð Þ<sup>X</sup> 2 *l*¼1 n 2*φ*<sup>0</sup> *l* ð Þ *<sup>x</sup> <sup>∂</sup>*<sup>2</sup> *<sup>x</sup>ξ<sup>l</sup>* þ *φ*<sup>00</sup> *<sup>l</sup>* ð Þ *<sup>x</sup> <sup>∂</sup>ξ<sup>l</sup>* h i*Y<sup>k</sup> <sup>N</sup><sup>l</sup>* � � þ 2*φ*<sup>0</sup> *l* ð Þ *<sup>x</sup> <sup>∂</sup>*<sup>2</sup> *<sup>x</sup>ξ<sup>l</sup>* þ *φ*<sup>00</sup> *<sup>l</sup>* ð Þ *<sup>x</sup> <sup>∂</sup>ξ<sup>l</sup>* h i*z<sup>k</sup> <sup>N</sup><sup>l</sup>* � �*<sup>e</sup> η* ,*ψ*ð Þ*t* o > *Lxuk*ð Þ¼ *<sup>M</sup>* <sup>&</sup>lt;*Lxvk*ð Þþ *<sup>x</sup>*, *<sup>t</sup>* <sup>X</sup> 2 *l*¼1 *LxY<sup>k</sup> N<sup>l</sup>* � � þ *Lxc k* ð Þþ *<sup>x</sup>*, *<sup>t</sup> Lx*<sup>Λ</sup> *<sup>P</sup><sup>k</sup>* ð Þ *<sup>x</sup>* � � <sup>þ</sup><sup>X</sup> 2 *l*¼1 *Lxz<sup>k</sup> <sup>N</sup><sup>l</sup>* � � " #*<sup>e</sup> η* ,*ψ*ð Þ*t* >*:*

Iterative equations are written in the form.

$$T\_0 u\_k(\mathcal{M}) = h^k(\mathcal{M}).\tag{9}$$

**Theorem 1:** Let conditions (1), (2), and *h<sup>k</sup>* ð Þ *M* ∈ *U*<sup>2</sup> hold. Then Eq. (9) is solvable in *U*.

**Proof:** Let *h<sup>k</sup>* ð Þ *M* ∈ *U*2:

$$h^k(\mathcal{M}) = \sum\_{l=1}^2 \left\{ h^{k,1}(N^l) + h^{k,2}(N^l)e^{\eta} \right\}, \mathcal{y}(t) > \text{.} \tag{10}$$

*A System of Singularly Perturbed Parabolic Equations with a Power Boundary Layer DOI: http://dx.doi.org/10.5772/intechopen.106239*

Function (7) will be the solution of Eq. (9) in *U*. If the functions *Y<sup>k</sup> N<sup>l</sup>* � �,*zk N<sup>l</sup>* � � are the solution of the equations

$$
\partial\_{\mathbf{r}} Y^k(\mathbf{N}^l) - \partial\_{\xi\_l}^2 Y^k(\mathbf{N}^l) = h^{k,1}(\mathbf{N}^l), \quad \partial\_{\mathbf{r}} z^k(\mathbf{N}^l) - \partial\_{\xi\_l}^2 z^k(\mathbf{N}^l) = h^{k,2}(\mathbf{N}^l). \tag{11}
$$

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]:

$$\left| \left| Y^k \left( N^l \right) \right| \right| < c e^{-\frac{\xi\_l^2}{8r}}, \ \left| \left| z^k \left( N^l \right) \right| \right| < c e^{-\frac{\xi\_l^2}{8r}}.$$

**Theorem 2:** Let conditions (1), (2), and *<sup>h</sup><sup>k</sup>*,3ð Þ¼ *<sup>x</sup>*, 0 0 hold. Then problem

$$ = \  \tag{12}$$

$$\overline{c^k(\mathbf{x},0)} = -\Lambda(v\_k(\mathbf{x},\ 0)) - \Lambda\left(P^k(\mathbf{x})\right) - \Lambda\left(\overline{c^k(\mathbf{x},\ 0)} \cdot \mathbf{1}\right), \ \mathbf{1} = col(\mathbf{1},\ \mathbf{1},\ \dots,\ \mathbf{1}), \quad \text{(13)}$$

$$\left(c^k\_{\vec{\boldsymbol{\mu}}}(\mathbf{x},\ \mathbf{t})\Big|\_{t=0} = -v\_{k,i}(\mathbf{x},\ \mathbf{0}) - P^k\_i(\mathbf{x}) - \sum\_{j=1(i\neq j)}^n c^k\_{\vec{\boldsymbol{\mu}}}(\mathbf{x},\ \mathbf{0})\right),$$

has a unique solution. Hereinafter, *c* denotes a matrix with nonzero diagonal, and *c* with zero diagonal elements, that is, *c* ¼ *c* þ *c*.

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

$$\begin{split} \sum\_{i,j=1}^{n} \Big[ \boldsymbol{t} \Big[ \boldsymbol{\partial} \boldsymbol{c}\_{ij}^{k}(\mathbf{x}, \ t) + \sum\_{\mu=1}^{n} \boldsymbol{a}\_{\mu,i}(\boldsymbol{t}) \Big( \boldsymbol{c}\_{\mu,i}^{k}(\mathbf{x}, \ t) + \boldsymbol{P}\_{\mu}^{k}(\mathbf{x}) \Big) \Big] + \Big( \boldsymbol{\lambda}\_{j}(\mathbf{0}) - \boldsymbol{\lambda}\_{i}(\mathbf{t}) \Big) \boldsymbol{c}\_{ij}^{k}(\mathbf{x}, \ t) \Big] \boldsymbol{\nu}\_{i}(\mathbf{t}) \Big] \Big] \\ + \sum\_{i=1}^{n} (\boldsymbol{\lambda}\_{i}(\mathbf{0}) - \boldsymbol{\lambda}\_{i}(\mathbf{t})) \boldsymbol{P}\_{i}^{k}(\mathbf{x}) \boldsymbol{\varepsilon}^{\eta\_{i}} \boldsymbol{\nu}\_{i}(\mathbf{t}) = \sum\_{i,j=1}^{n} \boldsymbol{h}\_{ij}^{k,3}(\mathbf{x}, \ t) \boldsymbol{\varepsilon}^{\eta\_{i}} \boldsymbol{\nu}\_{i}(\mathbf{t}). \end{split}$$

Removing the degeneracy of this equation, assuming that

$$\left. \left( \dot{\lambda}\_j(\mathbf{0}) - \dot{\lambda}\_i(t) \right) c\_{ij}^k(\mathbf{x}, t) \right|\_{t=0} = h\_{ij}^{k, 3}(\mathbf{x}, \mathbf{0}), \ \forall i \neq j. \tag{14}$$

Equating the coefficients *ψ*, we get

$$t\left[\partial\_t c\_{ii}^k(\mathbf{x},\ t) + a\_{ii}(t)c\_{ii}^k(\mathbf{x},t)\right] = (\lambda\_i(t) - \lambda\_i(0))P\_i^k(\mathbf{x}) + h\_{ii}^{k,3}(\mathbf{x},t) - t\sum\_{\mu=1}^n a\_{\mu i}(t)P\_\mu^k(\mathbf{x}),\ i = \overline{1,n},\tag{15}$$

$$t\left[\partial\_t c\_{\vec{\eta}}^k(\mathbf{x},\ t) + \sum\_{\mu=1(i\neq\mu)}^n a\_{\mu\dot{\imath}}(\mathbf{t}) c\_{\mu\dot{\jmath}}^k(\mathbf{x},t)\right] + \left(\lambda\_{\dot{\jmath}}(\mathbf{0}) - \lambda\_{\dot{\imath}}(\mathbf{t})\right) c\_{\vec{\eta}}^k(\mathbf{x},t) = h\_{\vec{\eta}}^k(\mathbf{x},t),\ i \neq j,\ i,j = \overline{1,n}.\tag{16}$$

Eq. (15), by virtue of condition *h<sup>k</sup> ii*ð Þ¼ *x*, 0 0, under the initial condition (13), and Eq. (16) with condition (14) have one-to-one solutions.

**Remarks.** *In iterative problems, the condition h<sup>k</sup> ii*ð Þ¼ *x*, 0 0 *will be provided by the choice of the function P<sup>k</sup> <sup>i</sup>* ð Þ *x .*

**Theorem 3**: Let conditions (1) and (2) be fulfilled. Then Eq. (9) has a unique solution satisfying the conditions: (a) *uk*ð Þ *<sup>M</sup> <sup>t</sup>*¼*τ*¼*η*¼<sup>0</sup> <sup>¼</sup> 0, *uk*ð Þ *<sup>M</sup>* � � � � *<sup>x</sup>*¼*l*�1,*ξl*¼<sup>0</sup> <sup>¼</sup> 0; (b)

*<sup>T</sup>*1*uk*ð Þþ *<sup>M</sup> hk* ð Þ *M* ∈ *U*2; (c) *Lξuk*ð Þ¼ *M* 0.

