S. A. LOMOV

A POWER BOUNDARY LAYER IN PROBLEMS WITH A SMALL PARAMETER

(Presented by Academician I. G. Petrovskii on 6 VII 1962)

In this note we consider the construction of an asymptotic expansion in powers of the small parameter \(\varepsilon > 0\) for the solution of the Cauchy problem and of a boundary-value problem for the equation

\[ L_\varepsilon y \equiv (\varepsilon + x)y^{(n)} + a_{n-1}(x)y^{(n-1)} + a_{n-2}(x)y^{(n-2)} + \cdots + a_0(x)y = h(x). \tag{1} \]

Although for \(\varepsilon=0\) the order of equation (1) is not lowered, nevertheless one boundary condition is lost, since the equation

\[ L_0 w \equiv xw^{(n)} + a_{n-1}(x)w^{(n-1)} + a_{n-2}(x)w^{(n-2)} + \cdots + a_0(x)w = h(x) \tag{2} \]

is degenerating at the left endpoint of the interval \([0,a]\), on which we consider equation (1). Therefore in such problems (including equations in partial derivatives) the phenomenon of a boundary layer arises naturally, as in problems in which, for \(\varepsilon=0\), the order of the equation is lowered. Here the boundary layer has a power order of decrease as \(\varepsilon \to 0\); more precisely, it has the form
\[ P_m[\ln(1+x/\varepsilon)]/(1+x/\varepsilon)^c, \]
where \(P_m(u)\) is a polynomial of degree \(m\), \(c>0\), \(P_m(0)\ne 0\). In considering the Cauchy problem we restrict ourselves to the case where \(a_{n-1}(0)\equiv \alpha\) is not an integer. In what follows this must be kept in mind. The boundary-value problem is considered for \(n=2\), both for integral and nonintegral \(a_1(0)>1\).

1. Let us consider the behavior of the solution \(y_\varepsilon(x)\) of equation (1) for sufficiently small \(\varepsilon\) under zero initial conditions:

\[ y_\varepsilon(0)=y'_\varepsilon(0)=\cdots=y_\varepsilon^{(n-1)}(0)=0. \tag{1'} \]

For fixed \(\varepsilon>0\), problem (1), (1′) has a unique solution. For \(\varepsilon=0\), equation (1) turns into equation (2), whose solution can no longer satisfy the conditions (1′), but does satisfy the conditions

\[ w(0)=w'(0)=\cdots=w^{(n-2)}(0)=0,\qquad |w^{(n-1)}(x)|\le M. \tag{2'} \]

Theorem 1. If the coefficients of equation (2) \(a_i(x), h(x)\in C^{(m+1)}(0,a)\), and \(a_{n-1}(0)\equiv \alpha>0\), then equation (2) has a unique solution \(w_0(x)\in C^{(m+n)}(0,a)\) satisfying the conditions:

\[ w_0^{(r)}(0)=D_r\quad (r=0,1,\ldots,n-2);\qquad |w_0^{(n-1)}(x)|\le M. \tag{2''} \]

