Reports of the Academy of Sciences of the USSR
1960, Volume 132, No. 5

MATHEMATICS

M. B. KAPILEVICH

ON MIXED BOUNDARY-VALUE PROBLEMS FOR SINGULAR EQUATIONS OF HYPERBOLIC TYPE

(Presented by Academician S. L. Sobolev on 29 I 1960)

Denote by \(u(x,y,\beta,\beta')\) and \(\bar u(x,y,\beta,\beta')\) the solutions of the equation

\[ (y-x)u_{xy}+\beta' u_x-\beta u_y=0 \qquad (0\leq \beta<1/2,\; 0\leq \beta'<1/2), \tag{1} \]

belonging to the class \(L_2\) in the domain \(D(0\leq x\leq y\leq x_0)\) of the half-plane \(y\geq x\) and satisfying, on two boundaries of this domain, the boundary conditions

\[ u(0,y)=0,\qquad u(x,x)=\tau(x),\qquad \tau(0)=0; \tag{2a} \]

\[ \bar u(0,y)=0,\qquad \bar u_\eta(x,x)=\nu(x). \tag{2b} \]

We assume here that \(\tau(x)\) and \(\nu(x)\) are twice continuously differentiable functions on the interval \((0,x_0)\),

\[ \eta=-\left(\frac{y-x}{2-a-a'}\right)^{1-\beta-\beta'} \qquad (a=2\beta,\; a'=2\beta'). \]

Theorem 1. Let the initial values \(\tau_1(x)\) and \(\tau_2(x)\) be connected by the equality \(\tau_2(x)=P(x)\tau_1(x)\), where \(P(x)\) is an arbitrary integrable function on \((0,\bar x_0)\), \((0<\bar x_0\leq x_0)\). Then for the corresponding solutions \(u_1,u_2\) of problem (2a), for \(\beta_2'>\beta_1'\geq 0\),

\[ \Gamma(\beta_2')\Gamma(1-\beta_1)\Gamma(1-\beta_1')\Gamma(1-\beta_2-\beta_2')\chi_1 = -\Gamma(1-\beta_2)\Gamma(1-\beta_1-\beta_1') \]

the relation

\[ u_2(x,y,\beta_2,\beta_2') = \int_0^x K_1(x,y,\xi,\beta_1,\beta_1',\beta_2,\beta_2')\, u_1(\xi,y,\beta_1,\beta_1')\,d\xi, \tag{3} \]

holds, in which

\[ K_1 = \chi_1 (y-x)^{1-\beta_2-\beta_2'} (y-\xi)^{\beta_1+\beta_1'-1}D_\xi\Omega(\xi), \]

\[ \Omega(\xi) = \int_\xi^x P(t)(y-t)^{-\beta}(t-\xi)^{-\beta_1'}(x-t)^{\beta_2'-1}\,dt \qquad (D_x=\partial/\partial x,\; \beta=\beta_1-\beta_2). \]

An analogous connection formula for \(\beta_2'=\beta_1'\),

\[ x_0\Gamma(1-\beta_1)\Gamma(1-\beta_2-\beta_1') = \Gamma(1-\beta_2)\Gamma(1-\beta_1-\beta_1'), \qquad K_2(\xi)=K_1\big|_{\beta_2'=\beta_1'}, \]

has the form

\[ u_2(x,y,\beta_2,\beta_1')-x_0P(x)u_1(x,y,\beta_1,\beta_1') = \int_0^x K_2(\xi)u_1(\xi,y,\beta_1,\beta_1')\,d\xi. \tag{4} \]

If, in particular, \(P(x)=1\), \(\beta^*=\beta_2'-\beta_1'\), \(\omega(x-y)=x-\xi\),

\[ -\bar\chi_1\Gamma(\beta^*)\Gamma(1-\beta_1)\Gamma(1-\beta_2-\beta_2') = \Gamma(1-\beta_2)\Gamma(1-\beta_1-\beta_1'), \]

then

\[ K_1 = \bar\chi_1 (y-x)^{1-\beta_1-\beta_2'} (y-\xi)^{\beta_1+\beta_1'-1} (x-\xi)^{\beta^*-1} F(\beta,\beta_2',\beta^*;\omega), \]

\[ K_2 = -x_0\beta\beta_1'(y-x)^{-\beta_1-\beta_1'} (y-\xi)^{\beta_1+\beta_1'-1} F(1+\beta,1+\beta_1',2;\omega). \]