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

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

$$<\boldsymbol{v}\_{k}(\boldsymbol{l}-\mathbf{1},\ \boldsymbol{t}) + \left[\boldsymbol{c}^{k}(\boldsymbol{l}-\mathbf{1},\boldsymbol{t}) + \Lambda\left(\boldsymbol{P}^{k}(\mathbf{x})\right)\right] \boldsymbol{e}^{\eta} + \sum\_{l=1}^{2} \left[\boldsymbol{Y}^{k}(\mathbf{N}^{l})\right]\_{\boldsymbol{x}=\boldsymbol{l}-1,\boldsymbol{t};=\boldsymbol{0}}$$

$$+ \boldsymbol{z}^{k}(\mathbf{N}^{l})|\_{\boldsymbol{x}=\boldsymbol{l}-1,\boldsymbol{t};=\boldsymbol{0}}\boldsymbol{e}^{\eta}|\boldsymbol{y}(\boldsymbol{t})> = \mathbf{0},$$

Hence, we define

$$\overline{c^{k}(\mathbf{x},\mathbf{0})} = -\Lambda(\boldsymbol{\nu}\_{k}(\mathbf{x},\ \mathbf{0})) - \Lambda(\boldsymbol{P}^{k}(\mathbf{x})) - \Lambda\left(\overline{c^{k}(\mathbf{x},\ \mathbf{0})}\mathbf{1}\right),$$

$$\begin{split} \left. Y^{k}(\mathbf{N}^{l}) \right|\_{t=\tau=0} &= \mathbf{0}, \ \boldsymbol{z}^{k}(\mathbf{N}^{l})\Big|\_{t=\tau=0} = \mathbf{0},\\ \left. z^{k}(\mathbf{N}^{l})\right|\_{\dot{\boldsymbol{\xi}}\_{l}=0} = \boldsymbol{W}^{k,l}(\mathbf{x},t), \boldsymbol{W}^{k,l}(\mathbf{x},t)|\_{\mathbf{x}=l-1} = \boldsymbol{c}^{k}(l-1,t) - \Lambda\left(\boldsymbol{P}^{k}(\mathbf{x})\right),\\ \left. Y^{k}(\mathbf{N}^{l})\right|\_{\dot{\boldsymbol{\xi}}\_{l}=0} = \boldsymbol{d}^{k,l}(\mathbf{x},t), \end{split} \tag{17}$$

$$\boldsymbol{d}^{k,l}(\mathbf{x},t)\Big|\_{\mathbf{x}=l-1} = -\nu\_{k}(l-1,t).$$

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

$$F\_k(\mathcal{M}) = T\_1 \mathfrak{u}\_k(\mathcal{M}) + h^k(\mathcal{M}) \in U\_{2,k}$$

based on the calculations (8) and the representation of

$$h^k(M) =  0$$

assuming that

$$ = -,\tag{18}$$

$$ = \mathbf{0}.\tag{19}$$

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

$$Y\_i^k(N^l) = d\_i^{k,l}(\infty, t) \text{erfc}\left[\xi\_l/\left(2\tau^{1/2}\right)\right] + h\_i^{k,3}(\infty, t) I\_1(\xi\_l, \tau),$$

*A System of Singularly Perturbed Parabolic Equations with a Power Boundary Layer DOI: http://dx.doi.org/10.5772/intechopen.106239*

$$z\_{ij}^k(N^l) = \mathcal{W}\_{ij}^{k,l}(\infty, t) \text{erfc}\left[\xi\_l/\left(2\pi^{1/2}\right)\right] + h\_{ij}^{k,4}(\infty, t) I\_2(\xi\_l, \tau),$$

where *erfc x*ð Þ¼ <sup>2</sup>*π*�1*=*<sup>2</sup> Ð *x* <sup>0</sup> *e*�*<sup>t</sup>* 2 *dt* is integral of the additional function and describes a parabolic boundary layer, *dk*,*<sup>l</sup>* ð Þ *<sup>x</sup>*, *<sup>t</sup>* ,*Wk*,*<sup>l</sup>* ð Þ *x*, *t* are arbitrary functions that are chosen, like solving the equations

$$D\_x d\_i^{k,l}(\mathbf{x}, t) = -D\_x h\_i^{k,3}(\mathbf{x}, t),\ D\_x \mathcal{W}\_{ij}^{k,3}(\mathbf{x}, t) = -D\_x h\_{ij}^{k,4}(\mathbf{x}, t),\ \ D\_x \equiv 2\rho\_l(\mathbf{x})\partial\_\mathbf{x} + \rho\_{l'}(\mathbf{x}),\tag{20}$$

with boundary conditions from (17). Eqs. (20) are obtained by satisfying condition c) and taking into account that the functions *erfc <sup>ξ</sup>l<sup>=</sup>* <sup>2</sup>*τ*<sup>1</sup>*=*<sup>2</sup> � � � � and *Il <sup>ξ</sup><sup>l</sup>* ð Þ , *<sup>τ</sup>* 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 *ν* ¼ 0,1 the equations for *uν*ð Þ *M* 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

$$F\_2(M) = f(\mathbf{x}, t) - T\_1 u\_0 = f(\mathbf{x}, t) - \, \text{<} \\ D\_1 v\_0(\mathbf{x}, t) + D\_2 \left[ \varepsilon^0(\mathbf{x}, t) + \Lambda \left( P^0(\mathbf{x}) \right) \right] \varepsilon^0$$

$$+ \sum\_{l=1}^2 \left[ D\_1 Y^0(N^l) + D\_2 \mathbf{z}^0(N^l) \varepsilon^l \right], \forall (t) > \cdot$$

We decompose *f x*ð Þ , *t* by the system *ψi*ð Þ*t* and substitute it with the previous relation. Further, providing the conditions of Theorem 1, we set

$$D\_1 \nu\_0(\mathbf{x}, t) = f(\mathbf{x}, t), \ t \left[ \partial\_t \nu\_{0, i}(\mathbf{x}, \ t) + \sum\_{\mu=1}^n a\_{i\mu}(t) \nu\_{0, \mu}(\mathbf{x}, t) \right] - \lambda\_i(t) \nu\_{0i}(\mathbf{x}, t) = \langle f(\mathbf{x}, t), \nu\_i \rangle,$$

$$\qquad < D\_2 \left[ c^0(\mathbf{x}, t) + \Lambda \langle P^0(\mathbf{x}) \rangle \right], \newline \qquad \qquad = 0.$$

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

$$t\left[\partial\_t c\_{ii}^0 + a\_{ii}(t)c\_{ii}^0(\mathbf{x}, t)\right] = (\lambda\_i(t) - \lambda\_i(\mathbf{0}))P\_i^0(\mathbf{x}) - t\sum\_{\mu=1}^n a\_{\mu, i}(t)P\_\mu^0(\mathbf{x}),$$

$$c\_{ii}^0(\mathbf{x}, t)|\_{t=0} = -\nu\_{i0}(\mathbf{x}, \mathbf{0}) - P\_i^0(\mathbf{x}), \tag{21}$$

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

$$(\lambda\_j(\mathbf{0}) - \lambda\_i(t))c\_{ij}^0(\mathbf{x}, \mathbf{0}) = \mathbf{0}, \ i \neq j, \ i, j = \overline{\mathbf{1}, \mu}. \tag{22}$$

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

*Boundary Layer Flows - Modelling, Computation, and Applications of Laminar,Turbulent…*

For *<sup>k</sup>* <sup>¼</sup> 3, by Theorem 3, ensuring condition (c) for *<sup>d</sup>*0,*<sup>l</sup>* ð Þ *<sup>x</sup>*, *<sup>t</sup>* and *<sup>W</sup>*0,*<sup>l</sup>* ð Þ *x*, *t* , we obtain problem

$$\begin{aligned} D\_{\mathbf{x}}d\_i^{0,l}(\mathbf{x},t) &= \mathbf{0}, \ \left. d\_i^{0,l}(\mathbf{x},\ t) \right|\_{\mathbf{x}=l-1} = -\nu\_{0,i}(l-\mathbf{1},t),\\ D\_{\mathbf{x}}W^{0,l}\_{\vec{\eta}}(\mathbf{x},t) &= \mathbf{0}, \ \left. W^{0,l}\_{\vec{\eta}}(\mathbf{x},\ t) \right|\_{\mathbf{x}=l-1} = -c^{0}\_{\vec{\eta}}(l-\mathbf{1},t). \end{aligned}$$

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

$$D\_1 \nu\_{1,i} = 0, \quad t \left[ \partial\_t c\_{ii}^1 + a\_{ii}(t) c\_{ii}^1 \right] = (\lambda\_i(t) - \lambda\_i(0)) P\_i^1(\mathbf{x}) - t \sum\_{\mu=1}^n a\_{\mu,i}(t) P\_\mu^1(\mathbf{x}),$$

$$c\_{ii}^1(\mathbf{x}, \ t)|\_{t=0} = -\nu\_{1i}(\mathbf{x}, \ \mathbf{0}) - P\_i^1(\mathbf{x}),$$

$$t \left[ \partial\_t c\_{ij}^1 + \sum\_{\mu=1(\mu \neq i)}^n a\_{\mu,i}(t) c\_{\mu j}^1(\mathbf{x}, t) \right] + \left( \lambda\_j(\mathbf{0}) - \lambda\_i(t) c\_{ij}^1(\mathbf{x}, t) = \mathbf{0}, \quad c\_{ij}^1(\mathbf{x}, \ t) \Big|\_{t=0}.$$

From here we define *<sup>v</sup>*1*<sup>i</sup>*ð Þ¼ *<sup>x</sup>*, *<sup>t</sup>* 0, below it will be shown that *<sup>P</sup>*<sup>1</sup> *<sup>i</sup>*ð Þ¼ *x* 0, therefore *c*1 *ii*ð Þ¼ *<sup>x</sup>*, *<sup>t</sup>* 0, and from the last equation we find *<sup>c</sup>*<sup>1</sup> *ij*ð Þ¼ *x*, *t* 0, *i* 6¼ *j*.