We take the solution \(w_0(x)\) of the limiting problem (2), (2′) as a certain approximation to the solution \(y_\varepsilon(x)\). Now it is necessary to eliminate the discrepancy in the last initial condition (1′). To this end we shall seek the solution of the problem (1), (1′) in the form \(y_\varepsilon(x)=w_\varepsilon(x)+v_\varepsilon(x)\), so that \(L_\varepsilon w_\varepsilon=h(x)\) and \(L_\varepsilon v_\varepsilon=0\). The point \(x=-\varepsilon\) for the equation \(L_\varepsilon v_\varepsilon=0\) is a singular point. The roots of the indicial equation, which is introduced in the same way as in the analytic case, are
\[ \rho_0=0,\ \rho_1=1,\ldots,\rho_{n-2}=n-2,\ \rho_{n-1}=n-1-a_{n-1}(-\varepsilon). \]
The root \(\rho_{n-1}\) gives grounds to suppose that the boundary layer should be sought in the form
\[ v_\varepsilon(x)=c\,v(x,\varepsilon)/(x+\varepsilon)^{a_{n-1}(-\varepsilon)-n+1}, \]
\[ (a_{n-1}(-\varepsilon)>0,\ 0\le \varepsilon<\varepsilon_0). \]
Subjecting \(v_\varepsilon^{(n-1)}(x)\) to the condition
\[ w^{(n-1)}(0)+v^{(n-1)}(0)=0, \]
we find that
\[ v_\varepsilon(x)=\varepsilon^{\,n-1}c_0 v(x,\varepsilon)/(1+t)^{a_{n-1}(-\varepsilon)-n+1}\qquad (t=x/\varepsilon). \]
Therefore in \(L_\varepsilon v_\varepsilon(x)\) we substitute \(v_\varepsilon(x)=v(x,\varepsilon)\,\bar v(t,\varepsilon)\). Taking into account that the function \(\bar v(t,\varepsilon)\) satisfies any of the equations of the form
\[ (1+t)\bar v^{(i)}+\bigl(a_{n-1}(-\varepsilon)-n+i\bigr)\bar v^{(i-1)}=0,\quad i=1,\ldots,n, \]
and grouping the terms—

Taking into account their dependence on \(\varepsilon\), we express the operator \(L_\varepsilon\) in terms of two operators \(M_\varepsilon\) and \(\overline L_\varepsilon\):

\[ L_\varepsilon v_\varepsilon(x)\equiv v M_\varepsilon v+ \sum_{i=0}^{n-1}\varepsilon^{-n+i+1}K_i v\cdot (\overline L_\varepsilon \overline v)^{(n-i-1)}, \tag{*} \]

where

\[ M_\varepsilon v\equiv (\varepsilon+x)v^{(n)}+ \sum_{k=0}^{n-1} b_{n-k-1}(x,\varepsilon)v^{(n-k-1)}, \]

\[ \overline L_\varepsilon \overline v\equiv (1+t)\overline v' +\bigl(a_{n-1}(-\varepsilon)-n+1\bigr)\overline v, \]

\[ K_0v\equiv v(x,\varepsilon),\qquad K_i v=C_n^{\,n-i}v^{(i)}+\sum_{k=0}^{i-1}d_{i-k-1}(x,\varepsilon)v^{(i-k-1)} \quad (i=1,\ldots,n-1), \]

\[ b_{n-k-1}(x,\varepsilon)=A_{n-k-1}(x,\varepsilon)/(x+\varepsilon)^k,\qquad d_{i-k-1}(x,\varepsilon)=A_{n-k-1}^{\,i}(x,\varepsilon)/(x+\varepsilon)^{k+1}. \]

For brevity we do not write out the dependence of the sufficiently smooth functions \(A_{n-k-1}(x,\varepsilon)\) and \(A_{n-k-1}^{\,i}(x,\varepsilon)\) on the coefficients \(a_i(x)\) and \(a_{n-1}(-\varepsilon)\). We can now obtain separately equations for \(v(x,\varepsilon)\) and \(\overline v(t,\varepsilon)\), by putting \((\varepsilon+x)^{n-2}M_\varepsilon v=0\) and \(\overline L_\varepsilon \overline v=0\). For \(\varepsilon=0\) these equations give zero approximations to the functions \(v(x,\varepsilon)\) and \(\overline v(t,\varepsilon)\):

\[ M_0v\equiv x^{n-1}v^{(n)}+x^{n-2}\overline b_{n-1}^{\,0}(x)v^{(n-1)}+\cdots \]
\[ \cdots+x\overline b_2^{\,0}(x)v''+\overline b_1^{\,0}(x)v' +\overline b_0^{\,0}(x)v=0, \tag{3} \]

\[ \overline L_0\overline v\equiv (1+t)\overline v' +(\alpha-n+1)\overline v=0, \tag{4} \]

