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MATHEMATICS
M. B. KAPILEVICH
ON SINGULAR GOURSAT PROBLEMS IN A NEIGHBORHOOD OF THE ZERO AND INFINITELY REMOTE SPECIAL CHARACTERISTIC
(Presented by Academician I. G. Petrovskii on 22 XI 1960)
Consider, in the domain \(D\) \((0 \le x \le x_0,\; 0 \le y \le y_0)\), the equation
\[ L(z,B,C)=xz_{xy}+A(x)z_x+B(x)z_y+C(x)z=0, \tag{1} \]
assuming that \(A(x)>0\), and that \(B(x)\) and \(C(x)\) are continuous together with their derivatives of the 2nd order on the interval \(X\) \((0 \le x \le x_0)\). Denote by \(z(x,y,B,C)\) the solution of (1), twice continuously differentiable in \(D\), for which
\[ z(0,y)=f(y),\qquad z(x,0)=0,\qquad f(0)=0. \tag{2} \]
Here, as in (¹), we shall assume that \(f(y)\) belongs to the class \(C_p^q\) on the segment \(Y\) \((0 \le y \le y_0)\) of the axis \(Oy\).
In the study of the singular Goursat problem (1), (2), the following Duhamel principle plays an important role: if \(U(x,y)\) is the integral of equation (1) with discontinuous boundary data \(U(0,y)=1\), \(U(x,0)=0\), then
\[ z(x,y,B,C)=D_y\int_0^y U(x,y-\eta)f(\eta)\,d\eta =\int_0^y U(x,y-\eta)\,df(\eta). \tag{3} \]
When \(A(x)\equiv A(0)=a,\; B(x)\equiv B(0)=b,\; C(x)\equiv 0\), i.e. \(z(x,y,B,C)=z(x,y,b)\) satisfies the equation (1)
\[ L(z,b)=xz_{xy}+az_x+bz=0\qquad (a>0), \tag{4} \]
we obtain \(U(x,y)\Gamma(b)=\gamma(b,ay/x)\), where \(\gamma(b,z)\) is Euler’s incomplete gamma function. The existence of discontinuous solutions \(U(x,y)\) in the general case (1) is proved by the method of successive approximations.
Fixing an arbitrary point \(M(x_1,y_1)\in D\), construct the rectangle \(G=OAMBO\) with vertices \(O(0,0)\), \(A(x_1,0)\), \(B(0,y_1)\). Let \(z=0\) on \(OA\) and \(OB\). We shall show that then \(z\equiv 0\) everywhere in \(\overline{G}\). Indeed, multiply (1) by \(2[xz_x+B(x)z]\), and then, integrating the result over the domain \(G\), apply Green’s theorem. Then we obtain
\[ \int_{AM} (AB+Cx)z^2\,dy +\int_{BM}(xz_x+Bz)^2\,dx +\iint_G \{2Axz_x^2+[2BC-(AB+Cx)_x]z^2\}\,dx\,dy=0. \]
Hence, under the conditions \(A>0,\; B\ge 0,\; C\ge 0,\; 2BC-(AB+Cx)_x\ge 0\), it follows that \(z\equiv 0\) in \(\overline{G}\). With the aid of connection formulas one can pass in (4) from negative \(b\) to positive \(b\), and thereby extend the uniqueness theorem proved above also to parameters \(b<0\). As the estimates obtained show, for small values of \(y/x\) for equation (1), as in the case (4), \(U=O[(y/x)^b]\); therefore the integrals in (3) converge only when \(b>-1\).
