REPRESENTATION OF THE SOLUTION OF A CERTAIN CAUCHY PROBLEM IN THE FORM OF AN INTEGRAL RESIDUE

N. A. Aliev, I. S. Zeinalov

Consider the problem of finding a solution of the differential equation

\[ \sum_{k=0}^{m} a_k \frac{\partial^n u}{\partial x^{\,n-k}\partial y^k}=0 \tag{1} \]

under the initial conditions:

\[ \left.\frac{\partial^k u}{\partial y^k}\right|_{y=0} =\varphi_k(x)\quad (k=0,\overline{m-1}), \tag{2} \]

where \(a_k\) are constant numbers, \(\varphi_k(x)\) are, generally speaking, complex-valued functions of the real variable \(x\in(-\infty,\infty)\), and \(m\) and \(n\) are natural numbers, with \(0<m\le n\), \(a_m\ne0\).

It turns out that for the solution of problem (1), (2) one can obtain a compact explicit representation in the form of an integral residue.

The formula obtained for the solution of this problem is very convenient for computation.

We have the following result.

Theorem. If \(\varphi_k(z)\) \((k=0,\overline{m-1})\) are analytic functions in the complex \(z\)-plane except for the infinitely distant point,* then for the solution of the Cauchy problem (1), (2) we have the following representation

\[ u(x,y)=E_\lambda\frac{1}{F(\lambda)} \sum_{p=1}^{m}\sum_{q=0}^{m-p} a_{m-q}\lambda^{m-p-q}\times \]

\[ \times \int_{0}^{x+\lambda y}d\xi_1 \int_{0}^{\xi_1}d\xi_2\cdots \int_{0}^{\xi_{p-2}} \varphi_{p-1}(\xi_{p-1})\,d\xi_{p-1}, \tag{3} \]

where

\[ F(\lambda)=\sum_{r=0}^{m} a_r\lambda^r \tag{4} \]

is the characteristic polynomial for equation (1), and \(E_\lambda\) denotes the total integral residue of the function standing under the sign \(E_\lambda\), over all \(\lambda\)-roots of the characteristic polynomial (4).

It is easy to verify that expression (3) is a solution of the Cauchy problem (1), (2). Indeed, differentiating (3) \(k\) times with respect to \(y\), we obtain**

* In particular, if all roots of the characteristic polynomial turn out to be real, it is sufficient to assume analyticity of the functions \(\varphi_k(x)\) only on the real axis.

** The differentiation may be carried out under the residue sign, because the function standing under the residue sign has a finite number of poles.

\[ \frac{\partial^k u}{\partial y^k} = E_\lambda \frac{\lambda^k}{F(\lambda)} \left\{ \sum_{p=1}^{k}\sum_{q=0}^{m-p} a_{m-q}\lambda^{m-p-q}\, \varphi_{p-1}^{(k-p+1)}(x+\lambda y) +\right. \]

\[ \left. +\sum_{q=0}^{m-k-1} a_{m-q}\lambda^{m-k-1-q}\, \varphi_k(x+\lambda y) +\right. \]

\[ \left. +\sum_{p=k+2}^{m}\sum_{q=0}^{m-p} a_{m-q}\lambda^{m-p-q} \int_0^{x+\lambda y} d\xi_{k+1} \int_0^{\xi_{k+1}} d\xi_{k+2}\cdots \int_0^{\xi_{p-2}}\varphi_{p-1}(\xi_{p-1})\,d\xi_{p-1} \right\}. \tag{5} \]

Differentiating (5) up to order \((n-k)\) with respect to \(x\), we obtain

\[ \frac{\partial^n u}{\partial x^{\,n-k}\partial y^k} = E_\lambda \frac{\lambda^k}{F(\lambda)} \sum_{p=1}^{m}\sum_{q=0}^{m-p} a_{m-q}\lambda^{m-p-q} \varphi_{p-1}^{(n-p+1)}(x+\lambda y). \tag{6} \]

It is clear from expression (6) that the function standing under the sign of the double sum does not depend on \(k\).

Taking the above into account, by direct verification we see that (3) satisfies equation (1).

The fact that the function defined by formula (3) satisfies the initial condition (2) can be shown as follows. The second of the terms on the right-hand side of (5) for \(y=0\) gives \(\varphi_k(x)\).