where \(\overline b_i^{\,0}(x)=x^{n-i-1}b_i^0(x)\), \(i=1,\ldots,n-1\), \(\overline b_0^{\,0}(x)=x^{n-2}b_0^0(x)\), and, for \(\alpha>0\), all \(\overline b_i^{\,0}(0)\ne0\), while \(b_{n-k-1}^0(x)=b_{n-k-1}(x,0)\). As the zero approximation to the function \(v(x,\varepsilon)\) we take the solution \(v_0(x)\) of equation (3) \((v_0(0)\ne0)\), continuous together with its derivatives up to a certain order; its existence will be established with the aid of the following facts. The indicial equation for equation (3) has the form:

\[ \rho\bigl[(\rho-1)(\rho-2)\cdots(\rho-n+1) +(\rho-1)(\rho-2)\cdots(\rho-n+2)\overline b_{n-1}^{\,0}(0)+\cdots \]
\[ \cdots+(\rho-1)\overline b_1^{\,0}(0)+\overline b_0^{\,0}(0)\bigr]=0. \tag{5} \]

Theorem 2. If the coefficients of the equation

\[ xw^{(n)}+\sum_{i=0}^{n-1}a_i(x)w^{(i)}=0, \tag{6} \]

\(a_i(x)\in C_{(0,a)}^{(m+1)}\), \(i=0,1,\ldots,n-1\); \(a_{n-1}(0)>0\), then a fundamental system of solutions of equation (6) admits the representation: \(w_i(x)=x^{\rho_i}\varphi_i(x)\), where \(\rho_i=i\) for \(i=0,1,\ldots,n-2\); \(\rho_{n-1}=n-1-a_{n-1}(0)\) are the roots of the indicial equation; \(\varphi_i(x)\in C_{(0,a)}^{(m+n-i)}\); \(\varphi_i(0)\ne0\), if \(\rho_{n-1}\) is not equal to an integer.

Corollary. A fundamental system of solutions of equation (3), for noninteger \(a_{n-1}(0)\), admits the representation \(v_i(x)=x^{\overline\rho_i}\overline\varphi_i(x)\), where \(\overline\rho_i\) are the roots of the indicial equation (5).

The function \(v_0(x)=\overline\varphi_0(x)\) \((\overline\rho_0=0)\) is the zero approximation to the function \(v(x,\varepsilon)\).

A solution of equation (4) is the function \(\overline v_0(t)=c_0/(1+t)^{\alpha-n+1}\). As a result, we obtain the zero approximation to the boundary-layer-type function \(v_\varepsilon(x)\):

\[ v_{\varepsilon0}(x)=v_0(x)\overline v_0(t)\equiv v_{00}(x,t), \]

with whose aid we satisfy the last of the initial conditions \((1')\) by the term of lowest order in \(\varepsilon\), i.e.

\[ v_0(x)\left.\frac{d^{\,n-1}\overline v_0(t)}{dx^{n-1}}\right|_{\substack{t=0\\ x=0}} = \frac{v_0(x)}{\varepsilon^{\,n-1}} \left.\frac{d^{\,n-1}\overline v_0(t)}{dt^{n-1}}\right|_{\substack{x=0\\ t=0}} =-w^{(n-1)}(0). \]

Hence we find the constant \(c_0=\varepsilon^{n-1}\overline c_0\), where \(\overline c_0\) is independent of \(\varepsilon\). Thus, finally,

\[ v_{\varepsilon0}(x)=\varepsilon^{n-1}\overline c_0 v_0(x)/(1+x/\varepsilon)^{\alpha-n+1} \]

and the zero approxi-

...to the solution \(y_\varepsilon(x)\) of problem (1), (1′) has the form \(y_{\varepsilon 0}(x)=w_0(x)+v_{\varepsilon 0}(x)\). It can be shown, using the representation (*), that \(L_\varepsilon y_{\varepsilon 0}(x)=h(x)+O(\varepsilon^{-n+2})\), while the residuals in the initial conditions have the form \(y_{\varepsilon 0}^{(k)}(0)=O(\varepsilon^{n-k-1})\), \(k=0,1,\ldots,n-2\), \(y_{\varepsilon 0}^{(n-1)}(0)=O(\varepsilon)\). To obtain the subsequent approximations we seek the functions

