MATHEMATICS

M. B. KAPILEVICH

ON CONNECTION FORMULAS FOR SINGULAR TRICOMI PROBLEMS

(Presented by Academician I. G. Petrovskii on 31 XII 1959)

We shall call singular Tricomi problems the problem of finding, in the domain \(D\) \((y>x>0)\), those solutions \(u(x,y,\beta)\) and \(\bar u(x,y,\beta)\) of the Euler–Poisson equation

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

which are continuous in \(D\) together with their second-order derivatives and satisfy, along the half-lines \(y=x\geq 0,\ x=0,\ y\geq 0\), respectively, the conditions

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

\[ \bar u_{\eta}(x,x)=f(x),\qquad \bar u(0,y)=0; \qquad \eta=-((y-x)/(2-2a))^{1-a}. \tag{3} \]

Here \(f(x)\) is assumed to be twice continuously differentiable on the semiaxis \(y=0,\ x\geq 0\), with \(f(0)=0\).

Theorem 1. If \(\beta_2>\beta_1\geq 0,\ \beta=\beta_2-\beta_1,\ a=a_2-a_1,\ \varkappa_1\Gamma(\beta)\Gamma(1/2-\beta_2)=2^a\Gamma(1/2-\beta_1)\), then the connection formula holds

\[ u(x,y,\beta_2)=(y-x)^{1-\beta_1-\beta_2} \int_0^x K_1(x,y,\xi,\beta_1,\beta_2)u(\xi,y,\beta_1)\,d\xi, \tag{4} \]

where

\[ K_1=\varkappa_1(y-\xi)^{a_1-1}(x-\xi)^{\beta-1}F(-\beta,\beta_2,\beta;\omega). \]

Theorem 2. Let \(\beta_1>\beta_2\geq 0,\ \bar\beta=\beta_1-\beta_2,\ \varkappa_2(1-a_1)^{a_1}\Gamma(\bar\beta)\Gamma(1/2-\beta_2)=(1-a_2)^{a_2}\Gamma(1/2+\beta_1)\). Then the functions \(\bar u(x,y,\beta_i)\) \((i=1,2)\) are connected by the equality

\[ \bar u(x,y,\beta_2)=(y-x)^{\bar\beta} \int_0^x K_2(x,y,\xi,\beta_1,\beta_2)\bar u(\xi,y,\beta_1)\,d\xi, \tag{5} \]

whose kernel is

\[ K_2=\varkappa_2(x-\xi)^{\bar\beta-1}F(-\bar\beta,1-\beta_2,\bar\beta;\omega). \]

Theorem 3. If \(\beta_1\geq 0,\ \beta_2\geq 0,\ \beta_1+\beta_2\neq 0,\ \varkappa_3(1-a_1)^{a_1}\Gamma(1/2-\beta_2)\times\Gamma(\beta_1+\beta_2)=2^{a_2}\Gamma(1/2+\beta_1)\), then \(\bar u(x,y,\beta_1)\) is transformed into \(u(x,y,\beta_2)\) by the relation

\[ u(x,y,\beta_2)=(y-x)^{\bar\beta} \int_0^x K_3(x,y,\xi,\beta_1,\beta_2)\bar u_{\xi}(\xi,y,\beta_1)\,d\xi, \tag{6} \]

where

\[ K_3=\varkappa_3(x-\xi)^{\beta_1+\beta_2-1}F(1-\beta_1-\beta_2,\beta_2,\beta_1+\beta_2;\omega). \]

The inverse connection is given by the solution of the integral equation (6):

\[ \bar u(x,y,\beta_2)=(y-x)^{\beta_1} \int_0^x K_4(x,y,\xi,\beta_1,\beta_2)u(\xi,y,\beta_1)\,d\xi, \tag{7} \]

valid for \(\beta_1 \ge 0,\ \beta_2 \ge 0,\ \beta_1+\beta_2<1,\ 2^{a_1}\chi_4\Gamma(1/2+\beta_2)\Gamma(1-\beta_1-\beta_2)= (1-a_2)^{a_2}\Gamma(1/2-\beta_1)\),
\(K_4=\chi_4(x-\xi)^{-\beta_1-\beta_2}(y-\xi)^{-\beta_2}F(2-a_1-a_2,\ -\beta_1,\ 1-\beta_1-\beta_2;\omega)\).