The simplest form is assumed by (3) when \(\beta_2=\beta_1\), \(\tau_2(x)=\tau_1(x)\). Namely, here

\[ \chi_2\Gamma(\beta')\Gamma(1-\beta_1-\beta_2') = \Gamma(1-\beta_1-\beta_1'), \qquad \beta_2'>\beta_1'\geq 0, \]

\[ u_2(x,y,\beta_1,\beta_2') = \chi_2 (y-x)^{1-\beta_1-\beta_2'} \int_0^x (x-\xi)^{\beta'-1} (y-\xi)^{\beta_1+\beta_1'-1} u_1(\xi,y,\beta_1,\beta_1')\,d\xi. \tag{5a} \]

Restricting ourselves to the class of initial values \(\tau(x)\) that are \((n+1)\)-times continuously differentiable on the interval \((0,x_0)\), one may replace (5a) by the expansion

\[ \begin{aligned} u_2(x,y,\beta_1,\beta_2')={}& x_2 \sum_{k=0}^{n} \frac{(x-y)^k}{k!}\, \mathrm{B}_{x/y}\bigl(k+\beta',\,1-k-\beta_1-\beta_2'\bigr) D_x^k u_1(x,y,\beta_1,\beta_1')+R_n, \end{aligned} \tag{5b} \]

in which \(\mathrm{B}_z(p,q)=p^{-1}z^p F(p,1-q,1+p;z)\) is the incomplete beta function, and

\[ R_n=\frac{(-1)^{n+1}x_2}{(n+1)!}(y-x)^{1-\beta_1-\beta_2'} \int_0^x \xi^{\,n+\beta'}(y-x+\xi)^{\beta_1+\beta_1'-1} D_\eta^{\,n+1}u_1(\eta,y,\beta_1,\beta_1')\,d\xi \]

\[ (\eta=x-\theta\xi,\;0<\theta<1). \]

In the case of an infinitely differentiable function \(\tau(x)\), (5b), as \(n\to\infty\), gives a double series, uniformly and absolutely convergent in the domain \(D\), which in symbolic notation reduces to Humbert’s function \(\Phi_1(\alpha,\beta,\gamma,X,Y)\) \(({}^1)\):

\[ u_2(x,y,\beta_1,\beta_2') = x_2\beta'\,\Phi_1\bigl(\beta',\,1-\beta_1-\beta_1',\,1+\beta';\,-\rho,\,-\delta_x\bigr) u_1(x,y,\beta_1,\beta_1'). \tag{6} \]

Here \(\rho(y-x)=x\), \(\delta_x=xD_x\), \(\overline{x_2\beta'}=x_2\). A direct consequence of the equalities (5) is:

Theorem 2. If on the interval \((0,x_0)\) the derivative \(\tau'(x)\) is positive, then, for \(\beta=\mathrm{const}\), at any fixed point \((x,y)\) of the domain \(D\) the function \(u(x,y,\beta,\beta')\) decreases as the parameter \(\beta'\) increases.

Theorem 3. The solutions \(\overline{u}_1,\overline{u}_2\), for which \(\beta_1>\beta_2\ge 0\), \(\nu_2(x)=P(x)\nu_1(x)\), are related by the equality

\[ \overline{u}_2(x,y,\beta_2,\beta_2') = \int_0^x K_3(x,y,\xi,\beta_1,\beta_1',\beta_2,\beta_2')\, \overline{u}_1(\xi,y,\beta_1,\beta_1')\,d\xi, \tag{7} \]

where now

\[ x_3\Gamma(\beta_1)\Gamma(\beta_1')\Gamma(1-\beta_2)\Gamma(\beta_2+\beta_2') (2-a_1-a_1')^{\beta_1+\beta_1'} \]

\[ = -\Gamma(\beta_2')\Gamma(\beta_1+\beta_1') (2-a_2-a_2')^{\beta_2+\beta_2'}, \]

\[ K_3 = x_3D_\xi \int_\xi^x P(t)(x-t)^{-\beta_2}(y-t)^{-\beta'}(t-\xi)^{\beta_1-1}\,dt. \]

