M. B. Kapilevich

On Singular Cauchy Problems for the Chaplygin Equation

(Presented by Academician I. G. Petrovskii, 13 IV 1962)

Let us consider, in the half-plane \(\sigma > 0\), the Chaplygin equation:

\[ z_{\theta\theta}-z_{\sigma\sigma}-b(\sigma)z_\sigma=0,\qquad b(\sigma)=(\ln \sqrt{K(\sigma)})_\sigma,\qquad \sqrt{K}=\sum_{n=0}^{\infty} k_n\sigma^{(2n+1)/3} \tag{1} \]

and call its integrals \(z(\theta,\sigma)\) and \(\bar z(\theta,\sigma)\) solutions of the first and second singular Cauchy problems if, respectively \(\left(\eta=-(\tfrac32\sigma)^{2/3}\right)\),

\[ z(\theta,0)=\tau(\theta),\qquad z_\eta(\theta,0)=\bar z(\theta,0)=0,\qquad \bar z_\eta(\theta,0)=\nu(\theta). \tag{2} \]

As shown in paper \((^1)\), for \(z\) and \(\bar z\) there are integral representations \((x=\theta-\sigma,\ y=\theta+\sigma)\):

\[ z(\theta,\sigma)=\int_x^y G(\theta-\alpha,\sigma)\tau(\alpha)\,d\alpha,\qquad \bar z(\theta,\sigma)=\int_x^y \bar G(\theta-\alpha,\sigma)\nu(\alpha)\,d\alpha, \tag{3} \]

where \(G(\theta,\sigma)\) and \(\bar G(\theta,\sigma)\) satisfy equation (1) and have the form \(G=G_0g(\theta,\sigma)\), \(\bar G=\bar G_0\bar g(\theta,\sigma)\), where \(g(\theta,\sigma)\) and \(\bar g(\theta,\sigma)\) are certain functions, bounded and nonzero on the characteristics \(\theta\pm\sigma=0\), while \(G_0\) and \(\bar G_0\) are the values of the kernels \(G\) and \(\bar G\) for \(b(\sigma)=\frac{1}{3\sigma}\): \(G_0=\gamma_1(2\sigma)^{2/3}r^{-5/3}\), \(\bar G_0=-\gamma_2r^{-1/3}\), \(r=\sqrt{\sigma^2-\theta^2}\), \(\gamma_1=\Gamma(1/3)/\Gamma^2(1/6)\), \(\gamma_2=(3/4)^{2/3}\Gamma(5/3)/\Gamma^2(5/6)\).

Divide the half-plane \(\sigma\ge 0\) into four infinite angular regions \(D_1(0\le\sigma\le\theta)\), \(D_2(0\le\theta\le\sigma)\), \(D_3(0\le-\theta\le\sigma)\), \(D_4(0\le\sigma\le-\theta)\), formed by the lines \(\theta=0\), \(\sigma=0\), \(\theta\pm\sigma=0\) \((\sigma\ge0)\). As in note \((^2)\), we shall agree to call the particular solutions \(V(\theta,\sigma)\), \(\bar V(\theta,\sigma)\), \(W(\theta,\sigma)\), \(\bar W(\theta,\sigma)\) of the problems (2) with discontinuous initial data the Duhamel functions:

\[ V(\theta,0)=\bar V_\eta(\theta,0)=\tfrac12(1-\operatorname{sign}\theta);\qquad W(\theta,0)=\bar W_\eta(\theta,0)=\tfrac12\operatorname{sign}\theta, \tag{4} \]

assuming that for all \(-\infty\le\theta\le\infty\),
\(V_\eta(\theta,0)=W_\eta(\theta,0)=\bar V(\theta,0)=\bar W(\theta,0)\equiv0\).
Along with this, put in (1) and (2) \(z=S(\sigma)\), \(\tau(\theta)=\nu(\theta)=\mu\) \((\mu=\mathrm{const},\ -\infty\le\theta\le\infty)\), and introduce into consideration two integrals of the equation \(z_{\sigma\sigma}+b(\sigma)z_\sigma=0\):

\[ S_\mu(\sigma)=\mu\quad\text{and}\quad \bar S_\mu(\sigma)=-\mu\int_0^\sigma \varkappa(\sigma)\,d\sigma, \]

\[ \varkappa(\sigma)=k_0(2/3)^{1/3}/\sqrt{K(\sigma)}. \]