Let us compare the Tricomi problems under consideration with the solutions \(z(x,y,\beta)\) and \(\bar z(x,y,\beta)\) of two singular Goursat problems (1):

\[ yz_{xy}+\alpha z_y+\beta z_x=0 \qquad (\alpha>0,\ \beta\ge 0,\ (x,y)\in D); \tag{8} \]

\[ z(0,y)=0,\qquad z(x,0)=f(x);\qquad \bar z(0,y)=0,\qquad \bar z_y(x,0)=f(x), \tag{8a} \]

assuming, as before, that \(f(0)=0,\ f(x)\subset L_2\).

Theorem 4. Assuming \(\beta_2>\beta_1\ge 0,\ \bar D_x=\partial/\partial x,\ y\omega=\alpha(x-\xi),\ \Gamma(1-\beta_2)=\mu_1\Gamma(1-a_2)\Gamma(1+\beta)\), we arrive at the transformation formula:

\[ u(x,y,\beta_2)=(y-x)^{-\beta_2}\left(\frac{y}{\alpha}\right)^{\beta_1}e^{-\alpha x/y} \int_0^x Q_1 z(\xi,y,\beta_1)\,d\xi . \tag{9} \]

This time the kernel \(Q_1\) reduces to Humbert’s confluent hypergeometric function \((^2)\):
\(Q_1=-\mu_1\exp(\alpha \xi/y)D_\xi[(x-\xi)^\beta\Phi_1(\beta_2,1-\beta_2,\ 1+\beta,\ \omega,\bar\omega)]\). When \(\beta_2=\beta_1=\beta\), and \(\bar Q_1=Q_1(x,y,\xi,\beta,\beta)\), formula (9) takes the form

\[ \left[\frac{\alpha(y-x)}{y}\right]^\beta u(x,y,\beta) = \frac{\Gamma(1-\beta)}{\Gamma(1-\alpha)}z(x,y,\beta) + e^{-\alpha x/y}\int_0^x \bar Q_1 z(\xi,y,\beta)\,d\xi . \]

Inverting the relations (9), we find, for \(\beta_2>\beta_1\ge 0,\ \mu_2\Gamma(1-\beta_1)\Gamma(1+\beta)=\Gamma(1-a_1)\),

\[ z(x,y,\beta_2)= \left(\frac{y}{\alpha}\right)^{-\beta_2}(y-x)^{1-\beta_1} \int_0^x Q_2 u(\xi,y,\beta_1)\,d\xi, \tag{10} \]

where

\[ Q_2=-\mu_2(y-\xi)^{a_1-1}D_\xi[(x-\xi)^\beta \Phi_1(\beta_2,\beta_1-1,\ 1+\beta;\omega,-\bar\omega)]. \]

For equal values \(\beta_2=\beta_1=\beta\) and \(Q_2(x,y,\xi,\beta,\beta)=Q_2\),

\[ \left[\frac{\alpha(y-x)}{y}\right]^{-\beta}z(x,y,\beta) = \frac{\Gamma(1-\alpha)}{\Gamma(1-\beta)}u(x,y,\beta) + (y-x)^{1-\alpha}\int_0^x \bar Q_2 z(\xi,y,\beta)\,d\xi . \]

Theorem 5. Let \(\bar\beta_1<1,\ \mu_3\Gamma(1+\beta_2)\Gamma^2(1-\beta_1)=\alpha\Gamma(1-a_1)\); \(\gamma(b,z)\) is Euler’s gamma function \((^3)\). Then:

\[ \bar z(x,y,\beta_2)=\int_0^x Q_3(x,y,\xi,\beta_1,\beta_2)u(\xi,y,\beta_1)\,d\xi, \tag{11} \]

\[ Q_3=\mu_3(y-\xi)^{a_1-1}D_\xi \int_\xi^x (y-t)^{1-\beta_1}(t-\xi)^{-\beta_1} \gamma\left[\beta_2,\frac{\alpha(x-t)}{y}\right]dt . \]