\[ w_\varepsilon(x)=\sum_{i=0}^{n+m-1}\varepsilon^i w_i(x)+O(\varepsilon^{n+m}), \]

\[ v(x,\varepsilon)=\sum_{i=0}^{n+m-1}\varepsilon^i v_i(x)+O(\varepsilon^{n+m}),\qquad \bar v(t,\varepsilon)=\sum_{i=0}^{n+m-1}\varepsilon^i \bar v_i(t)+O(\varepsilon^{n+m}) \]

and in the usual way obtain equations for determining \(w_k(x)\), \(v_k(x)\), \(\bar v_k(t)\):

\[ L_0 w_k(x)=-w_{k-1}^{(n)}(x),\qquad M_0 v_k=-\sum_{j=0}^{n}\sum_{i=1}^{k} b_j^i(x)v_{k-i}^{(j)},\qquad \bar L_0 \bar v_k(t)=-\sum_{i=1}^{k} a_{n-1}^{i}\bar v_{k-i}(t), \]

\[ k=1,\ldots,n+m-1, \]

where \(b_j^i(x)\) and \(a_{n-1}^i\) are the coefficients of the expansions of the corresponding coefficients in the operators \((\varepsilon+x)^{n-2}M_\varepsilon\) and \(\bar L_\varepsilon\).

We find the functions \(w_l(x)\), \(\bar v_l(t)\), and \(v_l(x)\) respectively under the initial conditions*:

\[ 1)\quad w_l^{(r)}(0)=0,\quad r=0,\ldots,n-l-2,\quad l=0,\ldots,n-2; \]

\[ w_l^{(r)}(0) = -\sum_{k=0}^{l-1} C_r^{\,n+k-l-1}\sum_{i=0}^{k} v_{k-i}^{(l+r-n-k+1)}(0)V_i^{(n+k-l-1)}(0), \]

\[ n-l-1\le r\le n-2,\quad 1\le l\le n+m-1;\qquad |w_l^{(n-1)}(x)|\le M_l,\quad l=0,1,\ldots,n+m-1; \]

\[ 2)\quad w_l^{(n-1)}(0)+ \sum_{k=0}^{l} C_{n-1}^{\,n+k-l-1}\sum_{i=0}^{k} v_{k-i}^{(l-k)}(0)V_i^{(n+k-l-1)}(0)=0,\quad 0\le l\le n+m-1; \]

\[ 3)\quad v_l(0)=1, \]

and the \(v_l(x)\) have the necessary number of continuous derivatives.

Here

\[ V_i(t)=\sum_{k=0}^{i} c_k \ln^k(1+t)/(1+t)^{\alpha-n+1}, \]

i.e., \(\bar v_i(t)=\varepsilon^{\,n-1}V_i(t)\) (the \(c_k\) do not depend on \(\varepsilon\)). As a result we can formulate the following theorem.

Theorem 3. If in equation (1)

\[ a_{n-1}(x)\in C^{((m+n)(n-1)+1+n)}(0,a), \]

\[ a_i(x)\in C^{((m+n)(n-1)+n)}(0,a),\quad i=0,1,\ldots,n-2,\quad a_{n-1}(0)>0,\quad h(x)\in C^{(n+m+1)}(0,a), \]

then the solution of problem (1), (1′) for sufficiently small \(\varepsilon>0\) can be represented in the form

\[ y_\varepsilon(x)=\sum_{k=0}^{n+m-1}\varepsilon^k\,[w_k(x)+v_{0k}(x;t)]+z_\varepsilon(x) = y_{\varepsilon,n+m-1}(x)+z_\varepsilon(x), \]

