MATHEMATICS

I. S. LYUBCHENKO

APPROXIMATE SOLUTION OF A BOUNDARY-VALUE PROBLEM FOR A NONLINEAR ORDINARY DIFFERENTIAL EQUATION OF SECOND ORDER WITH A SMALL PARAMETER AT THE HIGHEST DERIVATIVE ON THE BASIS OF NEWTON’S METHOD

(Presented by Academician S. L. Sobolev on December 6, 1960)

The problem of determining the temperature field of a cooled blade of an aviation gas turbine reduces to the following boundary-value problem ((^{1})) on the interval ([0,1]):

[
\varepsilon^{2}\frac{d}{dt}\left[f(t)\frac{dx}{dt}\right]-\Psi(x,t)=0;
\qquad
x(0)=\alpha,\quad x(1)=\beta .
\tag{1}
]

Assume the function (\Psi(u,t)) to be continuous and to have a continuous second derivative with respect to (u) in the domain (0\le t\le 1,\ |u-x_0(t)|\le \delta). The function (f(t)) is twice continuously differentiable and strictly positive on ([0,1]). As the initial approximation we take the twice continuously differentiable function (x_0(t))

[
x_0(t)=
\begin{cases}
x^{*}(t) & \text{on } [\varepsilon,\,1-\varepsilon],\
\bar{x}_0(t) & \text{on } [0,\varepsilon],\ [1-\varepsilon,1].
\end{cases}
]

Here (x^{*}(t)) is the solution of equation (1) for (\varepsilon=0), and (\bar{x}_0(t)) is a twice continuously differentiable function satisfying the boundary conditions of problem (1).

To apply the general theory of Newton’s method, we shall regard the differential equation (1) as a functional equation in the space (x=C^2) of twice continuously differentiable functions satisfying the boundary conditions, with norm

[
|\dot{x}|=\lambda \max_{t\in[0,1]}
\left|\varepsilon^{2}\frac{d}{dt}\left[f(t)\frac{dx}{dt}\right]\right|
+\max_{t\in[0,1]}|x(t)|,
\tag{2}
]

where (\lambda>0) will be determined below.

As the space (y) we consider the space (C) of functions continuous on the interval ([0,1]), with the usual definition of the norm:
[
|y|=\max_{t\in[0,1]}|y(t)|.
]

Consider the operation (\mathcal{P}),

[
y=\mathcal{P}(x),\qquad
y(t)=\varepsilon^{2}\frac{d}{dt}\left[f(t)\frac{dx}{dt}\right]-\Psi(x(t),t).
\tag{3}
]

It is not difficult to verify that (\mathcal{P}) maps the sphere (\Omega_0), (|x-x_0|\le \delta), into the space (C) and has in this sphere continuous first and sec-

of second order. In this case

[
\mathfrak{P}'(z)(\Delta x)(t)=\varepsilon^{2}\frac{d}{dt}\left[f(t)\frac{d(\Delta x)}{dt}\right]-\Psi'_u(z(t),t)\Delta x(t);
\tag{4}
]

[
\mathfrak{P}''(z)(\Delta x,\widetilde{\Delta x})(t)
=-\Psi''_{u^2}(z(t),t)\Delta x(t)\cdot\widetilde{\Delta x}(t).
\tag{5}
]

Consequently, from (4) the element (\Delta x=\Gamma_0(y)) is the solution of the boundary-value problem

[
\varepsilon^{2}\frac{d}{dt}\left[f(t)\frac{d(\Delta x)}{dt}\right]
-\Psi'_u(x_0(t),t)\Delta x=y(t);
\tag{6}
]

[
\Delta x(0)=0,\qquad \Delta x(1)=0.
]

Consider the homogeneous equation corresponding to (6), and by the substitution
(\Delta x=u(t)/\sqrt{f(t)}) reduce it to the form

[
\varepsilon^{2}u''(t)-\left[q^{2}(t)+\varepsilon^{2}r(t)\right]u(t)=0.
\tag{7}
]

Here

[
q^{2}(t)=-\frac{\Psi'_u(x_0(t),t)}{f(t)},\qquad
r(t)=\frac{f''(t)}{2f(t)}-\frac{f'^2(t)}{4f^{2}(t)}.
\tag{8}
]

Under the assumption of uniform boundedness of the exact solution of equation (7), it is easy to show that the approximate solution of equation (7), with accuracy up to a quantity (O(\varepsilon^3)), has the form

[
u(t)=\frac{c_1}{\sqrt{q(t)}}\exp\left(\frac{1}{\varepsilon}\int_{0}^{t}q(\tau)\,d\tau\right)
+\frac{c_2}{\sqrt{q(t)}}\exp\left(-\frac{1}{\varepsilon}\int_{0}^{t}q(\tau)\,d\tau\right).
\tag{9}
]

Now the linear operation (\Gamma), close to (\Gamma_0=[\mathfrak{P}'(x_0)]^{-1}), is easily determined. The Green’s function for the operator

[
\varepsilon^{2}\frac{d}{dt}\left[f(t)\frac{d(\Delta x)}{dt}\right]
-\Psi'_u(x_0(t),t)\Delta x
]

under the conditions (\Delta x(0)=\Delta x(1)=0), taking (8) and (9) into account, has the form:

[
\begin{aligned}
G(t,s)=&-\frac{\varepsilon f(0)}{2}
\left[
\exp\left(\frac{1}{\varepsilon}\int_{0}^{t}\left(\frac{\Psi'u}{f}\right)^{1/2}d\tau\right)
-\exp\left(-\frac{1}{\varepsilon}\int\left(\frac{\Psi'}^{tu}{f}\right)^{1/2}d\tau\right)
\right]\times\
&\times
\left[
\exp\left(-\frac{1}{\varepsilon}\int\left(\frac{\Psi'}^{su}{f}\right)^{1/2}d\tau\right)
-\exp\left(-\frac{2}{\varepsilon}\int\left(\frac{\Psi'}^{1u}{f}\right)^{1/2}d\tau\right)
\exp\left(\frac{1}{\varepsilon}\int\left(\frac{\Psi'}^{su}{f}\right)^{1/2}d\tau\right)
\right]\times\
&\times
\left{
[\Psi'_u(x_0,t)f(t)]^{1/4}[\Psi'_u(x_0,s)f(s)]^{1/4}
\left[
1-\exp\left(-\frac{2}{\varepsilon}\intd\tau\right)}^{1}\left(\frac{\Psi'_u}{f}\right)^{1/2
\right]
\right}^{-1}
\end{aligned}
]

for (0\leq t\leq s);

[
\tag{10}
]

[
\begin{aligned}
G(t,s)=&-\frac{\varepsilon f(0)}{2}
\left[
\exp\left(-\frac{1}{\varepsilon}\int_{0}^{t}\left(\frac{\Psi'u}{f}\right)^{1/2}d\tau\right)
-\exp\left(-\frac{2}{\varepsilon}\int\left(\frac{\Psi'}^{1u}{f}\right)^{1/2}d\tau\right)
\times
\right.\
&\left.\qquad\qquad\qquad\qquad\qquad\qquad\times
\exp\left(\frac{1}{\varepsilon}\int\left(\frac{\Psi'}^{tu}{f}\right)^{1/2}d\tau\right)
\right]
\left[
\exp\left(\frac{1}{\varepsilon}\int\left(\frac{\Psi'}^{su}{f}\right)^{1/2}d\tau\right)
-\exp\left(-\frac{1}{\varepsilon}\int\left(\frac{\Psi'}^{su}{f}\right)^{1/2}d\tau\right)
\right]\times\
&\times
\left{
[\Psi'_u(x_0,t)f(t)]^{1/4}[\Psi'_u(x_0,s)f(s)]^{1/4}
\left[
1-\exp\left(-\frac{2}{\varepsilon}\intd\tau\right)}^{1}\left(\frac{\Psi'_u}{f}\right)^{1/2
\right]
\right}^{-1}
\end{aligned}
]

for (s\leq t\leq 1).

Let us coordinate the norm of (X) with the norm of (Y), so that the operation (G) carries out an isometry between these spaces. Since the boundary-value problem (6) has a unique solution, we have

[
\Delta x=\int_{0}^{1}G(t,s)y(s)\,ds.
\tag{11}
]

Hence it follows that

[
\max_{t\in[0,1]}|\Delta x(t)|\leqslant
\max_{t,s\in[0,1]}|G(t,s)|\,|y|.
\tag{12}
]

From (6) we have

[
\max\left|\varepsilon^{2}\frac{d}{dt}\left[f(t)\frac{d(\Delta x)}{dt}\right]\right|\leqslant
\max_{t\in[0,1]}|\Psi'u(x_0(t),t)|\cdot
\max|\Delta x(t)|+|y|.
\tag{13}
]

Substituting (12) into (13), we obtain

[
\max\left|\varepsilon^{2}\frac{d}{dt}\left[f(t)\frac{d(\Delta x)}{dt}\right]\right|
\leqslant \theta|y|,
\tag{14}
]

where

[
\theta=\max_{t\in[0,1]}|\Psi'u(x_0(t),t)|\cdot
\max|G(t,s)|+1.
]

Substituting, finally, (12) and (14) into (2), we obtain

[
|\Delta x|=(\lambda\theta+\max_{t,s\in[0,1]}|G(t,s)|)|y|.
\tag{15}
]

From (15) it is clear that

[
|\Gamma|\leqslant(\lambda\theta+\max_{t,s\in[0,1]}|G(t,s)|).
\tag{16}
]

Estimating (\max\limits_{t,s\in[0,1]}|G(t,s)|), we have

[
\max_{t,s\in[0,1]}|G(t,s)|\leqslant
]

[
\leqslant
M\varepsilon\left|
\frac{
\exp\left(-\frac{2}{\varepsilon}\int_{t}^{1}\left(\frac{\Psi'u}{f}\right)^{1/2}d\tau\right)
+\exp\left(-\frac{2}{\varepsilon}\int\left(\frac{\Psi'}^{tu}{f}\right)^{1/2}d\tau\right)
-\exp\left(-\frac{2}{\varepsilon}\int\left(\frac{\Psi'}^{1u}{f}\right)^{1/2}d\tau\right)-1
}{
1-\exp\left(-\frac{2}{\varepsilon}\intd\tau\right)}^{1}\left(\frac{\Psi'_u}{f}\right)^{1/2
}
\right|
\leqslant
]