In the next *k* ¼ 4 step free member

$$\partial\_t,$$

$$F\_4(\mathcal{M}) = -T\_1\mu\_2 + L\_\xi\mu\_1 - \partial\_t\mu\_0.$$

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

$$D\_{\mathbf{x}}d\_{i}^{1,l}(\mathbf{x},t) = \mathbf{0}, \ \left.d\_{i}^{1,l}(\mathbf{x},\ t)\right|\_{\mathbf{x}=l-1} = -\nu\_{1,i}(l-\mathbf{1},t) = \mathbf{0},$$

$$D\_{\mathbf{x}}W^{1,l}\_{\vec{\eta}}(\mathbf{x},t) = \mathbf{0}, \ \left.W^{1,l}\_{\vec{\eta}}(\mathbf{x},\ t)\right|\_{\mathbf{x}=l-1} = -c^{1}\_{\vec{\eta}}(l-\mathbf{1},t) = \mathbf{0}.$$

That is, *d*1,*<sup>l</sup> <sup>i</sup>* ð Þ¼ *<sup>x</sup>*, *<sup>t</sup>* 0, *<sup>W</sup>*<sup>1</sup> *ij*,*l x*ð Þ¼ , *t* 0, and, therefore, *u*1ð Þ¼ *M* 0. Regarding *Y*3 *<sup>i</sup> <sup>N</sup><sup>l</sup>* � �,*W*<sup>3</sup> *ij <sup>N</sup><sup>l</sup>* � � we obtain homogeneous equations, therefore

$$\mathcal{W}\_i^3\left(\mathcal{N}^l\right) = d\_i^{3l}(\infty, t) \text{erfc}\left[\xi\_l/\left(2\tau^{1/2}\right)\right], \; \mathcal{W}\_{\vec{\eta}}^3\left(\mathcal{N}^l\right) = \mathcal{W}\_{\vec{\eta}}^{3,l}(\infty, t) \text{erfc}\left[\xi\_l/\left(2\tau^{3/2}\right)\right].$$

We calculate

$$F\_4(M) = - < D\_1 v\_2(\mathbf{x}, t) + D\_2 \left[ \varepsilon^2(\mathbf{x}, t) + \Lambda \left( P^2(\mathbf{x}) \right) \right] \mathbf{e}^\eta$$

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

$$+ \left[ \partial\_t \boldsymbol{\upepsilon}^0(\mathbf{x}, \ t) + A^T(t)\boldsymbol{\upepsilon}^0(\mathbf{x}, t) + A(t)\Lambda \left( P^0(\mathbf{x}) \right) \right] \boldsymbol{\upepsilon}^\eta$$

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

*A System of Singularly Perturbed Parabolic Equations with a Power Boundary Layer DOI: http://dx.doi.org/10.5772/intechopen.106239*

and ensuring the solvability in *U* of the iteration equation for *k* ¼ 4, we assume

$$ = -<\partial\_t v\_0(\mathbf{x}, t) + A(t)v\_0(\mathbf{x}, t), \boldsymbol{\psi}(t)>,\tag{23}$$

$$D\_2\left[\mathcal{c}^2(\mathbf{x},t) + \Lambda\left(P^2(\mathbf{x})\right)\right]\mathcal{y}(t) > \\\\ = - < \partial\_t \mathcal{c}^0(\mathbf{x},t) + A(t)\mathcal{c}^0(\mathbf{x},t) + A(t)\Lambda\left(P^0(\mathbf{x})\right), \mathcal{y}(t) > .$$

The solvability of the second system is ensured by the choice of *<sup>P</sup>*0ð Þ *<sup>x</sup>* :

$$\Lambda\left(P^{0}(\mathbf{x})\right) = -\overline{A^{-1}(t)[\partial\_t c^0(\mathbf{x},\ t) + A(t)c^0(\mathbf{x},\ t)]},$$

$$P^{0}\_i(\mathbf{x}) = a^{-1}\_{ii} \left[\partial\_t c^0\_{ii}(\mathbf{x},\ t) + \sum\_{k=1}^n a\_{ki}(t)c^0\_{ki}(\mathbf{x},\ t)\right]\_{t=0},$$

as well as

$$<\overline{c^2(\mathbf{x},t)\Lambda(\mathbf{0}) - \Lambda(t)c^2(\mathbf{x},t)\mathscr{y}(t)} > |\_{t=0}$$

$$= -\left.\overline{\partial\_t c^0(\mathbf{x},t) + A(t)c^0(\mathbf{x},t) + A(t)\Lambda(P^0(\mathbf{x}))}\right.$$

$$c^2\_{\vec{\eta}}(\mathbf{x},t)\Big|\_{t=0} = \frac{1}{\lambda\_j(\mathbf{0}) - \lambda\_i(\mathbf{0})} \left[\partial\_t c^0\_{\vec{\eta}}(\mathbf{x},t) + \sum\_{k=1}^n a\_{ki}(t)c^0\_{kj}(\mathbf{x},t) + a\_{ji}(t)P^0\_j(\mathbf{x})\right]\_{t=0}, \; i \neq j. \tag{24}$$

Under such assumptions, the second system of (23) is solvable. It is solved under the initial conditions (24) and the initial condition *c*<sup>2</sup> *ii*ð Þ *x*, *t* � � *<sup>t</sup>*¼<sup>0</sup> ¼ �*v*2*<sup>i</sup>*ð Þ� *<sup>x</sup>*, 0 *<sup>P</sup>*<sup>2</sup> *<sup>i</sup>*ð Þ *x* , 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 *<sup>R</sup><sup>ε</sup>n*ð Þ¼ *<sup>M</sup> u M*~ð Þ� , *<sup>ε</sup> <sup>u</sup><sup>ε</sup>n*ð Þ *<sup>M</sup>* , where *<sup>u</sup><sup>ε</sup>n*ð Þ¼ *<sup>M</sup>* <sup>P</sup>*<sup>n</sup> <sup>k</sup>*¼<sup>0</sup>*εku*2*<sup>k</sup>*ð Þ *<sup>M</sup> :* Substituting *u M*~ð Þ¼ , *ε u<sup>ε</sup>n*ð Þþ *M R<sup>ε</sup>n*ð Þ *M* into the extended problem (3), then, taking into account (6), with respect to *R<sup>ε</sup>n*ð Þ *M* , we obtain

$$\tilde{L}\_t R\_{\epsilon n(M)} = \epsilon^{n+1} \mathbf{g}\_{\epsilon n}(M), \ R\_{\epsilon n}(M)|\_{t=\tau=0} = R\_{\epsilon n}(M)|\_{\mathbf{x}=l-1, \xi\_l=0} = \mathbf{0}, \ l=1,2,3$$

where the function *g<sup>ε</sup>n*ð Þ *M* is expressed through *u<sup>ε</sup>*,*<sup>k</sup>*ð Þ *M* . Producing constriction this problem, taking into account the identity (4), with respect to *<sup>R</sup><sup>ε</sup>*,*<sup>n</sup>*ð Þ� *<sup>x</sup>*, *<sup>t</sup>*, *<sup>ε</sup>* ð Þj *<sup>R</sup><sup>ε</sup>n*ð Þ *<sup>M</sup> <sup>ζ</sup>*¼*γ*ð Þ *<sup>x</sup>*, *<sup>t</sup>*, *<sup>ε</sup>* we get the problem

$$L\_{\varepsilon}R\_{\varepsilon,n}(\infty,t,\ \varepsilon) = \varepsilon^{n+1} \mathbf{g}\_{\varepsilon n}(\infty,t,\ \varepsilon), \ R\_{\varepsilon n}|\_{t=0} = R\_{\varepsilon n}|\_{\mathbf{x}=l-1} = \mathbf{0}, \ l=\mathbf{1},2.$$

From the construction of solutions to iterative problems, it can be seen that the function *g<sup>ε</sup>n*ð Þ *x*, *t*, *ε* is uniformly bounded in Ω. Applying the maximum principle [28], we can establish a uniform estimate in Ω

$$\left\|\left|R\_{\epsilon n}(\mathbf{x},\ \mathbf{t},\ \ \epsilon)\right\|\right\| < \epsilon \varepsilon^{n+1}.\tag{25}$$

**Theorem 4**: Let conditions (1) and (2) be fulfilled. Then the function *u<sup>ε</sup>n*ð Þ *x*, *t*, *ε* 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

$$L\_{\varepsilon}u \equiv (\varepsilon + t)\partial\_t u - \varepsilon^2 A(\varkappa)\partial\_{\varkappa}^2 u - D(t)u = f(\varkappa, t), \ (\varkappa, t) \in \Omega,\tag{26}$$

$$u(\varkappa, \mathbf{0}, \varepsilon) = h(\varkappa), \ u(\mathbf{0}, t, \varepsilon) = u(\mathbbm{1}, t, \varepsilon) = \mathbf{0},$$

where Ω ¼ ð Þ� 0, 1 ð � 0, *T* , *ε*>0 is a small parameter,

$$u = u(\mathbf{x}, t, \varepsilon) = \operatorname{col}(u\_1(\mathbf{x}, t, \varepsilon), u\_2(\mathbf{x}, t, \varepsilon), \quad u\_n(\mathbf{x}, t, \varepsilon)), \\ \mathcal{A}(\mathbf{x}) \in \mathbb{C}^\infty([0, \ 1], \mathbb{C}^{n^2}),$$

$$D(t) \in \mathbb{C}^\infty([0, \ 1], \mathbb{C}^{n^2}) \\ f(\mathbf{x}, t) \in \mathbb{C}^\infty(\overline{\Delta}, \mathbb{C}^n).$$

The work is a continuation of [29], where instead of the matrix-function *A x*ð Þ, there was a scalar function and an asymptotic of the solution was constructed containing two functions describing the boundary layers along *x* ¼ 0 and *x* ¼ 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 2*m* parabolic boundary layer functions describing the boundary layers along *x* ¼ 0 and *x* ¼ 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), *erfc <sup>φ</sup>*ð Þ *<sup>x</sup>* <sup>2</sup>*<sup>ε</sup>* ffi *t* p � �, also contain the power boundary layer function

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

as well as their products, which describe the corner boundary layer in the vicinity of 0, 0 ð Þ.

Construction of the asymptotic solution of a singularly perturbed system of parabolic equations is devoted to works [8–10] 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].