Then it is not difficult to verify that the functions \(V,\bar V,W\), and \(\bar W\) possess the following properties:

1) They are related to \(S_\mu(\sigma)\) and \(\bar S_\mu(\sigma)\) by the equalities

\[ V(\theta,\sigma)+W(\theta,\sigma)=S_{1/2}(\sigma);\qquad \bar V(\theta,\sigma)+\bar W(\theta,\sigma)=\bar S_{1/2}(\sigma). \tag{5} \]

2) In the closed region \(\overline{D}_1\), \(V=\bar V\equiv0\), \(W=S_{1/2}\), \(\bar W=\bar S_{1/2}\), and at each point \((\theta,\sigma)\in\overline{D}_4\), \(V=S_1\), \(\bar V=\bar S_1\), \(W=S_{-1/2}\), \(\bar W=\bar S_{-1/2}\).

3) \(W(\theta,\sigma)\) and \(\bar W(\theta,\sigma)\) are odd functions of the variable \(\theta\), \(W(0,\sigma)=\bar W(0,\sigma)\equiv0\) and, consequently, by virtue of the equalities (5), \(V(0,\sigma)=S_{1/2}\), \(\bar V(0,\sigma)=\bar S_{1/2}\).

4) Substituting (4) into (3) and differentiating with respect to \(\theta\), we obtain in \(D_2\) and \(D_3\):

\[ W_\theta(\theta,\sigma)=-V_\theta(\theta,\sigma)=G(\theta,\sigma);\qquad \overline W_\theta(\theta,\sigma)=-\overline V_\theta(\theta,\sigma)=\overline G(\theta,\sigma). \tag{6} \]

Formulas (6) show that the derivatives \(G\) and \(\overline G\) of the odd functions \(W\) and \(\overline W\) will be even functions of the variable \(\theta\), and therefore, in order to find them in \(D_2+D_3\), it suffices to compute the values of \(G\) and \(\overline G\) only in the domain \(D_2\). In turn, from conclusions 2) and 3) it follows that, for the construction of the resolvents \(V,\overline V,W,\overline W\) in the domain \(D_2\), it is sufficient to find integrals of equation (1) which on the characteristic \(\theta=\sigma\) are equal to \(V=\overline V=0\), \(W=S_{1/2}\), \(\overline W=\overline S_{1/2}\), while on the line \(\theta=0\) \((\sigma\geq 0)\), conversely, they take the values \(V=S_{1/2}\), \(\overline V=\overline S_{1/2}\), \(W=\overline W=0\). These boundary data are continuous for the functions \(\overline V,\overline W\), and, conversely, in the case of \(V\) and \(W\) they undergo a discontinuity at the origin. Having obtained the solutions \(V,\overline V\) or \(W,\overline W\) of such mixed boundary-value problems, one can then easily determine, by formulas (6), the required singular kernels \(G\) and \(\overline G\).

Let us carry out, for example, such computations for the functions \(V\) and \(\overline V\). To this end we introduce in (1) the variables \(\sigma\) and \(t=r/\sigma\) \((r=\sqrt{\sigma^2-\theta^2})\),

\[ \frac{1-t^2}{\sigma^2}\frac{\partial^2 z}{\partial t^2} -\frac{2(1-t^2)}{\sigma t}\frac{\partial^2 z}{\partial t\,\partial\sigma} -\frac{\partial^2 z}{\partial\sigma^2} +\frac{1-2t^2}{\sigma^2t}\frac{\partial z}{\partial t} -b(\sigma)\left[\frac{1-t^2}{\sigma t}\frac{\partial z}{\partial t}+\frac{\partial z}{\partial\sigma}\right]=0 \tag{7} \]

and then set

\[ z(\theta,\sigma)=V(\theta,\sigma)=\sum_{n=0}^{\infty}V_n(\theta,\sigma) =\sum_{n=0}^{\infty}\sigma^{2n/3}f_n(t). \tag{8} \]

