MATHEMATICS

M. B. KAPILEVICH

ON THE PROBLEM OF ANALYTIC CONTINUATION OF THE PRINCIPAL SOLUTIONS OF AN EQUATION OF HYPERBOLIC TYPE WITH SINGULAR COEFFICIENTS

(Presented by Academician L. I. Sedov on 4 IV 1957)

Consider, in the half-plane \(y>x\), the equation

\[ (y-x)z_{xy}+\beta(z_x-z_y)+c(x,y)z=0,\qquad c(x,y)\geq 0,\qquad 0<\beta<\tfrac12, \tag{1} \]

assuming that

\[ c(x,y)=\sum_{k=0}^{\infty} c_{2k}(y-x)^{1+2k};\qquad c_{2k}=\mathrm{const}. \]

Let \(\overline D\) be the closed domain bounded by the segment \(MN\) of the line \(y=x\) and by the characteristics \(MP\) and \(NP\) of equation (1), issuing from the points \(M(x_1,x_1)\) and \(N(x_2,x_2)\). We shall call problem \(K\) for equation (1) the singular Cauchy problem

\[ z(x,x)=\tau(x),\qquad z_\zeta(x,x)=\nu(x);\qquad \zeta=-\left(\frac{y-x}{2-2a}\right)^{1-a},\qquad a=2\beta, \tag{2} \]

and the problems \(T_1\) and \(T_2\) the Tricomi problems

\[ z(x_1,y)=0,\quad z_\zeta(x,x)=\nu(x);\qquad z(x_1,y)=0,\quad z(x,x)=\tau(x),\quad \tau(x_1)=0. \tag{3} \]

Here \(\tau(x)\) and \(\nu(x)\) are twice continuously differentiable functions on the interval \(x_1\leq x\leq x_2\). Using as majorants the principal solutions constructed earlier \((^{1,2})\) in explicit form for the case \(4c(x,y)=b^2(y-x)\), \(b=\mathrm{const}\), one can prove the following theorems.

Theorem 1. There exist unique solutions of the problems \(K\), \(T_1\), and \(T_2\), twice continuously differentiable in the domain \(D\). These solutions depend continuously on the initial functions \(\tau(x)\) and \(\nu(x)\), and, moreover, for each of the problems \(K\), \(T_1\), and \(T_2\) zero-order correctness holds \((^3)\).

Theorem 2. The solution \(z_0\) of problem \(K\) can be represented in the form

\[ z_0=\gamma_1 (y-x)^{1-a}\int_x^y \tau(x')[(x'-x)(y-x')]^{\beta-1} R_{\beta-1}(x'-x,y-x')\,dx' \]

\[ {}-\gamma_2\int_x^y \nu(x')[(x'-x)(y-x')]^{-\beta} R_{-\beta}(x'-x,y-x')\,dx', \tag{4} \]

while the problems \(T_1\) and \(T_2\) have solutions \(z_1\) and \(z_2\) of the form

\[ z_1=\gamma\int_{x_1}^{x}\nu(x')[(x-x')(y-x')]^{-\beta} \overline R_{-\beta}(x-x',y-x')\,dx', \tag{5a} \]

\[ z_2=k (y-x)^{1-a}\int_{x_1}^{x}\tau(x')[(x-x')(y-x')]^{\beta-1} \overline R_{\beta-1}(x-x',y-x')\,dx'. \tag{5b} \]

Here \(\gamma_1=\Gamma(a)/\Gamma^2(\beta)\); \(\gamma_2=\Gamma(2-a)/\Gamma^2(1-\beta)\); \(k=\Gamma(1-\beta)/\Gamma(\beta)\Gamma(1-a)\); \(\gamma=k\gamma_2/\gamma_1\); \(R\) and \(\overline R\) are multiple power series having infinitely large radii of convergence and equal to one for \(x'=x\) and \(x'=y\). Moreover, if

\[ R_{-\beta}(x'-x,y-x')=\sum_{\nu=0}^{\infty}\sum_{s=0}^{\infty} a_{\nu s}(x'-x)^\nu(y-x')^s \quad (a_{\nu s}=\mathrm{const},\ a_{00}=1), \tag{6} \]

then \(R_{\beta-1}(x'-x,y-x')\) is obtained from series (6) by replacing \(\beta\) by \(1-\beta\).

