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SOLUTION OF AN INFINITE BOUNDARY VALUE PROBLEM FOR AN EQUATION OF THE THIRD ORDER
T. K. KRAVCHENKO, A. I. YABLONSKII
Consider the differential equation
\[ F''' + cF''F + mF'^2 = 0, \tag{1} \]
where \(c\) and \(m\) are constants and \(c \ne 0\), under the following boundary conditions:
\[ F(0)=0,\quad F'(0)=-1,\quad F'(\infty)=0. \tag{2} \]
Boundary value problems of this kind occur in the theory of the boundary layer.
In the present note a representation is given for the solution of this boundary value problem in the form of a uniformly and absolutely convergent series on the half-axis \(x > -\varepsilon\), where \(\varepsilon\) is a certain positive number.
We seek a formal solution of equation (1) in the form of a Dirichlet series
\[ F(x)=\frac{\gamma}{c}+\gamma\sum_{i=1}^{\infty} b_i a^i e^{-i\gamma x}. \tag{3} \]
Direct substitution of (3) into (1) gives
\[ -\sum_{i=1}^{\infty} i^3 b_i a^i e^{-i\gamma x} +\sum_{i=1}^{\infty} i^2 b_i a^i e^{-i\gamma x} + \]
\[ +c\sum_{i=2}^{\infty}\sum_{l=1}^{i-1} l^2 b_l b_{i-l} a^i e^{-i\gamma x} +m\sum_{i=2}^{\infty}\sum_{l=1}^{i-1} l(i-l)b_l b_{i-l} a^i e^{-i\gamma x}=0. \tag{4} \]
For \(i=1\) we obtain from (4)
\[ -b_1a+b_1a=0. \]
We set \(|b_1|=1\); then \(a\) is an arbitrary parameter. We rewrite (4) in the form
\[ \sum_{i=2}^{\infty} \left\{ -i^2(i-1)b_i+ \sum_{l=1}^{i-1} [cl^2+ml(i-l)]\,b_l b_{i-l} \right\} a^i e^{-i\gamma x}=0, \tag{4_1} \]
whence we obtain recurrence formulas for determining \(b_i\) \((i=2,3,\ldots)\):
\[ b_i=\frac{1}{i^2(i-1)} \sum_{l=1}^{i-1} [cl^2+ml(i-l)]\,b_l b_{i-l},\quad i=2,3,\ldots \tag{5} \]
If \(|a|<1\) and \(|b_i|\leq 1,\ i=1,2,\ldots,\) then the series (3) converges absolutely for any \(\gamma>0\) at \(x=-\varepsilon\), \(\varepsilon=-\left(\dfrac{\ln |a|}{\gamma}+\delta\right)>0\), where \(\delta>0\) is a sufficiently small number depending on \(a\) and \(\gamma\), and then this series converges absolutely and uniformly on the half-axis \(x>-\varepsilon\).*
From \(|b_i|\leq 1\ (i=1,2,\ldots)\) we obtain conditions on the coefficients \(c\) and \(m\) of equation (1) ensuring the absolute and uniform convergence of the series (3) on the half-axis \(x>-\varepsilon\):
\[ |b_1|=1,\quad |b_2|=\frac14 |c+m|, \]
whence
\[ |c+m|\leq 4, \tag{6} \]
\[ |b_3|=\frac1{18}|5c+4m|,\quad \text{taking (6) into account,} \]
\[ |c|\leq 2. \tag{7} \]
Suppose that, for \(|c+m|\leq 4\) and \(|c|\leq 2\), \(|b_n|\leq 1\). Applying induction, we see that
\[ |b_{n+1}|\leq \frac{1}{(n+1)^2 n} \left| \sum_{l=1}^{n} [cl^2+ml(n+1-l)] \right| = \]
\[ = \frac{1}{6(n+1)} \left|[(n+2)(m+c)+(n-1)c]\right| \leq \frac{(n+2)4+(n-1)2}{6(n+1)} = \]
\[ = \frac{3n+3}{3(n+1)}=1. \]
Consequently, for \(|c+m|\leq 4\) and \(|c|\leq 2\), \(|b_i|\leq 1,\ i=1,2,\ldots\).
We have obtained a two-parameter family of solutions of equation (1), defined on the half-axis \(x>-\varepsilon\) by the series (3) and satisfying the condition \(\lim\limits_{x\to+\infty} F'(x)=0\).
For the solution of the boundary-value problem (2), it is sufficient that the equalities
\[ F(0)=\frac{\gamma}{c}+\gamma \sum_{i=1}^{\infty} b_i a^i=0 \tag{8} \]
and
\[ F''(0)=\gamma^3 \sum_{i=1}^{\infty} i^2 b_i a^i=-1 \tag{9} \]
be fulfilled for \(\gamma>0\) and \(|a|<1\).
Let \(c\) and \(m\) satisfy conditions (6), (7) and, in addition, \(m^2-c^2<0\).
From (5) it is easy to see that
\[ b_i= \frac{1}{2i^2(i-1)} \sum_{l=1}^{i-1} [cl^2-2(c-m)il+2(c-m)l^2]\,b_l b_{i-l}, \quad i=2,3,\ldots. \tag{5₁} \]
Since the quadratic trinomial \(c\lambda^2-2(c-m)\lambda+2(c-m)\), for \(m^2-c^2<0\), preserves the sign of \(c\), it follows that
* The series converges in the half-plane \(\operatorname{Re} x>-\varepsilon\) of the complex \(x\)-plane [1].
for \(c>0\) and \(b_1=1\), \(b_i>0\) \((i=2,3,\ldots)\),
and
for \(c<0\) and \(b_1=-1\), \(b_i<0\) \((i=2,3,\ldots)\).
The first is obvious; in the second case the expression in square brackets is always negative, and if \(b_1<0\), then \(b_2<0\), and it is easy to see that \(b_i<0\) \((i=3,4,\ldots)\).
Taking into account the constancy of sign of \(b_i\) \((i=1,2,\ldots)\), condition (8) can be fulfilled only when \(a<0\); hence \(F(x)\), as well as \(F'(x)\) and \(F''(x)\), in our problem are represented by alternating series, which makes it possible easily to give an error estimate for computations performed by the proposed method.
In particular, for equation (1), when \(c=\dfrac{2}{3}\) and \(m=-\dfrac{1}{3}\), conditions (6), (7) are satisfied and \(b_i>0\) \((i=1,2,\ldots)\). In this case the parameters \(a\) and \(\gamma\) have been determined and a table of values for \(x:0(0.05)8\) has been compiled. The computations were carried out on the “Ural” computer.
References
- A. I. Markushevich. Theory of Analytic Functions. GITTL, 1950.
Received by the editors
December 1, 1964
Institute of Mathematics and Computational Technology
Academy of Sciences of the BSSR