Then, taking into account the expansion \(b(\sigma)=\sum_{n=0}^{\infty} b_n\sigma^{2n/3-1}\) \((b_0=1/3,\ b_1=2k_1/3k_0,\ldots)\), we arrive at the recurrent system of ordinary differential equations

\[ L_n[f_n]\equiv t(1-t^2)f_n'' +\frac{1}{3}\bigl[(4n-5)t^2+2-4n\bigr]f_n' -\frac{4}{9}n(n-1)t f_n \]

\[ = \sum_{m=1}^{n} b_m\left[(1-t^2)f_{n-m}' +\frac{2}{3}(n-m)t f_{n-m}\right],\qquad f_{-1}(t)\equiv 0,\quad n=0,1,2,\ldots . \tag{9} \]

In addition, in order to ensure satisfaction of the boundary conditions \(V(\sigma,\sigma)=0\), \(V(0,\sigma)=1/2\), we set

\[ f_0(0)=0,\quad f_0(1)=1/2;\qquad f_n(0)=f_n(1)=0\quad \text{for } n=1,2,\ldots . \tag{10} \]

Then first of all we find that
\[ f_0(t)=\frac12 I_{t^2}\!\left(\frac16,\frac12\right), \]
where
\[ I_z(p,q)=\frac{z^p F(1-q,p,p+1;z)}{pB(p,q)} \]
is the modified incomplete beta function of Euler. Next we obtain
\[ f_1(t)=\frac38 b_1\left[I_{t^2}\!\left(\frac56,\frac12\right)-I_{t^2}\!\left(\frac16,\frac12\right)\right]. \]
Continuing this process, one can show that also for \(n=2,3,\ldots\) the functions \(f_2(t),f_3(t),\ldots\) are uniquely determined by conditions (9) and (10). Since \(0\leq I_z(p,q)\leq 1\), if \(0\leq z\leq 1\), then for \(f_0(t)\) and \(f_1(t)\) the estimates
\[ 0\leq f_0(t)\leq \frac12,\qquad |f_1(t)|\leq \frac34|b_1|\quad (0\leq t\leq 1) \]
hold \((^3)\).

Let us now turn to the construction of the Riemann function \(\overline V(\theta,\sigma)\) of the second singular Cauchy problem. To this end we shall seek a solution of equation (7)

\[ \bar z=\overline V(\theta,\sigma)=\sum_{n=1}^{\infty}\overline V_{n-1}(\theta,\sigma) =\sum_{n=1}^{\infty}\sigma^{2n/3}f_n(t) =\sum_{n=1}^{\infty}\left(\frac49\right)^{n/3}(-\eta)^n f_n(t), \tag{11} \]

where, in order to satisfy the boundary conditions \(\overline V(\sigma,\sigma)=0\), \(\overline V(0,\sigma)=\overline S_{1/2}(\sigma)=\sum_{n=1}^{\infty}A_n\sigma^{2n/3}\), where \(A_1=-\frac12(3/2)^{2/3}\), \(A_2=-\frac34 b_1A_1,\ldots\), we require that

\[ f_n(0)=0,\qquad f_n(1)=A_n\quad \text{for } n=1,2,\ldots . \tag{12} \]

Then in this case as well we arrive at system (9), in which, however, \(f_0(t)\equiv 0\), while \(f_1(t)=A_1t^2(5/6,\,1/2)\) is uniquely determined from the homogeneous equation \(L_1[f_1]=0\) and equality (12). In addition, now for \(n=2\) the equation \(L_2[f_2]=b_1[(1-t^2)f'_1+\frac{2}{3}tf_1]\) and condition (12) can be satisfied by setting \(f_2(t)=-\frac{3}{4}b_1f_1(t)\). In the same way, step by step, the subsequent solutions \(f_3(t), f_4(t),\ldots\) are computed.