The functions \(R_{-\beta}\) and \(R_{\beta-1}\), which are strictly positive in the half-plane \(y>x\), satisfy the inequalities

\[ R_{-\beta}(P)\leq \overline I_{-\beta}(br),\qquad R_{\beta-1}(P)\leq \overline I_{\beta-1}(br), \tag{7} \]

\[ r=\sqrt{(x'-x)(y-x')},\qquad P=P(x'-x,y-x'),\qquad b=2\sqrt{\sup_{y>x}(c/(y-x))}. \]

Analogously to the case \(c=(\tfrac12 b)^2(y-x)\), where\({}^{(2)}\) \(\overline P=P(x-x',y-x')\),

\[ \overline R_{-\beta}(\overline P)=\overline J_{-\beta}(br_1),\qquad \overline R_{\beta-1}(\overline P)=\overline J_{\beta-1}(br_1),\qquad r_1=\sqrt{(x-x')(y-x')}, \tag{8} \]

the series \(\overline R_{-\beta}\) and \(\overline R_{\beta-1}\) give oscillating functions possessing an infinite number of nodal lines for \(y>x\). From Theorems 1 and 2 it follows:

1. Denote by \(u_1(x,y,x_0,y_0)\) the Riemann function of equation (1), and by \(H(x,y,x_0,y_0)\) and \(\overline H(x,y,x_0,y_0)\) the Hadamard functions, respectively, of problems \(T_1\) and \(T_2\). Fix in the half-plane \(y>x\) a point \(C(x_0,y_0)\) (the observation point) and draw through it two characteristics \(CA\) and \(CB\) (the incident characteristics). At the points \(A(x_0,x_0)\) and \(B(y_0,y_0)\) (reflection points) construct the straight lines \(y=x_0\) and \(x=y_0\) (the reflected characteristics). Then we obtain six regions: \(1(x_0<y_0<x<y)\), \(2(x<x_0<y_0<y)\), \(3(x<y<x_0<y_0)\), \(4(x_0<x<y_0<y)\), \(5(x<x_0<y<y_0)\), and \(6(x_0<x<y<y_0)\). By its initial values on the incident characteristics the function \(u_1\) is defined in regions 2, 4, 5, and 6. On the other hand, the known initial data on the reflected characteristic and the line \(y=x\), associated with \(H\) and \(\overline H\) \({}^{(4)}\), determine these functions in region 1 (or 3).

As formulas (4) and (5) show, the initial values \(u_1|_{y=x}\), \(u_{1\xi}|_{y=x}\), \(H|_{y=x}\), \(\overline H_\xi|_{y=x}\), after removal of the multiplicative power singularities at the reflection points, become entire holomorphic functions. Solving with their help problem (1), (2), we obtain integral representations for the functions \(u_1\), \(H\), and \(\overline H\) in regions 2, 6, 1 (or 3). Thus, for example, in region 6

\[ u_1=m(y-x)^{1-a}(y_0-x_0)^a \int_x^y r^{a-2}r_0^{-a}R_{-\beta}(P_0)R_{\beta-1}(P)\,dx' \;-\; \]

\[ {}-m(y_0-x_0)\int_x^y r^{-a}r_0^{a-2}R_{-\beta}(P)R_{\beta-1}(P_0)\,dx' =u_{36}-u_{46}; \tag{9} \]

in region 3

\[ H=n(y-x)^{1-a}(y_0-x_0)^a \int_x^y r^{a-2}r_{10}^{-a}R_{\beta-1}(P)\overline R_{-\beta}(\overline P_0)\,dx', \tag{10a} \]

\[ \overline H=n(y_0-x_0)\int_x^y r^{-a}r_{10}^{a-2}R_{-\beta}(P)\overline R_{\beta-1}(\overline P_0)\,dx'. \tag{10b} \]