*A System of Singularly Perturbed Parabolic Equations with a Power Boundary Layer DOI: http://dx.doi.org/10.5772/intechopen.106239*

#### **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 x*ð Þ , *t* and *h x*ð Þ, the inclusions

*f x*ð Þ , *<sup>t</sup>* <sup>∈</sup>*C*<sup>∞</sup> <sup>Ω</sup>, <sup>ℂ</sup>*<sup>n</sup>* � �, *h x*ð Þ∈*C*<sup>∞</sup> ½ � 0, 1 , <sup>ℂ</sup>*<sup>n</sup>* ð Þ,

are fulfilled for *n* � *n*-matrix-valued functions *D t*ð Þ and *A x*ð Þ-inclusions

$$D(t) \in \mathbb{C}^{\infty}([0, \ T], \mathbb{C}^{n \times n}), \ A(\varkappa) \in \mathbb{C}^{\infty}([0, \ 1], \mathbb{C}^{n \times n});$$


#### **2.2 Regularization of the problem**

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

$$\xi\_{i,l} = \frac{\rho\_{i,l}(\mathbf{x})}{\sqrt{\varepsilon^3}}; \rho\_{i,l}(\mathbf{x}) = \left(-\mathbf{1}\right)^{l-1} \int\_{l-1}^{\mathbf{x}} \frac{ds}{\sqrt{\lambda\_i(s)}}, l = \mathbf{1}, 2, i = \overline{\mathbf{1}, n}, \tag{27}$$

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

and an extended function such that

$$\left. \bar{u}(M, \ \varepsilon) \right|\_{\xi = \varrho(\mathbf{x})/\varepsilon} \equiv u(\mathbf{x}, t, \ \varepsilon), \mathcal{M} = (\mathbf{x}, t, \ \xi, \tau, \mu), \ \xi = (\xi\_1, \xi\_2),$$

$$\xi\_l = \begin{pmatrix} \xi\_{1,l}, \xi\_{2,l}, \ \dots, \xi\_{n,l} \end{pmatrix}, \varrho(\mathbf{x}) = (\varrho\_1(\mathbf{x}), \varrho\_2(\mathbf{x})), \tag{28}$$

$$\varrho\_l(\mathbf{x}) = \begin{pmatrix} \varrho\_{1,l}, \varrho\_{2,l}(\mathbf{x}), \ \dots, \varrho\_{n,l}(\mathbf{x}) \end{pmatrix}, \ l = \mathbf{1}, 2, \ \mu = (\mu\_1, \mu\_2, \dots, \mu\_n).$$

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

$$\begin{split} \partial\_t u \equiv & \left( \partial\_t \ddot{u} + \frac{1}{\varepsilon (t+\varepsilon)} \partial\_\tau \ddot{u} + \sum\_{j=1}^n \frac{\beta\_j(0)}{t+\varepsilon} \partial\_{\mu\_j} \ddot{u} \right) \bigg|\_{\theta = \chi(\mathbf{x}, \ t, \ \dot{\xi}, \ \mathbf{r}, \mu)} \\ \partial\_\mathbf{x} u \equiv & \left( \partial\_\mathbf{x} \ddot{u} + \sum\_{l=1}^2 \sum\_{i=1}^n \left[ \frac{1}{\sqrt{\varepsilon^3}} \rho'\_{i,l}(\mathbf{x}) \partial\_{\bar{\xi},i} \ddot{u} \right] \right) \bigg|\_{\theta = \chi(\mathbf{x}, \ t, \ \dot{\xi}, \ \mathbf{r}, \mu)} \\ \partial\_\mathbf{x}^2 u \equiv & \left( \partial\_\mathbf{x}^2 \ddot{u} + \sum\_{l=1}^2 \sum\_{i=1}^n \left[ \frac{1}{\varepsilon^3} \left( \rho'\_{i,l}(\mathbf{x}) \right)^2 \partial\_{\bar{\xi},i}^2 \ddot{u} + \frac{1}{\sqrt{\varepsilon^3}} L\_{i,l}^\xi \ddot{u} \right] \right) \bigg|\_{\theta = \chi(\mathbf{x}, \ t, \ \dot{\xi}, \ \mathbf{r}, \mu)} \end{split}$$

,

*Boundary Layer Flows - Modelling, Computation, and Applications of Laminar,Turbulent…*

$$L\_{i,l}^{\xi}\tilde{u} = 2\rho\_{i,l}'(\mathbf{x})\partial\_{\mathbf{x}\xi\_i\mathbf{j}}^2\tilde{u} + \rho\_{i,l}'(\mathbf{x})\partial\_{\tilde{\varepsilon},l}\tilde{u}, \ \chi(\mathbf{x},t,\varepsilon) = \left(\frac{\varrho(\mathbf{x})}{\sqrt{\varepsilon^3}}, \frac{1}{\varepsilon}\ln\left(\frac{t+\varepsilon}{\varepsilon}\right), K(t,\varepsilon)\right),$$

$$K(t,\varepsilon) = (K\_1(t,\ \varepsilon), K\_2(t,\ \varepsilon), \ \dots, K\_n(t,\ \varepsilon)), \ \theta = (\tau, \mu, \xi).$$

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

$$\bar{L}\_t \ddot{u} = \varepsilon \partial\_t \bar{u} + \frac{1}{\varepsilon} T\_0 \ddot{u} + T\_1 \bar{u} - \sqrt{\varepsilon} L\_{\dot{\varepsilon}} \bar{u} - t \partial\_t \bar{u} - \varepsilon^2 L\_{\mathcal{K}} \bar{u} = f(\mathbf{x}, t), \tag{29}$$

$$\bar{u}|\_{t=0} = \mathbf{0}, \ \bar{u}|\_{x=0, \dot{\varepsilon}\_t=0} = \bar{u}|\_{x=1, \dot{\varepsilon}\_t=0} = \mathbf{0}, \ \dot{u} = \overline{1, n},$$

$$T\_0 \ddot{u} = \partial\_t \bar{u} - \sum\_{l=1}^2 \sum\_{i=1}^n \left(\rho\_{i,l}^{\prime}(\mathbf{x})\right)^2 A(\mathbf{x}) \partial\_{\dot{\varepsilon}\_t}^2 \bar{u}, \ T\_1 \ddot{u} = t \partial\_t \bar{u} + \sum\_{j=1}^\infty \beta\_j(\mathbf{0}) \partial\_{\mu\_j} \bar{u} - D(t) \bar{u},$$

$$L\_{\dot{\varepsilon}} \ddot{u} = \sum\_{l=1}^2 \sum\_{i=1}^n A(\mathbf{x}) L\_{i,l}^{\dot{\varepsilon}\_t} \bar{u}, \ L\_{\dot{\varepsilon}} \bar{u} = A(\mathbf{x}) \partial\_{\mathbf{x}}^2 \bar{u}.$$

In this case, the identity holds

$$\left(\tilde{L}\_{\varepsilon}\tilde{u}(\mathbf{M},\ \varepsilon)\right)\_{\theta=\chi(\mathbf{x},\ t,\ \xi,\ \tau,\ \mu)} \equiv L\_{\varepsilon}u(\mathbf{x},t,\ \varepsilon). \tag{30}$$

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

$$
\tilde{u}(M,\varepsilon) = \sum\_{k=0}^{\infty} \varepsilon^{k/2} u\_k(M). \tag{31}
$$

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

$$T\_0 u\_0 = 0, \ T\_0 u\_1 = 0, \ T\_0 u\_2 = f(\mathbf{x}, t) - T\_1 u\_0, \ T\_0 u\_3 = L\_\xi u\_0 - T\_1 u\_1,\tag{32}$$

$$T\_0 u\_k = L\_\xi u\_{k-3} + L\_\mathbf{x} u\_{k-6} - \partial\_t u\_{k-4} - T\_1 u\_{k-2}.$$

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

$$\left.u\_k(\mathcal{M})\right|\_{t=\mu=\tau=0} = \mathbf{0},$$

$$\left.u\_k(\mathcal{M})\right|\_{\mathbf{x}=l-1, \xi\_{i,l}=\mathbf{0}} = \mathbf{0}, \ k \ge \mathbf{0}, \ i = \overline{\mathbf{1},n}, \ l = \mathbf{1}, 2.1$$

#### **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:

$$U\_1 = \left\{ V(\mathbf{x}, \ t) : V(\mathbf{x}, t) = \sum\_{i=1}^n v\_i(\mathbf{x}, t) \boldsymbol{\nu}\_i(t), \ \boldsymbol{v}\_i(\mathbf{x}, t) \in \mathbb{C}^\infty(\overline{\Omega}) \right\},$$