The functions \(\bar z(x,y,\beta_2)\), \(u(x,y,\beta_1)\) also satisfy an analogous relation. It is also of interest to compare (1) with the equation

\[ yv_{yy}+\beta v_y-\alpha v_x=0 \qquad (\alpha>0,(x,y)\in D). \]

If for \(v(x,y,\beta)\) the first of conditions (8a) are fulfilled, then, denoting by \(Q_4(x,y,\xi,\beta_1,\beta_2)\) \((0\le \xi\le x)\) the function

\[ Q_4=-\mu_4(y-\xi)^{a_1-1}D_\xi \int_\xi^x (x-t)^{\beta_2-2}(y-t)^{1-\beta_1}(t-\xi)^{-\beta_1} \exp\left(-\frac{\alpha y}{x-t}\right)dt, \]

where \(\mu_4\Gamma^2(1-\beta_1)\Gamma(1-\beta_2)=\Gamma(1-a_1)\), we obtain

\[ v(x,y,\beta_2)=(\alpha y)^{1-\beta_2} \int_0^x Q_4(x,y,\xi,\beta_1,\beta_2)u(\xi,y,\beta_1)\,d\xi . \tag{12} \]

The formulas relating \(v\) to \(\bar u\), and also \(\bar v\) to \(u(x,y,\beta_1)\), have an analogous form. Along with (1), the more general singular equation (4) was studied earlier:
\[ (y-x)w_{xy}+\beta(w_x-w_y)-b^2(y-x)w=0. \]

For its solutions \(w(x,y,\beta)\) satisfying the conditions (2), we obtain, putting
\(r=\sqrt{(x-t)(y-t)}\), \(z^\nu I_\nu(z)=2^\nu \Gamma(1+\nu)I_\nu(z)\):
\[ w(x,y,\beta_2)=(y-x)^{1-a_2}\int_0^x R_1u(\xi,y,\beta_1)\,d\xi, \tag{13} \]
if \(\beta_2>\beta_1\geq 0\),
\(\nu_1\Gamma(\beta_2)\Gamma(1-\beta_1)\Gamma(1/2-\beta_2)=2^a\Gamma(1/2-\beta_1)\),
\[ R_1=-\nu_1(y-\xi)^{a_1-1}D_\xi \int_\xi^x (x-t)^{\beta_2-1}(y-t)^{\bar\beta}(t-\xi)^{-\beta_1}\bar I_{\beta_2-1}(2br)\,dt. \]
In constructing similar relations for the integrals \(\bar w(x,y,\beta)\), satisfying the boundary data (3), one must, conversely, take \(\beta_1>\beta_2\geq 0\), which gives, if
\[ \nu_2(1-a_1)^{a_1}\Gamma(1-\beta_2)\Gamma(\beta_1)\Gamma(\beta_2+1/2) =(1-a_2)^{a_2}\Gamma(\beta_1+1/2): \]
\[ \bar w(x,y,\beta_2)=\int_0^x R_2(x,y,\xi,\beta_1,\beta_2)\bar u(\xi,y,\beta_1)\,d\xi, \]
\[ R_2=-\nu_2D_\xi\int_\xi^x (x-t)^{-\beta_2}(y-t)^{\bar\beta}(t-\xi)^{\beta_1-1}\bar I_{-\beta_2}(2br)\,dt. \tag{14} \]

At the same time, for an unchanged value of the parameter \(\beta\),
\[ w(x,y,\beta)=u(x,y,\beta)+(y-x)^{1-a}\int_0^x R_1(x,y,\xi,\beta,\beta)u(\xi,y,\beta)\,d\xi \]
\[ \bar w(x,y,\beta)=\bar u(x,y,\beta)+\int_0^x R_2(x,y,\xi,\beta,\beta)\bar u(\xi,y,\beta)\,d\xi. \]