Restricting ourselves to the first two terms in the corresponding expansions for \(G\) and \(\overline G\), we find, approximately,
\(G(\theta,\sigma)\cong P_2(\sigma)G_0-\frac{1}{2}(3/2)^{1/3}b_1\overline G_0\),
\(\overline G(\theta,\sigma)\cong P_2(\sigma)\overline G_0\), where
\(P_2(\sigma)=1-\frac{3}{4}b_1\sigma^{2/3}\) is the partial sum of the series
\(\frac{2}{\eta}\overline S_{1/2}(\sigma)=\frac{2}{\eta}\sum_{n=1}^{\infty}A_n\sigma^{2n/3}\).
With the aid of these estimates one can obtain the corresponding approximations for the Riemann function \(v(\theta,\sigma,\theta_0,\sigma_0)\) of Chaplygin’s equation. Indeed, if instead of \(v\) we introduce the symmetric function
\(R(\theta,\sigma,\theta_0,\sigma_0)=\frac{1}{2}\chi(\sigma_0)v(\theta,\sigma,\theta_0,\sigma_0)\), then in a neighborhood \(x_0<x<y<y_0\) of the line \(\sigma=0\)

\[ R=\int_x^y \rho(\theta-\alpha,\sigma,\theta_0-\alpha,\sigma_0)\,d\alpha, \]

where

\[ \rho(\theta,\sigma,\theta_0,\sigma_0) =G(\theta_0,\sigma_0)\overline G(\theta,\sigma) -G(\theta,\sigma)\overline G(\theta_0,\sigma_0). \]

Replacing here \(G\) and \(\overline G\) by the approximations found, we obtain
\(R\cong P_2(\sigma)P_2(\sigma_0)R_0+\frac{3}{4}b_1\sigma^{2/3}[P_2(\sigma_0)-P_2(\sigma)]R_1\),
where \(R_0\) is the value of \(R\) for \(K(\sigma)=k_0\sigma^{1/3}\), and

\[ R_1=\gamma_2[(x-x_0)(y_0-x)]^{-1/6} F_1(5/6,\,1/6,\,1/6,\,5/3;\,(y-x)/(x_0-x),\,(y-x)/(y_0-x)). \]

Analogous estimates can also be obtained in a neighborhood of the characteristics \(x=x_0\), \(y=y_0\), if one uses the integral representations found earlier \((^4)\) for the Riemann function in the domains \(x_0<x<y_0<y\) and \(x<x_0<y<y_0\). In such representations the integrands contain initial values on the lines \(\sigma=0\), \(\sigma_0=0\) of two modified Green–Hadamard functions
\(Q=\frac{1}{2}\chi(\sigma_0)H(\theta,\sigma,\theta_0,\sigma_0)\) and
\(\overline Q=\frac{1}{2}\chi(\sigma_0)\overline H(\theta,\sigma,\theta_0,\sigma_0)\), associated in \(D_1\) with the boundary-value problems

\[ z(\theta,0)=\tau(\theta);\qquad z(\sigma,\sigma)=0;\qquad \overline z_\eta(\theta,0)=\nu(\theta);\qquad \overline z(\sigma,\sigma)=0;\qquad \tau(0)=0. \tag{13} \]

As follows from Gellerstedt’s results, the solutions \(z(\theta,\sigma)\) and \(\overline z(\theta,\sigma)\) of problems (13) have the form

\[ z(\theta,\sigma)=\int_0^x T(\theta-\alpha,\sigma)\tau(\alpha)\,d\alpha,\qquad \overline z(\theta,\sigma)=\int_0^x \overline T(\theta-\alpha,\sigma)\nu(\alpha)\,d\alpha, \]

where \(T(\theta,\sigma)=T_0q(\theta,\sigma)\), \(\overline T(\theta,\sigma)=\overline T_0\overline q(\theta,\sigma)\), with \(q=O(1)\), \(\overline q=O(1)\) as \(\theta\to\sigma\), and
\(T_0=\sqrt{3}\gamma_1(2\sigma)^{2/3}r_1^{-5/3}\),
\(\overline T_0=\sqrt{3}\gamma_2 r_1^{-1/3}\),
\(r_1=\sqrt{\theta^2-\sigma^2}\).
Similarly to the preceding case, it is also convenient here to apply the idea of preliminary integration of the kernels \(T(\theta,\sigma)\), \(\overline T(\theta,\sigma)\) with respect to the variable \(\theta\), and instead of these functions, which possess power singularities as \(r_1\to0\), to consider the Doetsch resolvents \(U\) and \(\overline U\), which are solutions of the problems (13) with discontinuous boundary data

\[ U(\theta,0)=1,\qquad U(\sigma,\sigma)=0,\qquad \overline U_\eta(\theta,0)=1,\qquad \overline U(\sigma,\sigma)=0. \tag{14} \]