When \(\beta_2=\beta_1\), the left-hand side of formula (7) should be replaced by the expression

\[ \overline{u}_2+x_3\Gamma(\beta_1)\Gamma(1-\beta_1)(y-x)^{-\beta'}P(x)\overline{u}_1, \]

and for the values \(\nu_2(x)=\nu_1(x)\) one must put there

\[ \overline{K}_3 = x_4(y-x)^{-\beta'}(x-\xi)^{\beta-1} F(\beta',1-\beta_2,\beta;\omega); \qquad x_4\Gamma(\beta)=-\Gamma(\beta_1)\Gamma(1-\beta_2)x_3. \]

Under transformation of the integrals \(\overline{u}\), as above for \(u\), alongside the case of the unchanged parameter \(\beta\), one should single out another special case of constant values of \(\beta'\). Namely, if \(\beta_2'=\beta_1'\), the kernel \(\overline{K}_3\), being expressed, gives

\[ \overline{u}_2(x,y,\beta_2,\beta_1') = \overline{x}_4 \int_0^x (x-\xi)^{\beta-1} \overline{u}_1(\xi,y,\beta_1,\beta_1')\,d\xi \qquad (\overline{x}_4=x_4|_{\beta'=0}). \tag{8a} \]

Assuming here that \(\nu(x)\subset L_{n+1}(0,x_0)\), we further find

\[ \overline{u}_2(x,y,\beta_2,\beta_1') = \overline{x}_4 x^\beta \sum_{k=0}^{n} \frac{(-x)^k}{k!(k+\beta)} D_x^k\overline{u}_1(x,y,\beta_1,\beta_1') +R_n, \tag{8b} \]

\[ R_n = \frac{\overline{x}_4}{n!} \int_0^x (x-\xi)^{n+\beta} \mathrm{B}_{\xi/(\xi-x)}(1+n,\beta) D_\xi^{\,n+1}\overline{u}_1(\xi,y,\beta_1,\beta_1')\,d\xi. \]

In the limit, when \(n\to\infty\), the sum (8a) in the class \(L_\infty(0,x_0)\) of functions \(\nu(x)\) may be replaced by the infinite series

\[ \overline{u}_2(x,y,\beta_2,\beta_1') = \overline{x}_4\Gamma(\beta)x^\beta \gamma^*(\beta,\delta_x)\overline{u}_1(x,y,\beta_1,\beta_1'), \tag{8c} \]

where \(\beta_1>\beta_2\geqslant 0\); \(\gamma(p,z)=\Gamma(p)z^p\gamma^*(p,z)\) is Euler’s incomplete gamma function.

Theorem 4. Let \(\tau(x)=P(x)\nu(x)\), \(\beta_1>0\), \(\beta_1'>0\), \(\beta_2\geqslant 0\), \(\beta_2'\geqslant 0\),

\[ \mu_1\Gamma(\beta_1)\Gamma(\beta_1')\Gamma(\beta_2')\Gamma(1-\beta_2-\beta_2')(2-a_1-a_1')^{\beta_1+\beta_1'} =2\Gamma(1-\beta_2)\Gamma(\beta_1+\beta_1'), \]

and let the function \(M_1\) be defined by the expression

\[ M_1=\mu_1(y-x)^{1-\beta_2-\beta_2'}\int_\xi^x P(t)(x-t)^{\beta_2'-1}(y-t)^{\beta_2+\beta_1'-1}(t-\xi)^{\beta_1-1}\,dt. \]

Then \(\overline{u}_1(x,y,\beta_1,\beta_1')\) is transformed into \(u_2(x,y,\beta_2,\beta_2')\) by the relation

\[ u_2(x,y,\beta_2,\beta_2')=\int_0^x M_1(\xi)D_\xi\overline{u}_1(\xi,y,\beta_1,\beta_1')\,d\xi . \tag{9a} \]

In particular, if \(P(x)=1\),

\[ M_1=\overline{\mu}_1(y-x)^{-\beta'}(x-\xi)^{\beta_1+\beta_2'-1} F(1-\beta_1'-\beta_2,\beta_2',\beta_1+\beta_2';\omega), \]

\[ \overline{\mu}_1\Gamma(\beta_1+\beta_2')=\mu_1\Gamma(\beta_1)\Gamma(\beta_2'). \]