*A System of Singularly Perturbed Parabolic Equations with a Power Boundary Layer DOI: http://dx.doi.org/10.5772/intechopen.106239*

$$U\_2 = \left\{ Y(N) : Y(N) = \sum\_{l=1}^2 \sum\_{i=1}^n y\_i^l(N\_i^l) b\_i(\mathbf{x}), \ |p\_i^l(N\_i^l)| < c \exp\left( -\frac{\xi\_{il}}{8\pi} \right) \right\},$$

$$U\_3 = \left\{ C(\mathbf{x}, \ t) : C(\mathbf{x}, t) = \sum\_{i=1}^n \left[ \sum\_{j=1}^n c\_{\vec{v}}(\mathbf{x}, \ t) \exp\left(\mu\_j\right) + p\_i(\mathbf{x}) \right] \psi\_i(t), \ c\_{\vec{v}}(\mathbf{x}, t) \in C^\infty(\overline{\Omega}) \right\},$$

$$U\_4 = \left\{ Z(N) : Z(N) = \sum\_{l=1}^2 \sum\_{i, j=1}^n z\_{ij}^l(N\_i^l) b\_i(\mathbf{x}) \exp\left(\mu\_j\right), \ \left| z\_{ij}^l(N\_i^l) \right| < c \exp\left(-\frac{\xi\_{il}^2}{8\pi} \right) \right\},$$

where *N<sup>l</sup> <sup>i</sup>* ¼ *x*, *t*, *ξi*,*l*, *τ*, *μ<sup>i</sup>* � �, *<sup>i</sup>* <sup>¼</sup> 1,2, … ,*n*, *<sup>l</sup>* <sup>¼</sup> 1,2*:* From these classes of functions we construct a new class as a direct sum: *U* ¼ *U*<sup>1</sup> ⊕ *U*<sup>2</sup> ⊕ *U*<sup>3</sup> ⊕ *U*4*:* The function *uk*ð Þ *M* ∈ *U* is representable in vector form

$$u\_k(\mathcal{M}) = \Psi(t) \left[ V\_k(\mathbf{x}, \ t) + \mathbf{C}^k(\mathbf{x}, t) \exp(\mu) \right]$$

$$+ \sum\_{l=1}^2 B(\mathbf{x}) \left[ Y^{k,l} \left( \mathcal{N}^l \right) + Z^{k,l} \left( \mathcal{N}^l \right) \exp \left( \mu \right) \right], \ \mathcal{C}^k(\mathbf{x}, t) = C\_1^k(\mathbf{x}, t) + \Lambda(P(\mathbf{x})),$$

$$C\_1^k(\mathbf{x}, t) = \left( c\_{\vec{\eta}}(\mathbf{x}, \ t) \right), \ \ V\_k(\mathbf{x}, t) = col(v\_{k1}, v\_{k2}, \dots, v\_{kn}),$$

$$Y^{k,l} \left( N^l \right) = col \left( \mathcal{Y}\_1^{k,l} \left( N\_1^l \right), \mathcal{Y}\_2^{k,l} \left( N\_2^l \right), \dots, \mathcal{Y}\_n^{k,l} \left( N\_n^l \right) \right), \ Z^{k,l} \left( N^l \right) = \left( z\_{\vec{\eta}}^{k,l} \left( N\_1^l \right) \right),$$

$$\Psi(t) = \left( \psi\_1(t), \psi\_2(t), \dots, \psi\_n(t) \right), \ B(\mathbf{x}) = \left( b\_1(\mathbf{x}), b\_2(\mathbf{x}), \dots, b\_n(\mathbf{x}) \right),$$

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

or in coordinate form

$$u\_k(\mathcal{M}) = \sum\_{i=1}^n v\_{k,i}(\mathbf{x}, t)\boldsymbol{\mu}\_i(t) + \sum\_{l=1}^2 \sum\_{i=1}^n \boldsymbol{\mu}\_i^{k,l}(\mathbf{N}\_i^l)\boldsymbol{b}\_i(\mathbf{x}) \tag{33}$$

$$+ \sum\_{i,j=1}^n \left\{ c\_{ij}^k(\mathbf{x}, t)\boldsymbol{\mu}\_i(t) + \sum\_{l=1}^2 \boldsymbol{z}\_{i,j}^{k,l}(\mathbf{N}\_i^l)\boldsymbol{b}\_i(\mathbf{x}) \right\} \exp\left(\boldsymbol{\mu}\_j\right) + \sum\_{i=1}^n p\_i^k(\mathbf{x})\boldsymbol{\mu}\_i(t)\exp\left(\boldsymbol{\mu}\_i\right).$$

The vector-functions *bi*ð Þ *x* ,*ψi*ð Þ*t* included in these classes are eigenfunctions of the matrices *A x*ð Þ and *D t*ð Þ, respectively

$$A(\mathbf{x})b\_i(\mathbf{x}) = \lambda\_i(\mathbf{x})b\_i(\mathbf{x}),\ D(t)\varphi\_i(t) = \beta\_i(t)\varphi\_i(t),\ \mathbf{i} = \overline{\mathbf{1},n}.\tag{34}$$

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

Along with the eigenvectors *bi*ð Þ *<sup>x</sup>* and *<sup>ψ</sup>i*ð Þ*<sup>t</sup>* , the eigenvectors *<sup>b</sup>*<sup>∗</sup> *<sup>i</sup>* ð Þ *<sup>x</sup>* ,*<sup>ψ</sup>* <sup>∗</sup> *<sup>i</sup>* ð Þ*t* , *i* ¼ 1,*n* of the conjugate matrices *<sup>A</sup>*<sup>∗</sup> ð Þ *<sup>x</sup>* , *<sup>D</sup>*<sup>∗</sup> ð Þ*<sup>t</sup>* will be used

$$A^\*\left(\mathbf{x}\right)b\_i^\*\left(\mathbf{x}\right) = \overline{\lambda}\_i(\mathbf{x})b\_i^\*\left(\mathbf{x}\right), \ D^\*\left(\mathbf{t}\right)\boldsymbol{\mu}\_i^\*\left(\mathbf{t}\right) = \overline{\beta}\_i(\mathbf{t})\boldsymbol{\mu}\_i^\*\left(\mathbf{t}\right)$$

and they are selected biorthogonal

$$\left(b\_i(\infty), b\_j^\*\left(\infty\right)\right) = \delta\_{i,j}, \ \left(\psi\_i(t), \psi\_j^\*\left(t\right)\right) = \delta\_{i,j}, \ i, j = \overline{1,n}, \dots$$

where *δ<sup>i</sup>*,*<sup>j</sup>* is the Kronecker symbol.

We calculate the action of the operators *T*0,*T*1,*Lξ*,*Lx* on the function *uk*ð Þ *M*, *ε* from (34), taking into account relations (35) and

$$
\rho\_{i,l}^{\prime 2}(\mathbf{x}) = \frac{1}{\lambda\_i(\mathbf{x})}, i = \overline{1,n}
$$

we have

$$T\_0 u\_k(\mathcal{M}) \equiv \sum\_{i=1}^n \sum\_{l=1}^2 \left\{ \partial\_\tau p\_i^{k,l} \left( N\_i^l \right) - \partial\_{\xi\_{il}}^2 p\_i^{k,l} \left( N\_i^l \right) \right. \tag{35}$$

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

or vector form

$$T\_0 u\_k(M) \equiv \sum\_{l=1}^2 B(\mathbf{x}) \left\{ \partial\_\mathbf{r} Y^{k,l}(N^l) - \partial\_{\bar{\xi}\_l}^2 Y^{k,l}(N^l) + \left[ \partial\_\mathbf{r} Z^{k,l}(N^l) - \partial\_{\bar{\xi}\_l}^2 Z^{k,l}(N^l) \right] \exp\left(\mu\right) \right\},$$

*<sup>Y</sup><sup>k</sup>*,*<sup>l</sup> <sup>N</sup><sup>l</sup>* � � is *<sup>n</sup>*-vector, *<sup>Z</sup><sup>k</sup>*,*<sup>l</sup> <sup>N</sup><sup>l</sup>* � � is *<sup>n</sup>* � *<sup>n</sup>*-matrix. Here *B x*ð Þ is a matrix function ð Þ *<sup>n</sup>* � *<sup>n</sup>* whose columns are the eigenvectors *bi*ð Þ *x* of the matrix *A x*ð Þ. We calculate

*<sup>T</sup>*1*uk* <sup>¼</sup> *<sup>t</sup>∂tuk* <sup>þ</sup>X*<sup>n</sup> j*¼1 *<sup>β</sup>j*ð Þ <sup>0</sup> *<sup>∂</sup>μ<sup>j</sup> uk* � *D t*ð Þ*uk* ¼ *t* X*n i*¼1 *<sup>∂</sup>tvk*,*<sup>i</sup>* <sup>þ</sup>X*<sup>n</sup> r*¼1 *α<sup>r</sup>*,*<sup>i</sup>*ð Þ*t vk*,*<sup>r</sup>*ð*x*, *t*Þ " #*<sup>ψ</sup>i*ðÞþ*<sup>t</sup>* <sup>X</sup> 2 *l*¼1 *∂ty k*,*l <sup>i</sup> N<sup>l</sup> i* � �*bi*ð Þ *<sup>x</sup>* ( þ X*n j*¼1 *∂tc k ij*ð Þþ *<sup>x</sup>*, *<sup>t</sup>* <sup>X</sup>*<sup>n</sup> r*¼1 *αri*ð Þ*t c k r*,*j* <sup>ð</sup>*x*, *<sup>t</sup>*Þ þ *<sup>α</sup>ji*ð Þ*<sup>t</sup> pk <sup>j</sup>* ð Þ *x* !*ψi*ð Þ*t* " (36) þ X 2 *l*¼1 *z k*,*l ij N<sup>l</sup> i* � �*bi*ð Þ *<sup>x</sup>* # exp *μ<sup>j</sup>* � �) þ X*n i*,*j*¼1 *βj* ð Þ 0 *c k ij*ð Þ *x*, *t ψi*ð Þ*t* exp *μ<sup>j</sup>* � � þX*<sup>n</sup> i*¼1 *<sup>β</sup>i*ð Þ <sup>0</sup> *<sup>p</sup><sup>k</sup> <sup>i</sup>* ð Þ *x ψi*ð Þ*t* exp *μ<sup>i</sup>* ð Þ þ X 2 *l*¼1 X*n i*,*j*¼1 *βj* ð Þ 0 *z k*,*l <sup>i</sup>*,*<sup>j</sup>* ð Þ *x*, *t bi*ð Þ *x* exp *μ<sup>j</sup>* � � � X*n i*¼1 *<sup>β</sup>i*ð Þ*<sup>t</sup> vki*ð*x*, *<sup>t</sup>*Þ*ψi*ðÞþ*<sup>t</sup>* <sup>X</sup> 2 *l*¼1 X*n r*¼1 *γ<sup>r</sup>*,*<sup>i</sup>*ð*x*, *t*Þ*y k*,*l <sup>i</sup> N<sup>l</sup> i* � �*bi*ð Þ *<sup>x</sup>* ( ) � X*n i*,*j*¼1 *βj* ð Þ*t c k i*,*j* ð Þ *x*, *t ψi*ð Þ*t* exp *μ<sup>j</sup>* � � �X*<sup>n</sup> i*¼1 *<sup>β</sup>i*ð Þ*<sup>t</sup> pk <sup>i</sup>* ð Þ *x ψi*ð Þ*t* exp *μ<sup>i</sup>* ð Þ � X 2 *l*¼1 X*n i*,*j*¼1 X*n r*¼1 *γ<sup>r</sup>*,*<sup>i</sup>*ð Þ *x*, *t z k*,*l <sup>r</sup>*,*<sup>j</sup> N<sup>l</sup> i* � �*bi*ð Þ *<sup>x</sup>* exp *<sup>μ</sup><sup>j</sup>* � �*:*