Theorem 1. Let \(V(x,y)\) satisfy the equation
\[ xV_{xy}+AV_x+(B_2-B_1)V_y+(C_2-C_1)V=0, \]
and the discontinuous initial conditions \(V(x,0)=0,\; V(0,y)=1\), and let \(z_k=z(x,y,B_k,C_k)\) be the solutions of problem (2) for \(L(z,B_k,C_k)=0\) \((k=1,2)\). Then, if \(b_2-b_1>-1\),
\[ z_2(x,y)=D_y\int_0^y V(x,y-\eta)z_1(x,\eta)\,d\eta =\int_0^y V(x,y-\eta)z_{1\eta}(x,\eta)\,d\eta. \tag{5} \]
In this case the Riemann functions \(U_1\), \(U_2\), and \(V\) themselves are related by the equality
\[ U_2(x,y-\eta)=D_y\int_\eta^y V(x,y-t)\,U_1(x,t-\eta)\,dt. \tag{6} \]
In the case (4), \(V\Gamma(b_2-b_1)=\gamma(b_2-b_1,ay/x)\), and (6) reduces to the well-known addition theorem for the functions \(\gamma(a,z)\). Since the conditions defining \(V(x,y)\) are analogous to those under which the solutions \(U(x,y)\) were constructed, the conclusions obtained for \(U(x,y)\) carry over to \(V(x,y)\), and, in particular, for small \(y/x\) one has here \(V(x,y)=O[(y/x)^{b_2-b_1}]\). The equalities (5), by the same methods as the formulas (3), are generalized to the case when \(b_2-b_1<-1\). For example, using the symbol of Riemann–Liouville fractional differentiation and assuming \(f(y)\subset C_n^n(Y)\), we find, for \(b=b_2-b_1\ge -n\) and any \(n=0,1,2,\ldots\),
\[ z(x,y,b_2)=\left(x/a+D_y^{-1}\right)^{-b}D_y^{-(b+n)}\,[D_y^n z(x,y,b_1)]. \tag{7} \]
These series terminate when \(b=-m\) \((m=0,1,2,\ldots)\) and give an expansion corresponding to the recurrence relation \(z(b)=\left(1+\frac{x}{a}D_y\right)^m z(b+m)\) for contiguous functions \(z(x,y,b+m)\) \((m=0,1,2,\ldots)\) \((^1)\). To construct the transformation operators \(T_y^{-1}\), inverse with respect to (5), it is enough to find from (5) \(z_1(x,y)\), i.e., to solve (for \(x=\mathrm{const}\)) Volterra integral equations of the first kind of convolution type with singular kernels \(V(x,y-\eta)\). Since \(z(x,y,0)=f(y)\), the \(T_y^{-1}\) give, in particular, the inversion of the formulas (3) with respect to the initial function \(f(y)\). Thus, for example,
\[ f(y)=(x/a)^b \times \exp(-ay/x)D_y^{\,b-1}D_y\,[\exp(ay/x)z(x,y,b)]. \]
Having obtained similar inversions \(f(y)=T_y^{-1}[z(x,y,B,C)]\) for (1), one can then generalize (5) to the case when the initial values \(f_1(y)\) and \(f_2(y)\) of the integrals \(z_1\) and \(z_2\) are connected by a previously specified relation. For example,
\[ z_2(x,y,b_1)=a^{b_1-b_2}D_yD_y^{\,b_2-b_1-1}z_1(x,y,b_1)\quad (b_2>b_1>0), \tag{8} \]
if \(f_1(y)\) and \(f_2(y)\) satisfy the same equality. Along with \(T_y\), it is important also to consider the transformation operators \(T_x\), acting with respect to the singular variable \(x\).
Theorem 2. Let \(z(x,y,b+n+1)\) be a solution of the problem \(z(0,y)=(-a)^{-n-1}(b)_{n+1}f^{(n+1)}(y)\), \(z(x,0)=0\) for the equation \(L(z,b+n+1)=0\), where \(f(y)\subset C_{n+3}^{n+1}\). Then the equality holds
\[ z(b)=\sum_{k=0}^{n}(b)_k/k!\left(-\frac{x}{a}\right)^k f^{(k)}(y) +\frac{1}{n!}\int_0^x (x-\xi)^n z(\xi,y,b+n+1)\,d\xi . \tag{9} \]
One further class of singular Goursat problems deserves attention, where zero initial values are prescribed on the infinitely distant regular characteristic \(y=-\infty\). Namely, denote by \(z_0(x,y,B,C)\) the integral of equation (1) for which
\[ z_0(0,y)=f(y),\qquad \lim_{y\to-\infty} z_0(x,y)=0,\qquad f(-\infty)=0. \tag{10} \]
Assuming that \(f(y)\) is given everywhere on the half-line \(-\infty<y<y_0\), we obtain, for \(b>0\),
