Abstract
Full Text
MATHEMATICS
Yu. F. KOROBEINIK
ON ANALYTIC SOLUTIONS OF THE TRICOMI EQUATION
(Presented by Academician M. A. Lavrent’ev on 8 IX 1961)
§ 1. A real function \(z(x,y)\) of real variables \(x,y\) is called a regular solution of the Tricomi equation
\[ y\,\frac{\partial^{2}z}{\partial x^{2}}+\frac{\partial^{2}z}{\partial y^{2}}=0 \tag{1} \]
in a domain \(D\) if it is continuous in this domain together with its first partial derivatives and satisfies equation (1) there.
We shall call a regular solution \(z(x,y)\) a real analytic integral of equation (1) in a neighborhood of a point \((x_0,y_0)\) from \(D\) if the function \(z(x,y)\) is analytic in some rectangle \(|x-x_0|<a,\ |y-y_0|<b\), i.e. is expandable in a double power series
\[ \sum_{n,m} b_{n,m}(x-x_0)^n(y-y_0)^m, \]
absolutely convergent inside this rectangle. It is clear that every real integral \(z(x,y)\) of equation (1) is also defined for complex values of \(x,y\). It is an analytic function in the bicylinder \(|x-x_0|<a,\ |y-y_0|<b\), taking real values for real \(x\) and \(y\). If the latter condition is dropped, then it is natural to call a complex-valued function \(z(x,y)\), analytic in some bicylinder \(|x-x_0|<r_1,\ |y-y_0|<r_2\) and satisfying the equation there, a complex analytic (or simply analytic) integral of equation (1) in a neighborhood of \((x_0,y_0)\).
Every analytic integral \(z(x,y)\) in a neighborhood of \((x_0,y_0)\) can be represented in the form \(z(x,y)=v_1(x,y)+iv_2(x,y)\), where \(v_1\) and \(v_2\) are real analytic integrals in a neighborhood of \((x_0,y_0)\). If by \(R_1\) we denote the class of analytic integrals, and by \(R_2\) the class of real analytic integrals, then \(R_1 \supset R_2\).
Let \(z(x,y)\) be an analytic integral of equation (1) in a neighborhood of \((0,0)\). Then the function \(z(x,y)\) is analytic in some bicylinder \(|x|<\rho_1,\ |y|<\rho_2\) and is represented as follows:
\[ z(x,y)= \sum_{n=0}^{\infty} \frac{(-1)^n f_0^{(2n)}(x)}{(3n)!} \prod_{k=0}^{\,n-1}(3k+1)y^{3n} + \sum_{n=0}^{\infty} \frac{(-1)^{n+1} f_1^{(2n)}(x)}{(3n+1)!} \prod_{k=0}^{\,n}(3k-1)y^{3n+1}; \tag{2} \]
here \(f_0(x)\) and \(f_1(x)\) are arbitrary functions analytic in the disk
\[ |x|<\rho_1+\frac{2}{3}(\rho_2)^{3/2}. \]
Conversely, if the function \(z(x,y)\) has the form (2) and the functions \(f_0(x), f_1(x)\) are analytic in the disk \(|x|<\rho\), then \(z(x,y)\) is a solution of equation (1), analytic in any bicylinder
\[ |x|<R_1,\qquad |y|<\left[\frac{3}{2}(\rho-R_1)\right]^{2/3}, \]
whatever \(R_1<\rho\) may be. In particular, the general form of an integral entire in \(x,y\) (i.e. analytic in every bicylinder \(|x|<R_1,\ |y|<R_2\))
of the Tricomi equation is given by formula (2), where \(f_0(x)\) and \(f_1(x)\) are arbitrary entire functions. Formula (2) also gives the general form of a real analytic integral in a neighborhood of \((0,0)\), if the functions \(f_0(x)\) and \(f_1(x)\) are regarded as functions analytic at \(x=0\) with real Taylor coefficients.
In the present note we shall consider several problems for analytic solutions of the Tricomi equation that reduce to differential equations of infinite order.