where

\[ L_\varepsilon y_{\varepsilon,n+m-1}(x)=h(x)+O(\varepsilon^{m+1}),\qquad y_{\varepsilon,n+m-1}^{(i)}(0)=O(\varepsilon^{n+m}),\quad i=0,1,\ldots,n-1, \]

where \(w_k(x)\in C^{(2n+m-k-1)}(0,a)\); the derivatives of order \(n-1\) of

\[ v_{0k}(x;t)=\sum_{i=0}^{k} v_{k-i}(x)\bar v_i(t) \]

are boundary-layer-type functions of power order in a neighborhood of the point \(x=0\) and uniformly tend to zero as \(\varepsilon\to0\) on any interval \([a_1,a]\) \((0<a_1<a)\); the remainder term \(z_\varepsilon(x)\) and all its derivatives up to order \(n-1\), as \(\varepsilon\to0\), are \(O(\varepsilon^{m+1})\).

1. Consider the following boundary-value problem:

\[ L_\varepsilon y\equiv(\varepsilon+x)y''+a_1(x)y'+a_0(x)y=h(x), \tag{7} \]

\[ y_\varepsilon(0)=y_0,\qquad y_\varepsilon(1)=y_1, \tag{7′} \]

where \(a_1(0)>1\), \(a_0(x)\le 0\). Denote \(a_1(x)=1+k(x)\), \(k_0=k(0)>0\). For \(\varepsilon=0\), equation (7) takes the form

\[ L_0 w\equiv xw''+(1+k(x))w'+a_0(x)w=h(x). \tag{8} \]

Problem (8), (7′) has no solution, but for equation (8) there is a unique solution to the problem in the following formulation:

\[ |w_0(x)|\le M,\quad 0\le x\le 1;\qquad w_0(x_0)=y_1,\quad 0\le x_0\le 1. \tag{8′} \]

* Terms with negative upper indices are replaced by zeros.

Using the representation (*) for \(n=2\), we obtain the equation for \(v(x,\varepsilon)\) and \(\bar v(t,\varepsilon)\):

\[ M_\varepsilon v \equiv (\varepsilon+x)v''+\bigl(1+k(x)-2k(-\varepsilon)\bigr)v' +\left(a_0(x)+\frac{k(-\varepsilon)\bigl(k(-\varepsilon)-k(x)\bigr)}{x+\varepsilon}\right)v=0; \tag{9} \]

\[ \bar L_\varepsilon \bar v \equiv (1+t)\bar v' + k(-\varepsilon)\bar v=0. \]

1) In the case of nonintegral \(k_0\), as in the Cauchy problem, we find \(v_0(x)\) and \(\bar v_0(t)=c_0/(1+t)^{k_0}\) \((v_0(0)=1,\ \bar v_0(0)=y_0-w_0(0))\) and set
\(y_{\varepsilon 0}(x)=w_0(x)+v_0(x)\bar v_0(t)\).

Having found \(w_1(x)\), \(v_1(x)\), and \(\bar v_1(t)=\bigl(c_1+c\ln(1+t)\bigr)/(1+t)^{k_0}\), and subjecting \(\bar v_1(t)\) to the condition \(v_1(0)=-w_1(0)-\bar v_0(0)\), we observe that discrepancies of a new kind appear:
\(y_{\varepsilon 1}(1)=y_1+\varphi_0(\varepsilon)\varepsilon^{k_0}+\varphi_1(\varepsilon)\varepsilon^{k_0+1}\ln\varepsilon\); therefore it is necessary to introduce into the expansions of the functions \(w_\varepsilon(x)\) and \(v_\varepsilon(x)\) terms with \(\varepsilon^{ik_0+j}\ln^k\varepsilon\), and to expand the functions \(\varphi_i(\varepsilon)\) in powers of \(\varepsilon\).