If, alongside them, we bring in the solutions \(u(\theta,\sigma)\) and \(\overline u(\theta,\sigma)\) of two conjugate discontinuous problems

\[ u(\theta,0)=0,\qquad u(\sigma,\sigma)=S_1(\sigma);\qquad \overline u_\eta(\theta,0)=0,\qquad \overline u(\sigma,\sigma)=\overline S_1(\sigma), \tag{15} \]

then it is easy to verify that in the domain \(D_1\)

\[ u+U=S_1,\quad \overline u+\overline U=\overline S_1,\quad U_\theta(\theta,\sigma)=-u_\theta(\theta,\sigma)=T(\theta,\sigma), \]

\[ \overline U_\theta(\theta,\sigma)=-\overline u_\theta(\theta,\sigma)=\overline T(\theta,\sigma). \tag{16} \]

Thus, having constructed the functions \(U,\overline U,u,\overline u\) from the boundary data (14) and (15), one can then, by formulas (16), also find the singular kernels \(T,\overline T\). Such

The computations performed in (2) for \(U\) and \(\overline U\) make it possible to determine, with prescribed accuracy, the initial data \(Q_\eta|_{\sigma_0=0}\) and \(\overline Q|_{\sigma_0=0}\) of the modified Green—Hadamard functions, and then from them, with the aid of formula (3), to construct the corresponding approximations for \(Q(\theta,\sigma,\theta_0,\sigma_0)\) and \(\overline Q(\theta,\sigma,\theta_0,\sigma_0)\). In turn, they make it possible to find the function \(R\) in a neighborhood of the incident and reflected characteristics from the transition line, and then also to solve more general mixed problems of the form (13) with nonzero data on the characteristic \(\theta=\sigma\). The algorithms indicated above can also be applied in the study of other singular equations admitting a shift transformation in one of the coordinates. Of special interest, for example, is the equation
\(\sigma(z_{\theta\theta}-z_{\sigma\sigma})-b(\sigma)z_\sigma+c(\sigma)z=0\), if in a neighborhood of the line \(\sigma=0\) \(b(\sigma)\) and \(c(\sigma)\) are represented by series in positive integral or fractional powers of \(\sigma\), with \(0<b(0)<1\), and \(c(\sigma)=O(\sigma^\alpha)\) \((\alpha\geqslant0)\). Equation (1) is also reduced to such a form if, by the substitution \(z=\sigma^{1/6}K^{-1/4}u\), it is transformed into the form
\(u_{\theta\theta}-u_{\sigma\sigma}-\frac{1}{3\sigma}u_\sigma+c(\sigma)u=0\), where

\[ c(\sigma)=\sum_{n=-1}^{\infty} c_n\sigma^{2n/3}. \]

In this connection, for the approximation of the solutions \(u(\theta,\sigma)\), one may use as a standard not only the Euler—Poisson equation, but also the equation studied earlier\({}^{5}\)

\[ u_{\theta\theta}-u_{\sigma\sigma}-\frac{a}{\sigma}u_\sigma-b^2u=0 \qquad (0<a<1,\ b=\mathrm{const}). \tag{17} \]