In this case
\[ P_0=P_0(x'-x_0,\ y_0-x'),\quad \overline{P}_0=\overline{P}_0(x_0-x',\ y_0-x'), \]
\[ r_0=\sqrt{(x'-x_0)(y_0-x')},\quad r_{10}=\sqrt{(x_0-x')(y_0-x')},\quad m=\operatorname{tg}\beta\pi/2\pi. \]

The singular nature of the expressions (9) and (10), which are valid only in a neighborhood of the singular line \(y=x\), necessarily leads to the problem of analytically continuing them into a neighborhood of the incident and reflected characteristics. For this purpose, along with the solutions \(u_1, H, \overline{H}\), it is necessary to introduce new auxiliary principal solutions, whose number and the conditions serving for their unique construction are entirely determined by the number of singular lines and by the character of the singularity, on these lines, of the functions \(u_1, u_3, u_4, H, \overline{H}\). Thus, for example, the integrals \(u_1, H\), and \(\overline{H}\) have logarithmic singularities on the reflected characteristics, near which these functions have the form

\[ u=P\ln\Lambda+Q;\quad \Lambda=(y_0-x)(y-x_0)/(x_0-x)(y-y_0). \tag{11a} \]

Therefore the problem of analytic continuation of the Riemann and Hadamard functions into a neighborhood of the reflected characteristics requires the introduction of principal solutions \(u_5\) and \(u_6\), the first of which serves as the coefficient of \(\ln\Lambda\) in formula (11a), while the second contains the indicated logarithmic singularity of the functions \(u_1, H, \overline{H}\), but has a regular part \(Q_6(x,y,x_0,y_0)\) vanishing on the reflected characteristics.

The functions \(u_{5k}\) \((k=1,3,4,5)\) are uniquely determined by their values on the reflected characteristics, analogous to the known initial data of the Riemann function on the incident characteristics. These functions, having logarithmic singularities on the incident characteristics, give in regions 4 and 5 fundamental solutions of equation (1) that are bounded on the reflected characteristics. Analogously, for the investigation of the fundamental solutions \(u_{3k}\) and \(u_{4k}\) \((k=2,6)\) in a neighborhood of the incident characteristics, where

\[ u_{ik}=P_{ik}(x,y,x_0,y_0)\ln\Lambda^{-1}+Q_{ik}(x,y,x_0,y_0) \quad (i=3,4;\ k=2,6), \tag{11b} \]

it is necessary to construct branches \(u_{1k}\) \((k=2,4,5,6)\) of the Riemann function belonging to the incident characteristics in regions \(2,4,5,6\), and coinciding (up to a constant factor) with the coefficient of \(\ln\Lambda^{-1}\) in formulas (11b), and also to introduce fundamental solutions \(u_{2k}\) \((k=2,4,5,6)\), carrying the logarithmic singularity of the functions \(u_{ik}\) \((i=3,4,5,6)\) on the incident characteristics, but having there zero regular parts \(Q_{2k}\). The fundamental solutions \(u_{5k}\), just like \(u_{3k}, u_{4k}\), are constructed by combining singularities of the initial functions in the problems \(K, T_1\), and \(T_2\) with power singularities of the kernels in the solutions of these problems. As a result, for the functions \(u_{1k}, u_{5k}\) one obtains integral expressions analogous to those indicated earlier \((^2)\) for \(c=(1/2b)^2(y-x)\). However, the products of Bessel functions under the integral signs of such expressions are replaced, in case (1), by products of power series of the form \(R_\nu\) or \(\overline{R}_\nu\).

Having obtained the branches \(u_{1k}, u_{5k}\), bounded respectively on the incident and reflected characteristics, we can use them to construct the branches \(u_{2k}, u_{6k}\), which have logarithmic singularities simultaneously on both the incident and the reflected characteristics. To this end, considering the inhomogeneous differential equations satisfied by the regular parts \(Q_{2k}\) and \(Q_{6k}\) of the integrals \(u_{2k}\) and \(u_{6k}\), we solve for the functions \(Q_{2k}\) and \(Q_{6k}\) the Goursat problem with zero data respectively on the incident and reflected characteristics. The linear relations connecting each three of the four branches \(u_{ik}\), belonging to regions \(1\)—\(6\), analytically continue these branches from a neighborhood of one singular line into a neighborhood of the other singular line of the functions \(u_{ik}\). The same