2) In the case of integral \(k_0\), the first iterative process is carried out as in the nonintegral case, since the proof of Theorem 1 for \(n=2\) remains valid also for integral \(k_0>0\); only the expansion is conducted in powers of \(\varepsilon\ln^i\varepsilon\). It can be proved that equation (9) has two linearly independent solutions of the form
\(v_1(x,\varepsilon)=(x+\varepsilon)^{k(-\varepsilon)}w(x,\varepsilon)\) and

\[ v_2(x,\varepsilon)=z(x,\varepsilon)+v_1(x,\varepsilon) \int_{x_0}^{x}\frac{dt}{(t+\varepsilon)^{k(-\varepsilon)-k_0+1}}, \]

where \(w(-\varepsilon,\varepsilon)\ne0,\ z(-\varepsilon,\varepsilon)\ne0\)
\((0\le \varepsilon<\varepsilon_0)\). For \(z(x,\varepsilon)\) there arises a differential equation of second order, and \(z(x,\varepsilon)\) is sought in the form of a polynomial in powers of \(\varepsilon\ln^i\varepsilon\). Up to powers \(\varepsilon^{k_0-1}\), we shall take as approximations of a boundary-layer function

\[ v_\varepsilon(x)= z(x,\varepsilon)\bar v(t,\varepsilon) +w(x,\varepsilon)\int_{0}^{t} \frac{(1+u)^{k_0-1}\,du}{(1+u)^{k(-\varepsilon)}}, \]

the corresponding approximations for the product
\(z(x,\varepsilon)\cdot \bar v(t,\varepsilon)\). With the appearance in this product of the power \(\varepsilon^{k_0-1}\), it is also necessary to take the integral term into account. Thus, in cases 1) and 2) we arrive at the theorem:

Theorem 4. If the coefficients of equation (7) satisfy \(k(0)>0,\ a_0(x)\le 0\), and

a) for nonintegral \(k_0=k(0)\):
\(k(x)\in C^{(m+2)}(0,1);\ a_0(x),h(x)\in C^{(m+1)}(0,1)\);

b) for integral \(k_0\):
\(k(x)\in C^{(2m+3)}(0,1);\ a_0(x),h(x)\in C^{(2m+2)}(0,1)\),

then the solution of equation (7) under conditions (7′), for sufficiently small \(\varepsilon\), is representable in the form

\[ y_\varepsilon(x)= \sum_{i,j,k=0}^{l,m,p} \varepsilon^{ik_0+j}\ln^k\varepsilon\,[w_{ij}^{k}(x)+u_{ij}^{k}(x;t)]+z_\varepsilon(x) \]

in the case of nonintegral \(k_0\), and in the form

\[ y_\varepsilon(x)= \sum_{i,j=0}^{m,l} \varepsilon^i\ln^j\varepsilon\,[w_{ij}(x)+u_{ij}(x;t)]+z_\varepsilon(x) \]

in the case of integral \(k_0\), where the functions
\(w_{ij}^{k}(x)\in C^{(m-j+2)}(0,1)\),
\(w_{ij}(x)\in C^{(m-i+2)}(0,1)\);
\(u_{ij}^{k}(x;t)\) and \(u_{ij}(x;t)\) are boundary-layer functions of power type in a neighborhood of the point \(x=0\), and the remainder term \(z_\varepsilon(x)\), together with its derivative, has the following order in \(\varepsilon\):
\(z_\varepsilon(x)=O(\varepsilon^m)\), \(z_\varepsilon'(x)=O(\varepsilon^{m-1})\).

In conclusion I express my deep gratitude to Prof. M. I. Vishik, under whose guidance the present work was carried out.

Moscow Power Engineering Institute

Received
4 VII 1962

REFERENCES

1. E. L. Ains, Ordinary Differential Equations, Kharkov, 1939.
2. M. I. Vishik, L. A. Lyusternik, UMN, 12, no. 5 (77), 3 (1957).
3. M. V. Keldysh, DAN, 27, no. 2, 181 (1951).
4. S. A. Lomov, Tr. Moscow Power Engineering Inst., Mathematics, no. 42 (1962).