Here the equation \(u_{\sigma\sigma}+\frac{a}{\sigma}u_\sigma+b^2u=0\), which determines \(S_\mu(\sigma)\) and \(\overline S_\mu(\sigma)\), is solved (under the conditions \(S(0)=\overline S_\eta(0)=\mu,\ S_\eta(0)=\overline S(0)=0,\ \eta=-(\sigma/(1-a)^{1-a})\)) in Bessel functions
\(S_\mu(\sigma)=\mu\overline J_{\beta-1/2}(b\sigma)\),
\(\overline S_\mu(\sigma)=\mu\eta\overline J_{1/2-\beta}(b\sigma)\) \((a=2\beta)\), and therefore the corresponding expansions for \(V\) and \(\overline V\) must have the form

\[ V(\theta,\sigma)=\frac12\sum_{n=0}^{\infty}\sigma^{2n}\varphi_n(t), \qquad \overline V(\theta,\sigma)=\frac12\eta\sum_{n=0}^{\infty}\sigma^{2n}\overline\varphi_n(t). \]

They generate the systems
\[ \mathcal L_{2n}[\varphi_n,a] =t(1-t^2)\varphi_n''+[1-a-4n+(4n+a-2)t^2]\varphi_n' -2n(2n+a-1)t\varphi_n=b^2t\varphi_{n-1}, \]
\[ \mathcal L_{2n}[\overline\varphi_n,2-a] =b^2t\overline\varphi_{n-1}, \qquad (\varphi_{-1}=\overline\varphi_{-1}\equiv0,\ n=0,1,2,\ldots), \]
to which, by virtue of the equalities
\(V(\sigma,\sigma)=\overline V(\sigma,\sigma)=0\),
\(V(0,\sigma)=S_{1/2}(\sigma)\),
\(\overline V(0,\sigma)=\overline S_{1/2}(\sigma)\), one should adjoin the conditions
\(\varphi_n(0)=\overline\varphi_n(0)=0\),
\(\varphi_n(1)=A_n\),
\(\overline\varphi_n(1)=\overline A_n\)
\((n=0,1,2,\ldots)\), where
\(A_n=(-b^2/4)^n/n!(\beta+\tfrac12)_n\), and \(\overline A_n\) is obtained from \(A_n\) by replacing \(\beta\) by \(1-\beta\). Solving these boundary-value problems, we find the values
\(\varphi_n(t)=A_n I_{t^2}(n+\beta,\tfrac12)\),
\(\overline\varphi_n(t)=\overline A_n I_{t^2}(n+1-\beta,\tfrac12)\). They correspond to the double series
\[ \overline V=\eta V(\theta,\sigma,1-\beta), \]
\[ V(\theta,\sigma,\beta) =\frac{\gamma_1}{\beta}\left(\frac{t}{2}\right)^a \sum_{m=0}^{\infty}\sum_{n=0}^{\infty} \frac{(1/2)_m(\beta)_{m+n}}{m!\,n!\,(\beta)_n(\beta+1)_{m+n}} \,t^{2m}\left(-\frac{b^2r^2}{4}\right)^n. \]

However, these higher transcendental functions degenerate under differentiation with respect to \(\theta\) and give the values
\[ G=G_0\overline J_{\beta-1}(br), \qquad \overline G=\overline G_0\overline J_{-\beta}(br), \qquad G_0=\gamma_1(2\sigma)^{1-a}r^{a-2}, \]
\[ \overline G_0=-\gamma_2 r^{-a}, \qquad \gamma_1=\frac{\Gamma(a)}{\Gamma^2(\beta)}, \qquad \gamma_2=[2(1-a)]^{a-1}\frac{\Gamma(2-a)}{\Gamma^2(1-\beta)}. \]
Analogous constructions are also carried out for the example
\[ z_{\theta\theta}-z_{\sigma\sigma}-\left(\frac{a}{\sigma}-1\right)z_\sigma+\frac{c}{\sigma}z=0 \qquad (0<a<1,\ c=\mathrm{const}), \]
where
\[ S_\mu(\sigma)=\mu_1F_1(c,a,\sigma), \qquad \overline S_\mu(\sigma)=\mu\eta_1F_1(c-a+1,2-a,\sigma). \]

Received
11 IV 1962

REFERENCES

1. F. I. Frankl, Izv. AN SSSR, Ser. Matem., 8, No. 5, 195 (1944).
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4. M. B. Kapilevich, DAN, 91, No. 4, 719 (1953).
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