for the purpose of regular continuation from a neighborhood of one characteristic to a neighborhood of another there also serve two-term linear relations (symmetry relations) for the solutions \(u_{ik}\). For example, the analytic continuation of the function \(u_5\) from a neighborhood of the reflected characteristic \(y_0=x\) of domain 1 into a neighborhood of the transition line \(y_0=x_0\) gives the equality

\[ \begin{aligned} &n (y-x)^{1-a}\int_{y_0}^{x} r_{10}^{-a} r_1^{a-2}\,\overline{R}_{-\beta}(\overline{P}_0)\, \overline{R}_{\beta-1}(\overline{P})\,dx' \\ &= m (y_0-x_0)^{1-a}\int_{x_0}^{y_0} r_1^{-a} r_0^{a-2}\,\overline{R}_{-\beta}(\overline{P})\, R_{\beta-1}(P_0)\,dx' \\ &\quad - m (y-x)^{1-a}\int_{x_0}^{y_0} r_1^{a-2} r_0^{-a}\,\overline{R}_{\beta-1}(\overline{P})\, R_{-\beta}(P_0)\,dx'. \end{aligned} \tag{12} \]

1. The functions \(R_\nu\) and \(\overline{R}_\nu\), equal to unity respectively on the incident and reflected characteristics, can be constructed from these initial data of Goursat by the usual methods of iterations or power series. Their effective computation makes it possible to obtain expansions of the principal solutions \(u_{ik}\) of equation (1) in uniformly and absolutely convergent series in Appell hypergeometric functions \(F_1(\alpha,\beta,\beta',\gamma;X,Y)\) \((^5)\).

2. In the case under consideration \(0<a<1\), the logarithmic singularities of the fundamental solutions (11b) disappear when the pole of these solutions falls on the transition line. This shows that for \(0<a<1\) there do not exist solutions of equation (1) with logarithmic singularities at points of the transition line. Such solutions appear only for \(a=\pm1,\pm2,\pm3,\ldots\), when \(y=x\) becomes a branching line of logarithmic character. Thus, for example, if \(a=1\), then in order to obtain the general integral of the form (4), and also in order to construct the principal solutions \(u_{ik}\) of equation (1), it is necessary to use the solutions of equation (1)

\[ z_1(a,x,y)=\bigl[(x'-x)(y-x')\bigr]^{-a/2}R_{-a/2}(x'-x,y-x'), \]

\[ z_2=\lim_{a\to1} \frac{z_1(a,x,y)-(y-x)^{1-a}z_1(2-a,x,y)}{1-a}. \]

1. The results obtained can be carried over to linear equations of higher orders and to systems of such equations with one or several regular singular lines. The equation

\[ (\zeta-a)z_{\theta\theta}+(\zeta-b)z_{\zeta\zeta}=0 \qquad (a<b), \]

deserves attention; it is a prototype of the well-known Chaplygin equations \((^{6,7})\). The nature of the branching of the principal solutions of such an equation is complicated because of the multiple reflection of the incident characteristics from two parabolicity lines and the effect of interference of the singular lines.

Moscow Evening Metallurgical
Institute

Received
5 III 1957

CITED LITERATURE

\(^1\) M. B. Kapilevich, DAN, 81, No. 1 (1951).
\(^2\) M. B. Kapilevich, DAN, 91, No. 4 (1953).
\(^3\) F. I. Frankl, Izv. AN SSSR, ser. matem., 8, No. 5 (1944).
\(^4\) J. Hadamard, Bull. Soc. Math. de France, 31, 208 (1903).
\(^5\) P. Appell, J. Kampé de Fériet, Fonctions hypergéométriques et hypersphériques, Polynomes d’Hermite, Paris, 1926.
\(^6\) S. A. Chaplygin, On Gas Jets, Collected Works, 2, ch. V, Publ. House of the Academy of Sciences of the USSR, 1933.
\(^7\) V. V. Sokolovskii, Prikladn. matem. i mekh., 13, issue 2 (1949).