§ 2. Problem \(\mathrm{T}_1\). Find a solution of equation (1), entire in \(x,y\), satisfying, for all finite values of \(x\), the conditions
\[ u(x,0)=\varphi_0(x), \qquad u\bigl(x,c\sqrt[3]{x}\bigr)=\varphi_1(x). \tag{3} \]
Here \(\varphi_0(x)\), \(\varphi_1(x)\) are given functions, and \(c\) is a given complex number. From representation (2) we find that \(f_0(x)=\varphi_0(x)\), while the function \(f_1(x)\) is determined from the differential equation of infinite order
\[ f_1(x)+\sum_{n=1}^{\infty} \frac{(-1)^{n+1} f_1^{(2n)}(x)}{(3n+1)!} \prod_{k=0}^{n}(3k-1)c\sqrt[3]{x}\,c^{3n}x^n \ = \]
\[ =\varphi_1(x)-\sum_{n=0}^{\infty} \frac{(-1)^n \varphi_0^{(2n)}(x)}{(3n)!} \prod_{k=0}^{\,n-1}(3k+1)c^{3n}x^n . \tag{4} \]
It is easy to see that, for the solvability of problem \(\mathrm{T}_1\) in the class of integrals analytic in a neighborhood of \((0,0)\), it is necessary that the equality
\[ \varphi_1(x)-\sum_{n=0}^{\infty} \frac{(-1)^n \varphi_0^{(2n)}(x)}{(3n)!} \prod_{k=0}^{\,n-1}(3k+1)c^{3n}x^n = \sqrt[3]{x}\,\nu(x), \]
hold, where \(\nu(x)\) is a function analytic at the point \(x=0\). If this condition is satisfied, then equation (4) is rewritten as
\[ f_1(x)+\sum_{n=1}^{\infty} \frac{(-1)^n f_1^{(2n)}(x)}{(3n+1)!} \prod_{k=1}^{n}(3k-1)c^{3n}x^n = \nu_1(x), \]
where \(\nu_1(x)=\nu(x)/c\).
Investigating the last equation with the aid of work (1), we arrive at the following result:
Theorem 1. Problem \(\mathrm{T}_1\) is solvable if \(\varphi_0(x)\) is an entire function of order \(<1/3\), and \(\varphi_1(x)/\sqrt[3]{x}\) is an entire function of order \(<1/2\). Uniqueness holds in the class of solutions entire in \(x,y\) for which \(z(x,0)\) is an entire function of order \(<1/3\), and \(\partial z/\partial y|_{y=0}\) is an entire function of order \(<1/2\).
Analogous results hold for the problem with conditions
\[ \left.\frac{\partial z}{\partial y}\right|_{y=0}=\varphi_1(x), \qquad z\bigl(x,c\sqrt[3]{x}\bigr)=\varphi_0(x). \]
Remark. If \(c\) is a real number and the Taylor coefficients of the entire functions \(\varphi_0(x)\) and \(\varphi_1(x)\) are real, then the solution of problem \(\mathrm{T}_1\) will be a real integral entire in \(x,y\).
§ 3. Problem \(\mathrm{T}_2\). Find a solution \(z(x,y)\), analytic in a neighborhood of \((0,0)\), of equation (1), for which
\[ z(x,0)=\varphi_0(x); \qquad z\bigl(x,c\sqrt[3]{x^2}\bigr)=\varphi_1(x) \quad \text{for } |x|\le h. \tag{5} \]
As in problem \(\mathrm{T}_1\), \(\varphi_0(x)\), \(\varphi_1(x)\) are given functions, and \(c\) is a complex number.
The solution of problem \(T_2\) will be found if from conditions (5) we determine the functions \(f_0(x)\) and \(f_1(x)\). As before, \(f_0(x)=\varphi_0(x)\), while \(f_1(x)\) satisfies the equation
\[ c x^{2/3} f_1(x) = \sum_{n=1}^{\infty} \frac{(-1)^n f_1^{(2n)}(x)}{(3n+1)!} \prod_{k=1}^{n}(3k-1)c^{3n+1}x^{2n+2/3} = \]
\[ = \varphi_1(x) - \sum_{n=0}^{\infty} \frac{(-1)^n \varphi_0^{(2n)}(x)}{(3n)!} \prod_{k=1}^{n-1}(3k+1)c^{3n}x^{2n}. \tag{6} \]
For the existence of an integral of problem \(T_2\) analytic in a neighborhood of \((0,0)\), it is necessary that the function
\[ \nu(x) = c^{-1}x^{-2/3} \left[ \varphi_1(x) - \sum_{n=0}^{\infty} \frac{(-1)^n f_0^{(2n)}(x)}{(3n)!} \prod_{k=0}^{n-1}(3k+1)c^{3n}x^{2n} \right] \]
be analytic at \(x=0\). If this circumstance holds, then equation (6) can be rewritten as follows:
\[ f_1(x) + \sum_{n=1}^{\infty} \frac{(-1)^n f_1^{(2n)}(x)}{(3n+1)!} \prod_{k=1}^{n}(3k-1)c^{3n}x^{2n} = \nu(x). \tag{7} \]
Thus, in order to determine the sought function \(f_1(x)\), we have an Euler differential equation of infinite order. Let
\[ f_1(x)=\sum_{m=1}^{\infty}c_m x^m, \qquad \nu(x)=\sum_{m=0}^{\infty}g_m x^m. \]