As a result of replacing \(u\) and \(\bar u\) by the functions \(z\) and \(\bar z\), these formulas assume the form
\[ w(x,y,\beta_2)=\left(\frac{y}{\alpha}\right)^{\beta_1}(y-x)^{1-a_2} \int_0^x R_3D_\xi\left[e^{\alpha\xi/y}z(\xi,y,\beta_1)\right]\,d\xi, \tag{15a} \]
\[ \bar w(x,y,\beta_2)=\left(\frac{y}{\alpha}\right)^{\beta_1} \int_0^x R_4D_\xi\left[e^{\alpha\xi/y}\bar z_\xi(\xi,y,\beta_1)\right]\,d\xi, \tag{15b} \]
where for \(R_3\) we obtain the integral representation
\[ R_3=\nu_3\int_\xi^x (t-\xi)^{-\beta_1}r^{a_2-2}e^{-\alpha t/y}\bar I_{\beta_2-1}(2br)\,dt, \]
in which
\[ \nu_3\Gamma(\beta_2)\Gamma(1-a_2)\Gamma(1-\beta_1)=\Gamma(1-\beta_2), \]
while \(R_4\) is determined from the equality
\[ \alpha\cdot 2^{1-a_2}\Gamma(a_2)R_4 =-\beta_1(1-a_2)^{a_2}\Gamma(a_2-1)R_3(x,y,\xi,\beta_1,1-\beta_2). \]
After integration by parts, (15) generates four relations corresponding to the conditions
\(\beta_1>\beta_2\geq 0\), \(\beta_1+\beta_2>1\), and \(\beta_1=\beta_2=\beta\). Equalities transforming \(u\) and \(z\) into \(\bar w\), and also \(u\) and \(\bar z\) into \(w\), have an analogous form.

If \(f(x)\in L_\infty\), then, by analogy with (1), for the indicated transformation operators one can construct expansions into infinite series, converging absolutely and uniformly in the domain \(D\). Namely, we introduce the notation
\[ U_x^\beta=k_1\rho^\beta\Phi_1(\beta,1-\beta,1+\beta;-\rho,-\delta_x), \quad \bar U_x^\beta=k_2x^{1-a}\rho^\beta\Phi_1(1-\beta,\beta,2-\beta;-\rho,-\delta_x), \]
where
\[ \rho(y-x)=x,\qquad \delta_x=xD_x, \]
\[ k_1\Gamma(1+\beta)\Gamma(1-a)=\Gamma(1-\beta),\qquad k_2\,2^{1-a}\Gamma(2-\beta)\Gamma(a)=(1-a)^a\Gamma(\beta). \]

Moreover, consider the inverse operators \((U_x^{\beta})^{-1}=k_3(y-x)^{1-\beta}\times D_x[x^{1-\beta}(y-x)^{a-1}\Phi_1(1-\beta,1-a,2-\beta,-\beta,-\delta_x)]\), \((\overline U_x^{\beta})^{-1}=k_4(y-x)^\beta\times D_x[x^\beta \Psi(\beta,\delta_x)]\), \(k_3(1-\beta)\Gamma^2(1-\beta)=\Gamma(1-a)\), \(\sqrt\pi(1-a)^a k_1=\Gamma(\tfrac12+\beta)\). Then, using the notation of paper \((^1)\), we obtain \(u_2=U_x^{\beta_2}(U_x^{\beta_1})^{-1}u_1\); \(\overline u_2=\overline U_x^{\beta_2}(\overline U_x^{\beta_1})^{-1}u_1\); \(u_2=U_x^{\beta_2}Z_x^{1-\beta_1}\Delta z_1\); \(u_2=U_x^{\beta_2}\Delta_x^{\beta_1}z_1\); \(z_2=Z_x^{\beta_2}(U_x^{\beta_1})^{-1}u_1\); \(z_2=\Delta_x^{-\beta_2}(U_x^{\beta_1})^{-1}u_1\); \(v_2=V_x^{\beta_2}(U_x^{\beta_1})^{-1}u_1\); \(\overline v_2=\overline V_x^{\beta_2}(\overline U_x^{\beta_1})^{-1}\overline u_1\). Note that \(u(x,y,0)=u_x(x,y,0)=z(x,y,0)=f(x)\). Thus, the above equalities for \(\beta_1=0\) or \(\beta_2=0\) give explicit formulas for the solution of the corresponding boundary-value problems, or their inversion with respect to the initial functions.