Computing the first of these terms for \(y=0\), we obtain

\[ E_\lambda \frac{\lambda^k}{F(\lambda)} \sum_{p=1}^{k}\sum_{q=0}^{m-p} a_{m-q}\lambda^{m-p-q} \varphi_{p-1}^{(k-p+1)}(x) = \]

\[ = E_\lambda \sum_{p=1}^{k} \lambda^{k-p}\varphi_{p-1}^{(k-p+1)}(x) \frac{\displaystyle\sum_{q=0}^{m-p} a_{m-q}\lambda^{m-q}} {F(\lambda)} = \]

\[ = E_\lambda \sum_{p=1}^{k} \lambda^{k-p}\varphi_{p-1}^{(k-p+1)}(x) \left\{ 1- \frac{\displaystyle\sum_{q=m-p+1}^{m} a_{m-q}\lambda^{m-q}} {F(\lambda)} \right\} = \]

\[ = E_\lambda \sum_{p=1}^{k} \lambda^{k-p}\varphi_{p-1}^{(k-p+1)}(x) - E_\lambda \sum_{p=1}^{k} \lambda^{k-p}\varphi_{p-1}^{(k-p+1)}(x) \frac{\displaystyle\sum_{q=0}^{p-1} a_q\lambda^q} {F(\lambda)}. \]

The first term on the right-hand side of the last equality is zero as the residue of an analytic function. Each term of the second term is also zero, since in the second factor the degree of the numerator in \(\lambda\) is less than the degree of the denominator by at least \(2\) \((k\le m-1)\).

Similarly it is easy to show that the third term in the expression obtained from (5) for \(y=0\) also vanishes.

Further, applying Cauchy’s formula for reducing a multiple integral to a single one, instead of (3) we obtain the following compact expression for the solution of problem (1), (2):

\[ \begin{aligned} u(x,y)= {}& E_\lambda \frac{1}{F(\lambda)} \sum_{q=0}^{m-1} a_{m-q}\lambda^{m-q-1}\varphi_0(x+\lambda y)+ \\[4pt] &+ E_\lambda \frac{1}{F(\lambda)} \sum_{p=2}^{m}\sum_{q=0}^{m-p} a_{m-q}\lambda^{m-p-q}\frac{1}{(p-2)!}\times \\[4pt] &\times \int_{0}^{x+\lambda y}(x+\lambda y-\xi)^{p-2}\varphi_{p-1}(\xi)\,d\xi . \end{aligned} \tag{7} \]

Taking into account the conditions imposed on the functions \(\varphi_k(x)\), it is easy to see that

\[ F(\lambda,\varphi)= \sum_{p=1}^{m}\sum_{q=0}^{m-p} a_{m-q}\lambda^{m-p-q} \int_{0}^{x+\lambda y}d\xi_1 \int_{0}^{\xi_1}d\xi_2\ldots \int_{0}^{\xi_{p-2}}\varphi_{p-1}(\xi_{p-1})\,d\xi_{p-1} \tag{8} \]

is an analytic function of the complex variable \(\lambda\).

We shall show that the zeros of the characteristic polynomial (4) cannot at the same time be zeros of the function (8), i.e., if \(F(\lambda_k)=0\), then \(F(\lambda_k,\varphi)\ne 0\). We prove this by contradiction. Suppose

\[ F(\lambda_k,\varphi)=0. \tag{9} \]

Then (9) must be satisfied identically in \(\varphi\), i.e., for all values of \(x\) and \(y\). This will be the case when all the coefficients of \(\varphi_k\) in formula (8) vanish, i.e.,

\[ \sum_{q=0}^{m-p} a_{m-q}\lambda_k^{m-p-q}=0 \quad (p=\overline{1,m}). \tag{10} \]

Taking into account the equality \(F(\lambda_k)=0\), we arrive at the conclusion that all coefficients \(a_k\) \((k=\overline{1,m})\) vanish, which is impossible.

The theorem is completely proved.

The authors consider it their pleasant duty to express their gratitude to Prof. M. L. Rasulov for posing the problem and supervising the work.

Received by the editors
November 20, 1964

Azerbaijan State University
named after S. M. Kirov

AT 14538. Submitted for typesetting 29.VII.65. Signed for printing 18.IX.65. Format \(70\times108^{1/16}\).
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