If by a solution of equation (7) one understands a function \(f_1(x)\) such that, upon substituting it into the equation, its left-hand side converges uniformly to \(\nu(x)\) in some neighborhood of the origin of coordinates (and only such solutions are needed by us), then equation (7), in the class of these solutions, is equivalent to the system
\[ c_m\beta_m=g_m,\qquad m=0,1,2,\ldots, \tag{8} \]
where
\[ \beta_m = \sum_{n=0}^{m} \frac{\alpha_n m!}{(m-n)!}, \qquad \alpha_0=1, \qquad \alpha_{2n-1}=0, \]
\[ \alpha_{2n} = \frac{(-1)^n c^{3n}\prod_{k=1}^{n}(3k-1)}{(3n+1)!}, \qquad n=1,2,\ldots;\quad m=1,0,\ldots \tag{9} \]
Assume first that \(c\) is a real negative number. Then for the numbers \(\beta_m\) it is not difficult to obtain the estimate
\[ \frac{d}{m^{3/2}}(1+\bar{x})^m \leq |\beta_m| \leq A(1+\bar{x})^m, \]
\[ m=1,2,\ldots,\qquad A<\infty,\qquad d=-c^3,\qquad \bar{x}=\sqrt[2/3]{d}, \]
from which the estimate for the coefficients \(c_m\) also follows immediately. The final result can be formulated in the following form:
Theorem 2. Let \(c<0\), let the function \(\varphi_0(x)\) be analytic in the disk \(|x|<\rho\), and let \(\varphi_1(x)\) have the form \(\varphi_1(x)=\lambda(x)+x^{2/3}\mu(x)\), where \(\lambda(x)\) and \(\mu(x)\) are functions analytic in the disk
\[ |x|<\frac{\rho}{1+\sqrt[2/3]{|c|^3}}. \]
Then problem \(T_2\) has a solution \(z(x,y)\), analytic in the bicylinder
\[ |x|<R,\qquad |y|<\left[\frac{3}{2}(\rho-R)\right]^{2/3} \]
for ...
for any \(R<\rho\). The condition \(z(x,0)=\varphi_0(x)\) is satisfied in the disk \(|x|<\rho\), and the condition \(z(x,c\sqrt[3]{x^2})=\varphi_1(x)\) in the domain
\[ |x|<\frac{\rho}{1+\frac{2}{3}|c|^{3/2}}. \]
The solution of problem \(\mathrm{T}_2\) is unique in the class of all functions \(v(x,y)\) analytic in a neighborhood of \((0,0)\).
If the functions \(\varphi_0(x)\) and \(\varphi_1(x)\) have real Taylor coefficients, then the solution of problem \(\mathrm{T}_2\) will be a real analytic integral in a neighborhood of \((0,0)\).
Let us now consider the case where \(c\) is an arbitrary complex number. From formulas (9) it is not difficult to obtain that
\[ \lim_{m\to\infty}\sqrt[m]{|\beta_m|}\leq 1+\frac{2}{3}|c|^{3/2}. \]
Moreover, the numbers \(\beta_m/m!\) are the Taylor coefficients of the entire function
\[ h(z)=e^z\omega(c^{3/2}z), \qquad \omega(z)=1+\sum_{n=1}^{\infty} \frac{(-1)^n\prod_{k=1}^{n}(3k-1)}{(3n+1)!}\,x^{2n}, \]
an exponential function of type \(2/3\).
Theorem 3. Suppose that the following assumptions are satisfied:
1) all Taylor coefficients \(\beta_m/m!\) of the function \(h(z)\) are nonzero, and
\[ \lim_{m\to\infty}\sqrt[m]{|\beta_m|}=a>0; \]
2) the function \(\varphi_0(z)\) is analytic in the disk \(|x|<R\), and
\[ \varphi_1(x)=\lambda(x)+x^{2/3}\mu(x), \]
where \(\lambda(x)\), \(\mu(x)\) are functions analytic in the disk
\[ |x|<\frac{R}{1+\frac{2}{3}|c|^{3/2}}. \]
Then problem \(\mathrm{T}_2\) has a solution \(z(x,y)\), unique in the class of functions analytic in a neighborhood of \((0,0)\). The solution \(z(x,y)\) is analytic in the bicylinder
\[ |x|<r,\quad |y|< \left[ \frac{3}{2} \left( \frac{aR}{1+\frac{2}{3}|c|^{3/2}}-r \right) \right]^{2/3} \]
for any
\[ r<\frac{aR}{1+\frac{2}{3}|c|^{3/2}}. \]
In conclusion, we note that the requirement that the coefficients \(\beta_m\) not vanish is essential, since otherwise the uniqueness of the solution of the problem is violated. In particular, if \(c\) is a positive number, it can be shown that, if problem \(\mathrm{T}_2\) is solvable at all, then it necessarily has several solutions analytic in a neighborhood of \((0,0)\).
Rostov-on-Don State University
Received
19 VIII 1961
REFERENCES
- Yu. F. Korobeinik. Matem. sborn., 49 (91), no. 2, 191 (1959).