This kernel becomes elementary for \(\beta_2=1-\beta_1'\), when \(\mu_0=\overline{\mu}_1\big|_{\beta_2=1-\beta_1'}\),

\[ u_2(x,y,1-\beta_1',\beta_2') =\mu_0(y-x)^{-\beta'}\int_0^x (x-\xi)^{\beta_1+\beta_2'-1}D_\xi\overline{u}_1(\xi,y,\beta_1,\beta_1')\,d\xi . \tag{9b} \]

Solving the integral equations (9) with respect to \(\overline{u}\) and assuming \(\nu(x)=\overline{P}(x)\tau(x)\), \(\overline{P}(x)P(x)=1\), we obtain inverse relations. Thus, in the case \(\beta_2+\beta_1'<1\),

\[ \overline{u}_2(x,y,\beta_2,\beta_2')=\int_0^x M_2(\xi)u_1(\xi,y,\beta_1,\beta_1')\,d\xi, \tag{10a} \]

\[ M_2=\mu_2(y-\xi)^{\beta_1+\beta_1'-1}D_\xi \int_\xi^x \overline{P}(t)(x-t)^{-\beta_2}(y-t)^{1-\beta_1-\beta_2'}(t-\xi)^{-\beta_1'}\,dt, \]

\[ 2\mu_2\Gamma(1-\beta_1)\Gamma(1-\beta_1')\Gamma(1-\beta_2)\Gamma(\beta_2+\beta_2') = -\Gamma(\beta_2')\Gamma(1-\beta_1-\beta_1')(2-a_2-a_2')^{\beta_2+\beta_2'}. \]

For \(\beta_2+\beta_1'=1\), in the nonintegral term of the analogous relation formula there appears, as in (4), the function \(P(x)u_1\). Now the value \(P(x)=1\) corresponds to the kernel

\[ M_2=\overline{\mu}_2(y-x)^{1-\beta_1-\beta_2'}(x-\xi)^{-\beta_1'-\beta_2} (y-\xi)^{\beta_1+\beta_1'-1} F(\beta_1+\beta_2'-1,1-\beta_2,1-\beta_1'-\beta_2;\omega); \]

\[ \overline{\mu}_2\Gamma(1-\beta_1'-\beta_2) =-\Gamma(1-\beta_2)\Gamma(1-\beta_1')\mu_2, \]

which gives, for \(\beta_2=1-\beta_1\), \(\beta_1'\geqslant 0\), \(\beta_2\geqslant 0\), \(\beta_1'+\beta_2<1\), \(\mu_3=\overline{\mu}_2\big|_{\beta_2=1-\beta_1}\),

\[ \overline{u}_2(x,y,\beta_2,1-\beta_1) =\mu_3\int_0^x (x-\xi)^{-\beta_1'-\beta_2}(y-\xi)^{\beta_1+\beta_1'-1} u_1(\xi,y,\beta_1,\beta_1')\,d\xi . \tag{10b} \]

From the equalities (9b), (10b), under sufficient smoothness of the functions \(\tau(x)\) and \(\nu(x)\), expansions of the form (5b), (6), (8b), (8c) may be obtained.

More complicated forms are exhibited by similar representations for the other transformation operators found. Namely, introduce the notation

\[ X_\beta^{\beta'}=k_1\rho^{\beta'}\Phi_1(\beta',1-\beta,1+\beta';-\rho,-\delta_x); \]

\[ \overline{X}_\beta^{\beta'}=k_2x^{1-\beta-\beta'}\rho^\beta \Phi_1(1-\beta,\beta',2-\beta;-\rho,-\delta_x); \]

\[ k_1\Gamma(1+\beta')\Gamma(1-\beta-\beta')=\Gamma(1-\beta), \]

\[ 2k_2\Gamma(2-\beta)\Gamma(\beta+\beta') =\Gamma(\beta')(2-a-a')^{\beta+\beta'}, \]

and consider the inverse operators

\[ (X_\beta^{\beta'})^{-1} =k_3(y-x)^{1-\beta}D_x\left[x^\beta\rho^{1-\beta-\beta'} \Phi_1(1-\beta',1-\beta-\beta',2-\beta';-\rho,-\delta_x)\right], \]

\[ (\overline{X}_\beta^{\beta'})^{-1} =k_4(y-x)^\beta D_x\left[x^\beta\gamma(\beta,\delta_x)\right], \]

where

\[ k_3\Gamma(1-\beta)\Gamma(2-\beta')=\Gamma(1-\beta-\beta'), \]

\[ k_4\Gamma(\beta')(2-a-a')^{\beta+\beta'}=2\Gamma(\beta+\beta'). \]