\[ z_0(x,y,b)=\frac{1}{\Gamma(b)}\left(\frac{a}{x}\right)^b \int_{-\infty}^{y}(y-\eta)^{b-1}\exp\left[-\frac{a(y-\eta)}{x}\right]f(\eta)\,d\eta, \tag{11} \]
and when \(f(y)\in C_n^n(-\infty<y<y_0)\), \(b>-n-1\),
\[ z_0=\sum_{k=0}^n (b)_k/k!\left(-\frac{x}{a}\right)^k f^{(k)}(y)+R_n, \]
where
\[ a^n\Gamma(b)R_n=\int_0^\infty \exp(-a\xi)\Psi(1-b,1-b-n;a\xi)\, f^{(n+1)}(y-x\xi)\,d\xi . \]
For \(z_0\) there hold relations analogous to those satisfied by \(z(x,y,b)\). From (3) it follows that
\[ \lim_{x\to\infty}[x^bz(x,y,b)]=a^bD_y^{-b}f(y)=\Phi(y). \]
Thus, \(z(x,y,b)\) is an integral of equation (4), regular in a neighborhood of the singular characteristic \(x=\infty\), with the exponent of this singular line equal to \(b\). Therefore, in order to study \(z(x,y,b)\) in a neighborhood of \(x=\infty\), it is convenient to introduce the function \(u(x,y,b)=x^{-b}z(1/x,y,b)\) and thereby reduce the considerations to the Goursat problem:
\[ \mathscr L(u,b)=u_{xy}+axu_x+abu=0, \tag{12} \]
\[ u(0,y)=\Phi(y),\qquad u(x,0)=0,\qquad \Phi(0)=0. \tag{13} \]
Its solution is obtained from (3), if one passes from \(f(y)\) to \(\Phi(y)\), and has the form:
\[ u(x,y,b)=D_y\int_0^y U(x,y-\eta)\Phi(\eta)\,d\eta =\int_0^y U(x,y-\eta)\,d\Phi(\eta), \tag{14} \]
where this time \(U(x,y)={}_1F_1(b,1,-axy)\). Conversely, (3) arises from (14) and the relation \(\Phi(y)=a^bD_y^{-b}f(y)\). For \(b=1\), \(b=1/2\), \(b=-m\), and \(b=1+m\) \((m=0,1,2,\ldots)\), \({}_1F_1\) in formulas (14) is replaced by the resolvents
\(\exp(-ar)\), \(\exp(-ar/2)I_0(ar/2)\), \(L_m(-ar)\), and \(\exp(-ar)L_m(ar)\) \((r=x(y-\eta))\), respectively. Setting in (12) \(ab=k^2\), and then passing to the limit as \(a\to0\), we arrive, for \(w=\lim_{a\to0}u(x,y,k^2/a)\), at the telegraph equation
\[ w_{xy}+k^2w=0. \]
The same limiting transition in (14) gives
\[ U(x,y)=\lim_{a\to0}{}_1F_1(k^2/a,1,-axy)=J_0(2k\sqrt{xy}). \]
In the form (14) one also writes the solution of problem (13) for the more general equation
\[ \mathscr L(u,b)=u_{xy}+A(x)u_x+B(x)u=0, \tag{15} \]
where \(A(x)\) and \(B(x)\in C_2(X)\), and \(A(x)=O(x)\) as \(x\to0\). Using (12) as a majorant, one can show that here \(U(x,y)\) is represented by the uniformly and absolutely convergent series
\[ U(x,y)=\sum_{n=0}^\infty U_n(x)y^n, \]
in which \(U_n(x)\) are expressed in terms of \(A(x)\) and \(B(x)\). Moreover, if \(|\Phi(y)|\le K\) on \(Y\), and on the segment \(X\), \(A(x)=xa(x)\), \(|a(x)|\le M\), \(|B(x)|\le MN\), where \(K,M\), and \(N\) are positive constants, then
\[ |u|\le K\,{}_1F_1(N,1,Mxy),\qquad (x,y)\in D. \]
The same estimate, but with \(K=1\), also holds for \(U(x,y)\). Considering solutions \(u_k=u(x,y,B_k)\) of the equations \(\mathscr L(u,B_k)=0\) \((k=1,2)\), we again arrive at equalities (5), but now the kernel \(V(x,y)\) is determined from
\[ V_{xy}+AV_x+(B_2-B_1)V=0 \]
and the initial conditions
\[ V(x,0)=V(0,y)=V(0,0)=1, \]
which contain no jumps; hence \(V(x,y)\) is continuous in \(\overline D\) and is computed by means of the same type of series as \(U(x,y)\). In the case (12),
\[ V={}_1F_1(b_2-b_1,1,-axy), \]
and to the solution \(w\) there corresponds
\[ V=J_0\!\left[2\sqrt{(k_2^2-k_1^2)xy}\right]. \]
For (15), if \(|B_2-B_1|\le MN_0\), then
\[ |V|\le {}_1F_1(N_0,1;Mxy) \]
in \(D\); moreover, here too \(U_1,U_2,V\) are related by formula (6), which in the particular cases indicated above …
reduce in special cases to the known integrals of Bateman—Erdelyi and Sonin with the functions \({}_1F_1\), \(J_0(z)\) \((^2)\).