[
\leqslant M\varepsilon,
\tag{17}
]

where

[
M=\frac{f(0)}{2}\max_{t\in[0,1]}|(\Psi'_u f)^{-1/2}|.
]

Thus,

[
|\Gamma|\leqslant(M\varepsilon+\lambda\theta).
\tag{18}
]

Applying theorem 1 (2.XVIII)(^{(2)}) of L. V. Kantorovich, we carry out the estimates

[
1.\qquad [\Gamma(P(x_0))]y=\int_{0}^{1}G(t,s)y_0(s)y(s)\,ds=
]

[
=\left{\int_{0}^{\varepsilon}+\int_{\varepsilon}^{1-\varepsilon}+\int_{1-\varepsilon}^{1}\right}
(G(t,s)y_0 y\,ds)
\leqslant MN\varepsilon^{3}(1-2\varepsilon)|y|+2MQ\varepsilon^{2}|y|=
]

[
=[MN\varepsilon^{3}(1-2\varepsilon)+2MQ\varepsilon^{2}]|y|.
]

But

[
|[\Gamma(P(x_0))]y|\leqslant|\Gamma(P(x_0))||y|.
]

Consequently,

[
|\Gamma(P(x_0))|\leqslant
[MN\varepsilon^{3}(1-2\varepsilon)+2MQ\varepsilon^{2}]=O(\varepsilon^{2}).
\tag{19}
]

Here

[
N=\max_{t\in[0,1]}\left|\frac{d}{dt}\left[f(t)\frac{dx^*(t)}{dt}\right]\right|,
]

[
Q=\max_{t\in[0,1]}\left|\varepsilon^2\frac{d}{dt}\left[f(t)\frac{d\bar x_0(t)}{dt}\right]-\Psi(\bar x_0(t),t)\right|.
]

2.

[
|\Gamma P''(x)|\leq |\Gamma|\,|P''(x)|\leq (M\varepsilon+\lambda\theta)\max_{t\in[0,1]}|\Psi_{u}'(x,t)|=
]
[
=K(M\varepsilon+\lambda\theta),
\tag{20}
]

where

[
K=\max_{t\in[0,1]}|\Psi_{u}'(x(t),t)|\quad (x\in\Omega_0).
]

3.

[
|\Gamma P'(x_0)-I|=|\Gamma[P'(x_0)-P'(x)]|
=\left|\int_x^{x_0}\Gamma P''(x)\,dx\right|\leq
]
[
\leq K(M\varepsilon+\lambda\theta)|x_0-x|\leq K\delta(M\varepsilon+\lambda\theta).
\tag{21}
]

The solvability conditions for the boundary-value problem (1) are represented in the form

[
\bar h=\frac{2MQK\varepsilon^2(M\varepsilon+\lambda\theta)}
{[1-K\delta(M\varepsilon+\lambda\theta)]^2}\leq \frac12;\qquad
K\delta(M\varepsilon+\lambda\theta)<1.
\tag{22}
]

For sufficiently small (\lambda) (in absolute value), the solvability conditions (22) are satisfied.

The solution of the boundary-value problem (1), (x^{**}(t)), exists for

[
r\geq \bar r_0=\frac{1-\sqrt{1-2\bar h}}{\bar h}\,
\frac{O(\varepsilon^2)}{1-K\delta(M\varepsilon+\lambda\theta)}
]

and is unique for

[
r<\bar r_1=\frac{1+\sqrt{1-2\bar h}}{\bar h}\,
\frac{O(\varepsilon^2)}{1-K\delta(M\varepsilon+\lambda\theta)}.
]

At the same time, the estimate of convergence of the initial approximation (x_0(t)) to the exact solution (x^{**}(t)), according to Theorem 1 (2.XVIII) ({}^{(2)}) of L. V. Kantorovich, will be

[
|x^{**}-x_0|\leq
\frac{4MQ\varepsilon^2}{1-K\delta(M\varepsilon+\lambda\theta)}
=O(\varepsilon^2).
\tag{23}
]

But it follows from this that, as (\varepsilon\to0), the solution of the boundary-value problem (1), (x^{*}(t)), on the interior interval ([\varepsilon,1-\varepsilon]) converges uniformly to the solution (x^(t)) of the degenerate equation (\Psi(x(t),t)=0) with rate (O(\varepsilon^2)).

It is easy to see that by the indicated method one can find an approximate solution of boundary-value problems for the equation (\varepsilon^2 y''=f(x,y,y')) and give the corresponding estimates.

The author expresses gratitude to S. N. Slugin for valuable advice in writing this note.

Received
11 XI 1960

CITED LITERATURE

({}^{1}) G. A. Tirskii, V. A. Trenogin, Izv. AN SSSR, OTN, No. 2 (1959).

({}^{2}) L. V. Kantorovich, G. P. Akilov, Functional Analysis in Normed Spaces, 1959.