Since \(K_1\ge0\), it follows from (4) that if \(f'(x)>0\), then, as the parameter \(\beta\) increases, \(u(x,\bar y,\beta)\) decreases. Above, the integrals of two equations with an unchanged initial function were related. Let now \(f_2(x)=P(x)f_1(x)\), where \(P(x)\) is an arbitrary function integrable on every finite interval of the semiaxis \(y=0,\ x\ge0\). In this case, for the corresponding solutions formulas (11)—(15) also hold, but the integrands of the kernels \(Q_3,Q_4,R_i\ (i=1,2,3,4)\) acquire the additional factor \(P(t)\). Examples of such relations can also be constructed by writing \(u,z,v,w\) in the form of Duhamel integrals containing discontinuous solutions of the same problems for \(f_1(x)=1\). Without presenting other results arising from pairwise combinations of the solutions \(u_i,\overline u_i,v_i,\overline v_i,w_i,\overline w_i,z_i,\overline z_i\ (i=1,2)\) under the condition \(f_2(x)=P(x)f_1(x)\), we note only that if \(P(x)=(x-x_1)^{-k_1}\ldots(x-x_n)^{-k_n}\), the kernels \(Q_5\) and \(R_i\ (i=1,\ldots,4)\) are represented in the form of infinite series in the hypergeometric functions of Lauricella \((^2)\). When \(b=0\), such series terminate at their first term, and therefore, for example, \(\Gamma(1+\beta)R_1=\Gamma(\beta_2)\Gamma(1-\beta_1)v_1(y-\xi)^{a_1-1}(y-x)^\beta P(x)\times D_\xi[(x-\xi)^\beta F_D(\beta_2,\beta,k_1,\ldots,k_n,1+\beta,\omega,X_1,\ldots,X_n)]\), \((x-x_i)X_i=x-\xi\).

On the other hand, for \(P(x)=\exp(kx)\), \(b=0\), \(R_1\) is expressed through the confluent hypergeometric function \(\Phi_1\). Similar transformation formulas can in turn be generalized to the case when two or more initial functions \(f_i(x)\) are connected by an arbitrary, prescribed-in-advance relation. A number of other generalizations are provided by products of the transformation operators found. Thus, for example, in order to solve the integral equation (12) with respect to \(u(x,y,\beta_1)\), it suffices in formula (9) to replace \(z(x,y,\beta)\) by expression (10) from paper \((^1)\), which transforms \(z(x,y,\beta)\) into \(v(x,y,\beta)\). Substituting the same expression in (15), we connect \(w(x,y,\beta)\) with \(v(x,y,\beta)\).

With the aid of the connection formulas, for a suitable choice of the values \(f_i(x)\), it is also possible to compute a number of integrals with special functions. For example, if \(f_1(x)=x^m(1-cx)^{-k}\), \(f_2(x)=x^m e^{kx}\) \((m>0,\ k<1)\), then in the corresponding relations there appear the expressions* \(u_1=MF_1(1+m,1-\beta,k,1+m+\beta,x/y,cx)\), \(u_2=M\Phi_1(1+m,1-\beta,1+m+\beta,x/y,kx)\), where \(M\Gamma(1-a)\Gamma(1+m+\beta)=\Gamma(1-\beta)\Gamma(1+m)x^{m+\beta}y^{\beta-1}(y-x)^{1-a}\). In particular, when \(k=0\), \(u_1\) and \(u_2\) reduce to the hypergeometric functions of Gauss, and for \(f(x)=x^m(x_1-x)^{-k_1}\ldots(x_n-x)^{-k_n}\) the solutions \(u(x,y,\beta),\overline u(x,y,\beta)\) are expressed through the Lauricella functions \(F_D\).

Received
28 XII 1959

CITED LITERATURE

\(^1\) M. B. Kapilevich, DAN, 130, No. 3 (1960).
\(^2\) P. Appell, J. Kampé de Fériet, Fonctions hypergéométriques et hypersphériques, Polynomes d’Hermite, Paris, 1926.
\(^3\) A. Erdélyi, W. Magnus, F. Oberhettinger, F. Tricomi, Higher Transcendental Functions, 1, 2, N. Y., 1953.
\(^4\) M. B. Kapilevich, DAN, 81, No. 1, 13 (1951).

* Analogous values for \(z\) and \(v\) are indicated in \((^4)\).