By means of transformation (8) one can also pass directly from formulas (5) for \(z(x,y,b_k)\) to the corresponding relations for \(u(x,y,b_k)\), and conversely. More general transformation operators of convolution type, where \(\Phi_2(y)\) may differ from \(\Phi_1(y)\), can be constructed if, putting \(x=\mathrm{const}\), one inverts the Volterra integral equations (14). Thus, for example,
\[ \Phi(y)=\exp(-axy)\int_0^y {}_1F_1(b,1;ar)D_\eta[\exp(ax\eta)u(x,\eta,b)]\,d\eta, \]
\[ \Phi(y)=\int_0^y I_0(2k\sqrt r)\, w_\eta(x,\eta,k)\,d\eta . \]
Similar relations for the example are constructed when \(\Phi_2(y)=P(y)\Phi_1(y)\), where \(P(y)\) is an arbitrary weight function; moreover, together with the integral forms one obtains the expansions
\[ u_2=\nabla_a^{-b_2}\bigl[P(y)\exp(-axy)\nabla_{-a}^{-b_1}(e^{axy}u_1)\bigr], \]
\[ w_2=\chi_2\bigl[P(y)\chi_1^{-1}w_1\bigr], \]
\[ u_2=\nabla_a^{-b_2}\bigl[P(y)\chi_1^{-1}w_1\bigr], \]
\[ w_2=\chi_2\bigl[P(y)\exp(-axy)\nabla_{-a}^{b_1}(e^{axy}u_1)\bigr], \]
where \(\nabla_a=1+axD_y^{-1}\), \(\chi=\exp(-k^2xD_y^{-1})\). The functions \(u(x,y,b_i)\), \(w(x,y,k_i)\) are also related with respect to the variable \(x\). For example,
\[ u(b+n)(b)_n=x^{1-b}D_x^n[x^{b+n-1}u(b)]\quad(n=1,2,\ldots), \]
and, for
\[ b>0,\quad k\ne0\quad u(x,y,b)=\frac{1}{\Gamma(b)}\left(\frac{k^2}{a}\right)^b \int_0^\infty \xi^{b-1}\exp\left(-\frac{k^2\xi}{a}\right)w(x\xi,y,k)\,d\xi . \]
In addition, for \(u(x,y,b_k)\) the relations (9) and equalities of the form \(u_2=T_x[u_1]\), found in \((^1)\) (see in \((^1)\), (6) and (11)), are fulfilled.
We arrive at similar conclusions also by studying the solutions \(u_0(x,y,b_k)\) of problem (10), (12). Let us further note that (12) possesses the following property: if \(u_1(x,y,a,b)\) is an integral of this equation, then the function \(u_2=\exp(-axy)u(y,x,-a,1-b)\) also satisfies it. By means of this transformation, the results obtained for \(u(x,y,b)\) are transferred to the solution \(\tilde u(x,y,b)\) of the Goursat problem \(\tilde u(x,0)=\Phi(x)\), \(\tilde u(0,y)=0\), \(\Phi(0)=0\). Finally, denote by \(v(x,y,b)\) a solution of the parabolic-type equation
\[ xv_{xx}+bv_x-av_y=0\quad(a>0) \]
with boundary data (2) \((^{1,3,4})\). The function \(v(x,y,b)\) is also represented in the form of a Duhamel integral (3) with kernel \(\Gamma(1-b)U(x,y)=\Gamma(1-b,ax/y)\).
Theorem 3. If
\[ c_1\Gamma(b)\Gamma(1-b_2)=\Gamma(1-b_1), \]
\[ c_2\Gamma(1-b_1)\Gamma(1+\bar b)=-\Gamma(1-b_2) \]
and for arbitrary noninteger values \(b_1>0\), \(b_2>0\), \(b=b_2-b_1>0\), \(\bar b=b_1-b_2>-1\), the relations
\[ v(x,y,b_2)=c_1\int_1^\infty \xi^{b_1-1}(\xi-1)^{b-1}v(x\xi,y,b_1)\,d\xi; \]
\[ v(x,y,b_1)=c_2x^{1-b_1}D_x\int_x^\infty \xi^{b_2-1}(\xi-x)^{\bar b}v(\xi,y,b_2)\,d\xi, \]
hold, and for \(m=1,2,\ldots\)
\[ v(b)\Gamma(m+b)=\Gamma(b)x^{1-b}D_x^m[x^{m+b-1}v(m+b)]. \]
These formulas define transformation operators of Delsarte type \(\mathfrak B_x\) and \(\mathfrak B_x^{-1}\), satisfying the identities
\[ \mathfrak B_x L_x^{(1)}f(x)=L_x^{(2)}\mathfrak B_x f(x), \]
\[ \mathfrak B_x^{-1}L_x^{(2)}f(x)=L_x^{(1)}\mathfrak B_x^{-1}f(x), \]
\[ L_x^{(1)}f(x)=\mathfrak B_x^{-1}L_x^{(2)}\mathfrak B_x f(x), \]
\[ L_x^{(2)}f(x)=\mathfrak B_x L_x^{(1)}\mathfrak B_x^{-1}f(x), \]
if \(f(0)=f(\infty)=0\), \(L_x^{(k)}f=xf_{xx}+b_kf_x\) \((k=1,2)\).
Analogous results are obtained also for the integral \(v_0(x,y,b)\) with boundary data (10).
Moscow Evening
Metallurgical Institute
Received
19 XI 1960
CITED LITERATURE
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