Here *α<sup>i</sup>*,*<sup>r</sup>* ¼ *ψ*<sup>0</sup> *i* ð Þ*<sup>t</sup>* , *<sup>ψ</sup>* <sup>∗</sup> *<sup>r</sup>* ð Þ*<sup>t</sup>* � �, *<sup>γ</sup><sup>i</sup>*,*<sup>r</sup>*ð Þ¼ *<sup>x</sup>*, *<sup>t</sup> D t*ð Þ*bi*ð Þ *<sup>x</sup>* , *<sup>b</sup>*<sup>∗</sup> *<sup>r</sup>* ð Þ *<sup>x</sup>* � �*:* *A System of Singularly Perturbed Parabolic Equations with a Power Boundary Layer DOI: http://dx.doi.org/10.5772/intechopen.106239*

It will be shown below that the scalar functions *y k*,*l <sup>i</sup> <sup>N</sup><sup>l</sup>* � � and *<sup>z</sup>k*,*<sup>l</sup> <sup>i</sup>*,*<sup>j</sup> <sup>N</sup><sup>l</sup>* � � are representable in the form

$$\rho\_i^{k,l}(\mathcal{N}\_i^l) = d\_i^{k,l}(\infty, t)\mathcal{y}\_i^{k,l}(\xi\_l, \mathfrak{r}), \; z\_{i,j}^{k,l}(\mathcal{N}\_i^l) = o\_{i,j}^{k,l}(\infty, t)z\_{i,j}^{k,l}(\xi\_l, \mathfrak{r}).$$

Given these representations, we calculate

*<sup>L</sup>ξuk*ð Þ¼ *<sup>M</sup>* <sup>X</sup> 2 *l*¼1 *A x*ð ÞX*<sup>n</sup> i*¼1 2*φ*<sup>0</sup> *i* ð Þ *<sup>x</sup> bi*ð Þ *<sup>x</sup> dk*,*<sup>l</sup> <sup>i</sup>* ð*x*, *t*Þ � � *x* nh (37) þ*φ*<sup>00</sup> *i* ð Þ *<sup>x</sup> bi*ð Þ *<sup>x</sup> dk*,*<sup>l</sup> <sup>i</sup>* ð*x*, *t*Þ � �i*∂<sup>ξ</sup>i*,*<sup>l</sup> y k*,*l <sup>i</sup> <sup>ξ</sup>i*,*l*, *<sup>τ</sup>* � � þ X*n j*¼1 2*φ*<sup>0</sup> *i* ð Þ *<sup>x</sup> bi*ð Þ *<sup>x</sup> <sup>ω</sup><sup>k</sup>*,*<sup>l</sup> <sup>i</sup>*,*<sup>j</sup>* ð*x*, *t*Þ � � *x* h þ*φ*<sup>00</sup> *i* ð Þ *<sup>x</sup> bi*ð Þ *<sup>x</sup> <sup>ω</sup><sup>k</sup>*,*<sup>l</sup> <sup>i</sup>*,*<sup>j</sup>* ð*x*, *t*Þ � �i*∂ξi*,*<sup>l</sup> z<sup>k</sup>*,*<sup>l</sup> <sup>i</sup>*,*<sup>j</sup> <sup>ξ</sup><sup>i</sup>*,*<sup>l</sup>*, *<sup>τ</sup>* � � exp *<sup>μ</sup><sup>j</sup>* � �o, *Lxuk*ð Þ¼ *<sup>M</sup> A x*ð Þ *<sup>∂</sup>*<sup>2</sup> *<sup>x</sup>*ð Þþ *Vk*ð Þ *<sup>x</sup>*, *<sup>t</sup>* <sup>X</sup> 2 *l*¼1 *∂*2 *<sup>x</sup> B x*ð Þ*Y<sup>l</sup>*,*<sup>k</sup> <sup>N</sup><sup>l</sup>* � � � � " <sup>þ</sup>*∂*<sup>2</sup> *<sup>x</sup>* <sup>Ψ</sup>ð Þ*<sup>t</sup> <sup>C</sup><sup>k</sup>*,0ð*x*, *<sup>t</sup>*<sup>Þ</sup> � � exp ð Þþ *<sup>μ</sup>* <sup>X</sup> 2 *l*¼1 *∂*2 *<sup>x</sup> B x*ð Þ*Z<sup>l</sup>*,*<sup>k</sup> <sup>N</sup><sup>l</sup>* � � � � exp ð Þ *<sup>μ</sup>* # *:*

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

$$\left.y\_{i}^{k,l}\left(N\_{i}^{l}\right)\right|\_{t=r=\mu=0} = \mathbf{0}, \quad \left.x\_{i,j}^{k,l}\left(N\_{i}^{l}\right)\right|\_{t=r=\mu=0} = \mathbf{0},\tag{38}$$

$$\mathcal{E}\_{ii}^{k}(\mathbf{x}, \mathbf{0}) = -\nu\_{k,i}(\mathbf{x}, \mathbf{0}) - p\_{i}^{k}(\mathbf{x}) - \sum\_{i \neq j} c\_{ij}^{k}(\mathbf{x}, \mathbf{0}),$$

$$\left.y\_{i}^{k,l}\left(N\_{i}^{l}\right)\right|\_{\substack{\boldsymbol{\xi}, \boldsymbol{\mu} = \boldsymbol{0} \\ \boldsymbol{\mu} \neq \boldsymbol{0}}} = d\_{i}^{k,l}(\mathbf{x}, \mathbf{t}), \quad d\_{i}^{k,l}(\mathbf{x}, \mathbf{t})b\_{i}(\mathbf{x})\Big|\_{\substack{\boldsymbol{\xi} = \boldsymbol{I} \\ \boldsymbol{x} = l-1}} = -v\_{i}(l-1, \mathbf{t})\boldsymbol{\nu}\_{i}(\mathbf{t}),$$

$$\left.x\_{i,j}^{k,l}\left(N\_{i}^{l}\right)\right|\_{\boldsymbol{\xi} \boldsymbol{\mu} = \boldsymbol{0}} = o\_{i,j}^{k,l}(\mathbf{x}, \mathbf{t}), \quad o\_{i,j}^{k,l}(\mathbf{x}, \mathbf{t})b\_{i}(\mathbf{x})\Big|\_{\substack{\boldsymbol{\xi} = \boldsymbol{I} \\ \boldsymbol{x} = l-1}} = -\left[c\_{ij}(l-1, \ \mathbf{t}) + p\_{i}(l-1)\right]\boldsymbol{\nu}\_{i}(\mathbf{t}).$$

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

$$T\_0\mathfrak{u}\_k(\mathcal{M}) = h\_k(\mathcal{M}).\tag{39}$$

**Theorem 1**: Let *hk*ð Þ *M* ∈ *U* and conditions (2) and (3) on hold. Then Eq. (40) has a solution *uk*ð Þ *M* ∈ *U*, if the equations are solvable

$$
\partial\_{\tau} y\_i^{k,l} \left( \mathbf{N}\_i^l \right) - \partial\_{\xi\_i \boldsymbol{j}}^2 y\_i^{k,l} \left( \mathbf{N}\_i^l \right) = h\_i^{k,1} \left( \mathbf{N}\_i^l \right) \equiv \overline{h}\_i^{k,1} \left( \mathbf{x}, t \right) \overline{\overline{h}}\_i^{k,1} \left( \boldsymbol{\xi}\_{i,1}, \boldsymbol{\tau} \right), \tag{40}
$$

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

**Proof:** Let *hk*ð Þ¼ *<sup>M</sup>* <sup>P</sup><sup>2</sup> *l*¼1 P*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup> *<sup>h</sup><sup>k</sup>*,1 *<sup>i</sup> N<sup>l</sup> i* � � <sup>þ</sup> <sup>P</sup>*<sup>n</sup> <sup>j</sup>*¼<sup>1</sup>*h<sup>k</sup>*,2 *ij N<sup>l</sup> i* � � exp *μ<sup>j</sup>* h i � � *bi*ð Þ *<sup>x</sup>* <sup>∈</sup> *<sup>U</sup>*. Pose (34) into Eq. (40), then, on the basis of calculations (38), with respect *y k*,*l <sup>i</sup> N<sup>l</sup> i* � �, *z k*,*l ij N<sup>l</sup> i* � �, we obtain Eqs. (41). These equations, under appropriate boundary value conditions

$$\left.y\_i^{k,l}\left(N\_i^l\right)\right|\_{t=\tau=\mu=0} = \mathbf{0}, \ \left.y\_i^{k,l}\left(N\_i^l\right)\right|\_{\xi\_{il}=0} = d\_i^{k,l}(\mathbf{x},t),$$

$$\left.z\_{i,j}^{k,l}\left(N\_i^l\right)\right|\_{t=\tau=\mu=0} = \mathbf{0}, \ \left.z\_{i,j}^{k,l}\left(N\_i^l\right)\right|\_{\xi\_{il}=0} = o\_{i,j}^{k,l}(\mathbf{x},t).$$

