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:

\[ \begin{aligned} \overset{(0)}{y}_0\bigm|_{t=0}&=0, & \overset{(0)}{y}_0\bigm|_{t=1}&=\bar y(0), \\ \overset{(1)}{y}_0\bigm|_{t=1}&=0, & \overset{(1)}{y}_0\bigm|_{t=0}&=\bar y(1). \end{aligned} \tag{13} \]

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

\[ \overset{(0)}{y}_k\bigm|_{t=0}=0, \]

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

\[ \overset{(1)}{y}_k\bigm|_{t=1}=0, \]

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

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

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

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

\[ k!\,\overline{y}_k(t)=\lim_{\mu\to 0}\frac{\partial^k}{\partial\mu^k}\,y(t,\mu) \]

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

\[ U_n=\overline{y}_0(t)+\mu^2\overline{y}_2(t)+\cdots+\mu^{2n}\overline{y}_{2n}(t), \tag{18} \]

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

\[ \begin{aligned} |y(t,\mu)-Y_n|&<C\mu^{n+1} &&(0\leqslant t\leqslant 1),\\ |y(t,\mu)-U_n|&<C\mu^{2(n+1)} &&(\varepsilon\leqslant t\leqslant 1-\varepsilon), \end{aligned} \]

hold, where the functions \(Y_n, U_n\) are defined respectively by formulas (15) and (18); \(\varepsilon\) is an arbitrarily small number independent of \(\mu\); \(C\) is a constant independent of \(\mu\) for \(\mu\leqslant\mu^0\).

Moscow State University
named after M. V. Lomonosov

Received
30 VI 1960

REFERENCES

1. A. B. Vasil’eva, DAN, 124, No. 3 (1959).
2. N. I. Brish, DAN, 95, No. 3 (1954).