Then, assuming \(\tau(x)\) and \(\nu(x)\in L_\infty(0,x_0)\), we obtain, for example,

\[ u_2=X_{\beta_2}^{\beta_2'}\left[P(x)(X_{\beta_1}^{\beta_1'})^{-1}u_1\right], \qquad \overline{u}_2=\overline{X}_{\beta_2}^{\beta_2'}\left[P(x)(\overline{X}_{\beta_1}^{\beta_1'})^{-1}\overline{u}_1\right]. \]

It is of interest, as in \((^1)\), to investigate the confluent cases of equation (1). Let us replace \(x\) in (1) and (2a) by \(\varepsilon x\), and then, putting \(\beta=-\alpha/\varepsilon\), pass to the limit as \(\varepsilon\to 0\). This will give, for the function \(z(x,y,\beta')=\lim_{\varepsilon\to0}u(\varepsilon x,y,-\alpha/\varepsilon,\beta')\),

\[ y z_{xy}+\beta' z_x+\alpha z_y=0;\qquad z(0,y)=0,\quad z(x,0)=\tau(x),\quad \tau(0)=0. \tag{11} \]

Carrying out the same passage to the limit in the equalities (5), (6) and taking into account that \(\lim_{z\to\infty}[z^a\Gamma(z)]^{-1}\Gamma(z+a)=1\), we find connection formulas \((^2)\) for solutions of the singular Goursat problem (11):

\[ z_2(x,y,\beta'_2)= \left(\frac{\alpha x}{y}\right)^{\beta'} \gamma^*\left(\beta',\frac{\alpha x}{y}+\delta_x\right) z_1(x,y,\beta'_1) \qquad (\beta'=\beta'_2-\beta'_1>0), \]

\[ z_2(x,y,\beta'_2)= \frac{1}{\Gamma(\beta')} \left(\frac{y}{\alpha}\right)^{-\beta'} \int_0^x (x-\xi)^{\beta'-1} \exp\left[-\frac{\alpha(x-\xi)}{y}\right] z_1(\xi,y,\beta'_1)\,d\xi . \]

The functions \(u\) and \(z\) can also be connected in another way. For example, the relation

\[ u_2(x,y,\beta_2,\beta'_2)= \lambda_0\left(\frac{y}{\alpha}\right)^{\beta'_1} (y-x)^{1-\beta_2-\beta'_2} \int_0^x Q(\xi)z_1(\xi,y,\beta'_1)\,d\xi, \tag{12} \]

holds, where
\(\tau_2(x)=P(x)\tau_1(x)\),
\[ \lambda_0\Gamma(\beta'_2)\Gamma(1-\beta'_1)\Gamma(1-\beta_2-\beta'_2) =-\Gamma(1-\beta_2), \]
\[ \beta'_2>\beta'_1\ge 0,\qquad Q=e^{\alpha\xi/y}D_\xi \int_\xi^x (x-t)^{\beta'_2-1}(y-t)^{\beta_2-1} (t-\xi)^{-\beta'_1}P(t)e^{-\alpha t/y}\,dt . \]
For \(\beta'_2=\beta'_1\), an additional term appears on the right-hand side of (12):
\[ \frac{\Gamma(1-\beta_2)}{\Gamma(1-\beta_2-\beta'_1)} \left[\frac{y}{\alpha(y-x)}\right]^{\beta'_1} P(x)z_1; \]
if \(P(x)=1\), then \(Q(\xi)\) reduces to the function \(\Phi_1\). By similar equalities one also effects the inverse transformation of the integrals \(u_1\) into the solutions \(z_2\). The connection formulas obtained, as a result of a suitable particular choice of the values of \(\tau(x)\), \(\nu(x)\), and \(P(x)\), make it possible to compute a number of integrals with known or new special functions.

Also worthy of attention are the initial problems considered for the functions
\[ (u-x)^{-\beta'}u\left(x,x+\frac{1}{y-x}\right) \quad\text{and}\quad y^{-\beta'}z\left(x,\frac{1}{y}\right). \]

Received
27 I 1960

CITED LITERATURE

\(^1\) P. Humbert, Proc. Roy. Soc. Edinburgh, 41, Part I, 73 (1920–1921).
\(^2\) M. B. Kapilevich, DAN, 130, No. 3 (1960).