Solutions are represented in the form

$$\begin{split} y\_{i}^{k,l}(N\_{i}^{l}) &= d\_{i}^{k,l}(\mathbf{x},t) \text{erfc}\left(\frac{\xi\_{i,l}}{2\sqrt{\pi}}\right) + \overline{h}\_{i}^{k,1}(\mathbf{x},t)I\_{1}\left(\xi\_{i,l},\tau\right), \\ z\_{i,j}^{k,l}(N\_{i}^{l}) &= o\_{i,j}^{k,l}(\mathbf{x},t) \text{erfc}\left(\frac{\xi\_{i,l}}{2\sqrt{\pi}}\right) + \overline{h}\_{ij}^{k,2}(\mathbf{x},t)I\_{2}\left(\xi\_{i,l},\tau\right), \\ I\_{r}\left(\xi\_{i,l},\tau\right) &= \frac{1}{2\sqrt{\pi}} \int\_{0}^{\tau} \left| \frac{\overline{h}\_{i}^{k,r}(\eta,s)}{\sqrt{\tau-s}} \right| \exp\left(-\frac{\left(\xi\_{i,l}-\eta\right)^{2}}{4\left(\tau-s\right)}\right) \\ & \qquad - \exp\left(-\frac{\left(\xi\_{i,l}+\eta\right)^{2}}{4\left(\tau-s\right)}\right) \Big| d\eta ds, \ r = \mathbf{1},2, \end{split}$$

where *h k*,*r <sup>i</sup>* ð Þ *x*, *t* ,*h k*,*r <sup>i</sup>* ð Þ *η*, *s* are known functions. Evaluation of the integral

$$\left| I\_r \left( \xi\_{i,l}, \ \pi \right) \right| \leq c \exp \left( -\frac{\xi\_{i,l}^2}{8\pi} \right).$$

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

1.*uk <sup>t</sup>*¼*μ*¼*τ*¼<sup>0</sup> ¼ 0, *uk* � � � � *<sup>x</sup>*¼*l*�1,*ξi*,*l*¼<sup>0</sup> <sup>¼</sup> 0, *<sup>l</sup>* <sup>¼</sup> 1,2; 2.�*T*1*uk* � *<sup>∂</sup>tuk*�<sup>2</sup> <sup>þ</sup> *Lxuk*�<sup>4</sup> <sup>∈</sup> *<sup>U</sup>*<sup>2</sup> <sup>⊕</sup> *<sup>U</sup>*4; 3.*Lξuk* ¼ 0,

has a unique solutionin *U*.

**Proof:** Satisfying the function *uk*ð Þ *M* ∈ *U* with the boundary conditions from (26) we obtain (39). Based on calculations (36–38) we have

$$-T\_1\mu\_k - \partial\_t\mu\_{k-2} + L\_\mathbf{x}\mu\_{k-4} = -t\sum\_{i=1}^n \left\{ \partial\_t v\_{k,i}(\mathbf{x},\ t)\psi\_i(t) \right.$$

$$+ \sum\_{r=1}^n a\_{ri}(t)v\_{k,r}(\mathbf{x},t)\psi\_i(t) + \sum\_{r=1}^n a\_{ri}(t)p\_r^k(\mathbf{x})\psi\_i(t) \exp\left(\mu\_r\right)$$

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

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

$$t\left[\partial\_t V\_k + A^T(t)V\_k\right] = -\partial\_t V\_{k-2} - L\_x V\_{k-4}(\mathbf{x}, t), \tag{41}$$

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

$$-\Lambda(\beta(t)) \left[\mathcal{C}^k(\mathbf{x}, t) + \Lambda \left(\mathcal{P}^k(\mathbf{x})\right)\right] \tag{42}$$

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

$$\overline{\mathcal{C}^{k}(\mathbf{x},\ t)\Lambda(\boldsymbol{\beta}(\mathbf{0})) - \Lambda(\boldsymbol{\beta}(\mathbf{t}))\mathcal{C}^{k}(\mathbf{x},\ t)}\bigg|\_{t=0}$$

$$=\overline{-\partial\_{t}\overline{\mathcal{C}^{k-2} - A^{T}(t)\mathcal{C}^{k-2}(\mathbf{x},\ t) - A^{T}(t)\Lambda(\boldsymbol{P}^{k-2}) + L\_{\mathbf{x}}(\mathbf{C}^{k-4})}\bigg|\_{t=0}$$

$$\overline{A^{T}(t)\Lambda\Big(\boldsymbol{P}^{k-2}(\mathbf{x})\big)} = \overline{\left[-\partial\_{t}\mathcal{C}^{k-2} - A^{T}(t)\mathcal{C}^{k-2} - L\_{\mathbf{x}}(\mathbf{C}^{k-4}(\mathbf{x},\ t) + \Lambda(\boldsymbol{P}^{k-4}(\mathbf{x})))\right]}$$

$$c\_{ij}^k(\mathbf{x},\ t)\Big|\_{t=0} = -\frac{\mathbf{1}}{\beta\_j(\mathbf{0}) - \beta\_i(t)} \left[\partial\_t c\_{ij}^{k-2} + \sum\_{r \neq j} a\_{ri}(t)c\_{rj}^{k-2}(\mathbf{x},t) - q\_{ij}(\mathbf{x},t)\right]\_{t=0},$$

$$p\_i^{k-2}(\mathbf{x}) = -\frac{\mathbf{1}}{a\_{ii}(t)} \left[\partial\_t c\_{ii}^{k-2} + a\_{ii}(t)c\_{ii}^{k-2} - q\_{ii}(\mathbf{x},t)\right]\_{t=0},$$

where *qij*ð Þ *<sup>x</sup>*, *<sup>t</sup>* is known function included in *A x*ð Þ*∂*<sup>2</sup> *<sup>x</sup> <sup>C</sup>k*�4ð Þþ *<sup>x</sup>*, *<sup>t</sup>* <sup>Λ</sup> *Pk*�4ð Þ *<sup>x</sup>* � � � � .

On the basis of (38), condition (3), Theorem 2 is ensured if arbitrary functions *dl*,*<sup>k</sup> <sup>i</sup>* ð Þ *x*, *t bi*ð Þ *x* ,.

*ωk*,*<sup>l</sup> ij* ð Þ *x*, *t bi*ð Þ *x* are solutions to the problems

$$2\varrho\_{i,l}^{\prime}(\mathbf{x})\left(d\_i^{l,k}(\mathbf{x},\ t)b\_i(\mathbf{x})\right)\_{\mathbf{x}^{\prime}} + \varrho\_{i,l}^{\prime\prime}(\mathbf{x})\left(d\_i^{l,k}(\mathbf{x},\ t)b\_i(\mathbf{x})\right) = \mathbf{0},$$

$$d\_i^{l,k}(\mathbf{x},\ t)b\_i(\mathbf{x})\Big|\_{\mathbf{x}=l-1} = -\nu\_k(l-1,\mathbf{t})\wp\_i(\mathbf{t}),\ \ i=\overline{1,n},$$

$$2\varrho\_{i,l}^{\prime}(\mathbf{x})\Big(\varrho\_{i,j}^{l,k}(\mathbf{x},\ t)b\_i(\mathbf{x})\Big)\_{\mathbf{x}^{\prime}} + \varrho\_{i,l}^{\prime\prime}(\mathbf{x})\Big(\varrho\_{i,j}^{l,k}(\mathbf{x},\ t)b\_i(\mathbf{x})\Big) = \mathbf{0},$$

$$\varrho\_{i,j}^{l,k}(\mathbf{x},\ t)b\_i(\mathbf{x})\Big|\_{\mathbf{x}=l-1} = -\left[c\_{ij}^{k}(l-1,\ \mathbf{t}) + p\_i(l-1)\right]\wp\_j(\mathbf{t}).$$

Thus, arbitrary functions *dl*,*<sup>k</sup> <sup>i</sup>* ð Þ *<sup>x</sup>*, *<sup>t</sup>* , *<sup>ω</sup><sup>k</sup>*,*<sup>l</sup> ij* ð Þ *x*, *t* , *vki*ð Þ *x*, *t* , *c k*,*l ij* ð Þ *x*, *t* included in (34) are uniquely determined.

Iterative Eq. (32) for *k* ¼ 0,1 is homogeneous; therefore, by Theorem 1, it has a solution *uk*ð Þ *M* ∈ *U* if the functions *y l*,*k <sup>i</sup> N<sup>l</sup> i* � �,*z<sup>l</sup>*,*<sup>k</sup> <sup>i</sup>*,*<sup>j</sup> N<sup>l</sup> i* � � are solutions of the equations

$$
\partial\_{\mathbf{r}} \mathcal{y}\_i^{k,l}(\mathbf{N}\_i^l) = \partial\_{\xi\_{i,l}^2} \mathcal{y}\_i^{k,l}(\mathbf{N}\_i^l), \quad \partial\_{\mathbf{r}} \mathcal{z}\_{i,j}^{k,l}(\mathbf{N}\_i^l) = \partial\_{\xi\_{i,l}^2} \mathcal{z}\_{i,j}^{k,l}(\mathbf{N}\_i^l).
$$

for boundary value conditions

$$\begin{aligned} \left. \mathcal{Y}\_i^{k,l} \left( \mathcal{N}\_i^l \right) \right|\_{t=\tau=0} &= \mathbf{0}, \ \left. \mathcal{Y}\_i^{k,l} \left( \mathcal{N}\_i^l \right) \right|\_{\xi\_{i,l}=0} = d\_i^{k,l}(\varkappa, t), \\\left. z\_{i,j}^{k,l} \left( \mathcal{N}\_i^l \right) \right|\_{t=\tau=0} &= \mathbf{0}, \ \left. z\_{i,j}^{k,l} \left( \mathcal{N}\_i^l \right) \right|\_{\xi\_{i,l}=0} = o\_{i,j}^{k,l}(\varkappa, t). \end{aligned}$$

