MATHEMATICS

A. B. VASIL’EVA and V. A. TUPCHIEV

ASYMPTOTIC FORMULAS FOR THE SOLUTION OF A BOUNDARY-VALUE PROBLEM FOR A SECOND-ORDER EQUATION WITH A SMALL PARAMETER AT THE HIGHEST DERIVATIVE

(Presented by Academician I. G. Petrovskii, July 1, 1960)

In the paper (¹) asymptotic formulas were given for the solution of the boundary-value problem

[
y(0)=0,\qquad y(1)=0
\tag{1}
]

for the equation

[
\mu y''=F(y',y,t),
\tag{2}
]

when (\mu>0) tends to zero. One of the essential requirements of the method proposed in (¹) was the condition (F_{y'}\ne 0) at least in some neighborhood of the solution under consideration.

In the present note it will be shown that, for the solution of an equation of the form

[
\mu^2 y''=F(y,t)
\tag{3}
]

(instead of (\mu), for convenience we write (\mu^2), since, as will be seen below, the asymptotic formulas will contain a quantity equal to the square root of the factor multiplying (y'')), i.e., for the case when (F_{y'}\equiv 0), one can construct asymptotic formulas which in their structure resemble the formulas presented in (¹).

Let us prescribe for (3) the boundary conditions (1). In the paper (²), under the condition of continuity of (F) and (F_y) in some domain (D), (0\le t\le 1), (|y|\le M) ((M) is a certain constant determined by the form of the right-hand side), and under the condition (F_y>0) in (D), the existence was proved, for sufficiently small (\mu), of a solution (y(t,\mu)) of the boundary-value problem (1), and it was also proved that

[
\lim_{\mu\to 0} y(t,\mu)=\varphi(t)\qquad (0<t<1),
\tag{4}
]

where (\varphi(t)) is the solution of the degenerate equation

[
F(y,t)=0,
\tag{5}
]

lying in the domain (D).

Consider the curve consisting of the segment of the curve (y=\varphi(t)) ((0\le t\le 1)) and two straight-line segments (t=0) (from the point (y=0) to the point (y=\varphi(0))) and (t=1) (from the point (y=0) to the point (y=\varphi(1))). Denote by (\alpha) an arbitrarily small, but (\mu)-independent, neighborhood of this curve. For sufficiently small (\mu), the solution (y(t,\mu)) of the boundary-value problem under consideration lies in (\alpha). Suppose that in (\alpha) the function (F(y,t)) has continuous partial derivatives up to order ((n+1)) inclusive, and that (F_y) has continuous partial derivatives up to order (2n) inclusive. Denote by (m) some positive constant such that (F_y\ge m) in (\alpha).

We introduce for consideration a series of auxiliary functions. We write the formal solution of (3) in the form of the expansion

[
y=\bar y_0(t)+\mu \bar y_1(t)+\mu^2\bar y_2(t)+\cdots .
\tag{6}
]

In this case (\bar y_0(t)=\varphi(t)), (\bar y_1(t)=0) (and, in general, (\bar y_{2k+1}(t)=0)), (\bar y_2(t)=\dfrac{\varphi''(t)}{F_y(\varphi(t),t)}), etc. Next, in (3), make the change of variables (\tau_0=\dfrac{t}{\mu}) and (\tau_1=\dfrac{t-1}{\mu}). We obtain two auxiliary systems

[
\frac{d^2 \overset{(0)}{y}}{d\tau_0^2}
=
\overset{(0)}{F}(y,\tau_0\mu),
\tag{7}
]

[
\frac{d^2 \overset{(1)}{y}}{d\tau_1^2}
=
\overset{(1)}{F}(y,1+\tau_1\mu).
\tag{8}
]

Let us construct formal solutions of systems (7) and (8) in the form, respectively, of the expansions

[
\overset{(0)}{y}
=
\overset{(0)}{y}_0+\mu \overset{(0)}{y}_1+\cdots,
\tag{9}
]

[
\overset{(1)}{y}
=
\overset{(1)}{y}_0+\mu \overset{(1)}{y}_1+\cdots.
\tag{10}
]

Then

[
\frac{d^2 \overset{(0)}{y}_0}{d\tau_0^2}
=
\overset{(0)}{F}\bigl(\overset{(0)}{y}_0,0\bigr),
\tag{11}
]

[
\frac{d^2 \overset{(1)}{y}_0}{d\tau_1^2}
=
\overset{(1)}{F}\bigl(\overset{(1)}{y}_0,1\bigr).
\tag{12}
]

The equations for (\overset{(0)}{y}_k) and (\overset{(1)}{y}_k) ((k>0)) will be linear. In order to determine (\overset{(0)}{y}_0) and (\overset{(1)}{y}_0) from (11) and (12), it is necessary to prescribe additional conditions. We prescribe them as follows:

[
\begin{aligned}
\overset{(0)}{y}0\bigm|&=0,
&
\overset{(0)}{y}0\bigm|&=\bar y(0),
\
\overset{(1)}{y}0\bigm|&=0,
&
\overset{(1)}{y}0\bigm|&=\bar y(1).
\end{aligned}
\tag{13}
]

The functions (\overset{(0)}{y}_k), (\overset{(1)}{y}_k) will also be determined by boundary conditions of the following form:

[
\overset{(0)}{y}k\bigm|=0,
]

[
(\mu^k\overset{(0)}{y}k)\bigm|
=
\left[
\mu^k\bar y_k(0)
+
t\mu^{k-1}\bar y_{k-1,t}(0)
+\cdots+
\frac{t^k}{k!}\bar y_{0,t^k}(0)
\right]_{t=1};
\tag{14^0}
]

[
\overset{(1)}{y}k\bigm|=0,
]

[
(\mu^k\overset{(1)}{y}k)\bigm|
=
\left[
\mu^k\bar y_k(1)
+
(t-1)\mu^{k-1}\bar y_{k-1,t}(1)
+\cdots+
\frac{(t-1)^k}{k!}\bar y_{0,t^k}(1)
\right]_{t=0}.
\tag{14^1}
]

From the auxiliary functions thus constructed, we form the combinations:

[
\begin{aligned}
Y_n={}&
\bar y_0+\mu\bar y_1+\cdots+\mu^n\bar y_n
+\overset{(0)}{y}0+\mu\overset{(0)}{y}_1+\cdots+\mu^n\overset{(0)}{y}_n
\
&+\overset{(1)}{y}_0+\mu\overset{(1)}{y}_1+\cdots+\mu^n\overset{(1)}{y}_n
-\bigl(\bar y_0(0)+\mu\bar y_1(0)+t\bar y(0)+\cdots
\
&\qquad\cdots+\mu^n\bar y_n(0)+t\mu^{n-1}\bar y_{n-1,t}(0)+\cdots+\frac{t^n}{n!}\bar y_{0,t^n}(0)\bigr)
\
&-\bigl(\bar y_0(1)+\mu\bar y_1(1)+(t-1)\bar y_{0,t}(1)+\cdots
\
&\qquad\cdots+\mu^n\bar y_n(1)+(t-1)\mu^{n-1}\bar y_{n-1,t}(1)+\cdots+\frac{(t-1)^n}{n!}\bar y_{0,t^n}(1)\bigr).
\tag{15}
\end{aligned}
]

This expression will be the asymptotic formula for the solution (y(t,\mu)) of the boundary-value problem (1) for equation (3), in the sense that the inequality

[
|y(t,\mu)-Y_n|<C\mu^{n+1},
\tag{16}
]

holds, where (C) is some constant independent of (t) and (\mu) for (0\leqslant t\leqslant 1), provided (\mu) is sufficiently small, (\mu\leqslant \mu^0).

Estimate (16) is uniform on the whole interval (0\leqslant t\leqslant 1). If, however, one is interested in the asymptotics on the interval (\varepsilon\leqslant t\leqslant 1-\varepsilon), where (\varepsilon) is an arbitrarily small but fixed number independent of (\mu), then instead of (Y_n) one may use the simpler expression—the partial sum of series (6)

[
\overline{Y}_n=\overline{y}_0(t)+\mu\overline{y}_1(t)+\cdots+\mu^n\overline{y}_n(t).
\tag{17}
]

It can be shown that

[
k!\,\overline{y}k(t)=\lim\,y(t,\mu)}\frac{\partial^k}{\partial\mu^k
]

and, moreover, the derivatives (\dfrac{\partial^k}{\partial\mu^k}y(t,\mu)) are uniformly bounded on the interval (\varepsilon\leqslant t\leqslant 1-\varepsilon). Hence it follows that

[
|y(t,\mu)-\overline{Y}_n|<C\mu^{n+1},
]

where (C) is a constant independent of (t) and (\mu) for (\varepsilon\leqslant t\leqslant 1-\varepsilon) and (\mu\leqslant \mu^0). Since (\overline{y}_{2k+1}=0), it is more convenient, instead of (\overline{Y}_n), to use the expression

[
U_n=\overline{y}0(t)+\mu^2\overline{y}_2(t)+\cdots+\mu^{2n}\overline{y}(t),
\tag{18}
]

which is related to (y(t,\mu)) by the inequality

[
|y(t,\mu)-U_n|<C\mu^{2(n+1)}.
]

The results obtained may be formulated as the following theorem:

Theorem. If (F(y,t)) has, in the domain (\alpha), continuous partial derivatives up to order ((n+1)) inclusive, and (F_y\geqslant m>0) in this domain and has continuous partial derivatives up to order (2n) inclusive, then for the solution (y(t,\mu)) of the boundary-value problem (1) for equation (3) the inequalities

[
\begin{aligned}
|y(t,\mu)-Y_n|&