From this problem, we find

$$\mathcal{Y}\_{i}^{0,l}(N\_{i}^{l}) = d\_{i}^{0,l}(\infty, t) \text{erfc}\left(\frac{\xi\_{i,l}}{2\sqrt{\pi}}\right), \quad \mathcal{z}\_{i,j}^{0,l}(N\_{i}^{l}) = o\_{i,j}^{0,l}(\infty, t) \text{erfc}\left(\frac{\xi\_{i,l}}{2\sqrt{\pi}}\right).$$

The functions *d*0,*<sup>l</sup> <sup>i</sup>* ð Þ *<sup>x</sup>*, *<sup>t</sup>* ,*ω*0,*<sup>l</sup> <sup>i</sup>*,*<sup>j</sup>* ð Þ *x*, *t* are determined from problems (44) which ensure that condition *L<sup>ξ</sup> u*<sup>0</sup> ¼ 0 is satisfied. Using calculations (37), the free term of iterative Eq. (32) at *k* ¼ 2 is written as *F*2ð Þ¼� *M T*1*u*0ð Þþ *M f x*ð Þ , *t* by Theorem 1, an equation with such a free term is solvable in *U*, if

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

*A System of Singularly Perturbed Parabolic Equations with a Power Boundary Layer DOI: http://dx.doi.org/10.5772/intechopen.106239*

$$t\sum\_{i,j=1}^{n}\left\{\left[\partial\_{t}c\_{ij}^{0}(\mathbf{x},\ t)+\sum\_{r=1}^{n}a\_{ri}(t)c\_{rj}^{0}(\mathbf{x},\ t)\right]+a\_{ji}(t)p\_{j}^{0}(\mathbf{x})\right\}$$

$$+\sum\_{i,j=1}^{n}\left[\beta\_{j}(\mathbf{0})-\beta\_{i}(t)\right]c\_{\vec{\eta}}^{0}(\mathbf{x},t)+\sum\_{i=1}^{n}[\beta\_{i}(\mathbf{0})-\beta\_{i}(t)]p\_{i}^{0}(\mathbf{x})=\mathbf{0}.\tag{43}$$

From (47) we uniquely determine *c*<sup>0</sup> *i*,*j* ð Þ¼ *<sup>x</sup>*, *<sup>t</sup>* 0,∀*<sup>i</sup>* 6¼ *<sup>j</sup>* and the function *<sup>c</sup>*<sup>0</sup> *i*,*i* ð Þ *x*, *t* is determined from the equation

$$t\left[\partial\_t c\_{ii}^0(\mathbf{x},\ t) + a\_{ii}(\mathbf{t})c\_{ii}^0(\mathbf{x},\ t)\right] + (\beta\_i(\mathbf{0}) - \beta\_i(\mathbf{t}))c\_{ii}^0(\mathbf{x},\ t) + [\beta\_i(\mathbf{0}) - \beta\_i(\mathbf{t})]p\_i^0(\mathbf{x}) = \mathbf{0}$$

under the initial condition

$$c^0\_{ii}(\mathbf{x}, \mathbf{0}) = -v\_{0,i}(\mathbf{x}, \mathbf{0}) - p^0\_i(\mathbf{x}).$$

The first equation from (47), by virtue of condition (2), has a solution satisfying the condition [27] *<sup>v</sup>*<sup>0</sup>ð Þ *<sup>x</sup>*, 0 � � � � < ∞*:* We calculate the free term of Eq. (32) at *k* ¼ 3:

$$F\_3(\mathbf{M}) = -T\_1 \mathbf{u}\_1,$$

which has the same view as *T*1*u*0. Providing the solvability of equation *T*0*u*<sup>3</sup> ¼ �*T*1*u*<sup>1</sup> in *<sup>U</sup>*, with respect to *<sup>c</sup>*<sup>1</sup> *ij*ð Þ *x*, *t* ,*v*1*<sup>i</sup>*ð Þ *x*, *t* we obtain Eqs. (47).

In the next step, ð Þ *k* ¼ 4 the free term has the view

$$F\_4(M) = -T\_1\mu\_2 - \partial\_t\mu\_0 + L\_\xi\mu\_1.$$

The functions *d*1,*<sup>l</sup> <sup>i</sup>* ð Þ *<sup>x</sup>*, *<sup>t</sup>* ,*ω*1,*<sup>l</sup> <sup>i</sup>*,*<sup>j</sup>* ð Þ *x*, *t* entering the *u*1ð Þ *M* provide the condition *Lξu*<sup>1</sup> ¼ 0. Providing the solvability of the iterative equation at *k* ¼ 4, we set

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

For *c*<sup>2</sup> *ij*ð Þ *x*, *t* we obtain the same equation of the form (47), but with the right-hand side *∂tc*<sup>0</sup> *i*,*j* ð Þþ *<sup>x</sup>*, *<sup>t</sup>* <sup>P</sup>*<sup>n</sup> <sup>k</sup>*¼<sup>1</sup>*α<sup>k</sup>*,*<sup>i</sup>*ð Þ *<sup>x</sup> <sup>c</sup>*<sup>0</sup> *k*,*j* ð Þþ *<sup>x</sup>*, *<sup>t</sup> <sup>α</sup><sup>j</sup>*,*<sup>i</sup>*ð Þ *<sup>x</sup> <sup>p</sup>*<sup>0</sup> *<sup>j</sup>* ð Þ *x* � �. Taking off the degeneracy of this equation as *<sup>t</sup>* <sup>¼</sup> 0, we set *<sup>p</sup>*<sup>0</sup> *<sup>i</sup>* ð Þ¼� *<sup>x</sup>* <sup>1</sup> *<sup>α</sup>ii*ð Þ*<sup>t</sup> <sup>∂</sup>tc*<sup>0</sup> *ii* <sup>þ</sup> *<sup>α</sup>ii*ð Þ *<sup>x</sup> <sup>c</sup>*<sup>0</sup> *ii* � � *<sup>t</sup>*¼<sup>0</sup>. Further repeating the described process, using Theorems 1 and 2, sequentially determining *uk*ð Þ *M* , *k* ¼ 0,1, … ,*n*, we construct a partial sum

$$u\_{\epsilon n}(M) = \sum\_{k=0}^{n} \epsilon^{k/2} u\_k(M). \tag{44}$$

#### **2.4 Estimate for the remainder**

For the remainder term

$$R\_{\epsilon n}(M) = \mathfrak{u}(M, \varepsilon) - \mathfrak{u}\_{\epsilon n}(M) = \mathfrak{u}(M, \varepsilon) - \sum\_{k=0}^{n+2} \varepsilon^{k/2} \mathfrak{u}\_k(M) + \sum\_{l=1}^{2} \varepsilon^{\frac{n+l}{2}} \mathfrak{u}\_{n+l}(l)$$

we get the problem

*Boundary Layer Flows - Modelling, Computation, and Applications of Laminar,Turbulent…*

$$\begin{aligned} \tilde{L}\_{\iota}R\_{\iota\iota}(\mathcal{M}) &= e^{\frac{\pi+1}{2}} \mathcal{g}\_{\iota\iota}(\mathcal{M}), \\ \left. R\_{\iota\iota}(\mathcal{M}) \right|\_{l=\tau=\mu=0} &= \left. R\_{\iota\iota} \right|\_{\iota=l-1,\xi=0} = \mathbf{0}, \; l=1,2, \end{aligned}$$

where

$$\mathcal{g}\_{nr}(\mathcal{M}) = -\sum\_{l=0}^{1} (T\_1 u\_{n+1+l}) \varepsilon^{l/2} - \sum\_{l=0}^{3} \varepsilon^{l/2} \partial\_l u\_{n-1+l} + \sum\_{l=0}^{5} \varepsilon^{l/2} L\_x u\_{n-3+l} - \sum\_{l=0}^{1} \varepsilon^{l/2} \tilde{L}\_t u\_{n+1+l}.$$

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

$$\left\| \mathbf{g}\_{n\epsilon}(\mathbf{M}) \right\| < \mathbf{C}.$$

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

$$L\_{\varepsilon}R\_{\varepsilon n}(\varkappa,t,\varepsilon) = \varepsilon^{(n+1)/2} \mathfrak{g}\_{n\varepsilon}(\varkappa,t,\varepsilon),$$

$$R\_{\varepsilon n}|\_{t=0} = R\_{\varepsilon n}|\_{\varkappa=0} = R\_{\imath n}|\_{\varkappa=1} = \mathbf{0}.$$

We divide both sides of this equation by ð Þ *t* þ *ε* , while the properties of the matrices A are preserved. Therefore, using Theorem 5.5 of [33], we obtain the estimate

$$||R\_{\epsilon n}(\mathfrak{x},\ t,\ \mathfrak{e})|| < \varepsilon \varepsilon^{(n+1)/2}.$$

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

$$\left| \left| R\_{en}(\mathbf{x}, \ t, \ \varepsilon) \right| \right| < c\varepsilon^{\frac{n+1}{2}}.\tag{45}$$

**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, ∀*n* ¼ 0,1, … the estimate (45) holds at *ε* ! 0.

**Proof:** Let us rewrite the problem (31)

$$
\partial\_t u - \varepsilon^2 \frac{1}{t+\varepsilon} A(\varkappa) \partial\_\varkappa^2 u - \frac{1}{t+\varepsilon} D(t) u = \frac{1}{t+\varepsilon} f(\varkappa, t)
$$

here, the expression ð Þ *t* þ *ε* does not affect sufficiently small *ε* the properties of the matrices *A x*ð Þ,*D t*ð Þ 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).

*A System of Singularly Perturbed Parabolic Equations with a Power Boundary Layer DOI: http://dx.doi.org/10.5772/intechopen.106239*
