Full Text
M. N. OLEVSKII
ON RELATIONS BETWEEN SOLUTIONS OF A SINGULAR CAUCHY PROBLEM CONCERNING A GENERALIZED DIFFERENTIAL-OPERATOR EULER–POISSON–DARBOUX EQUATION
(Presented by Academician I. N. Vekua on 1 VIII 1964)
1. In a number of recent works \((^{1-9})\), for the equation
\[ \frac{\partial^2 u}{\partial t^2}+\frac{k}{t}\frac{\partial u}{\partial t}+cu=Xu \tag{1} \]
(\(k\) and \(c\) are constants), in which \(X\) is the Laplace operator with respect to the variables \(x_1,\ldots,x_m\) \((^{1-6})\), or, more generally \((^{7,8})\), a linear differential operator of second order of elliptic type in the same variables, the singular Cauchy problem
\[ u\big|_{t=0}=f(x),\qquad \frac{\partial u}{\partial t}\bigg|_{t=0}=0;\qquad x=(x_1,\ldots,x_m). \tag{2} \]
was studied.
Independently of the nature of the linear operator \(X\) in the variables \(x\), in \((^9)\) formulas were established expressing the solution \(u(t,x;k,c;f)\) of problem (1)—(2) in terms of \(u(t,x;k',c';f)\).
In the present work there are established (Sec. 2) the corresponding relations between solutions of the more general problem
\[ \frac{\partial^n \widetilde u}{\partial t^n} +\frac{k_1}{t}\frac{\partial^{n-1}\widetilde u}{\partial t^{n-1}} +\cdots+ \frac{k_{n-1}}{t^{\,n-1}}\frac{\partial \widetilde u}{\partial t} +c\widetilde u = X\widetilde u, \tag{3} \]
\[ \widetilde u\big|_{t=0}=f(x);\qquad \frac{\partial^l \widetilde u}{\partial t^l}\bigg|_{t=0}=0,\quad l=1,\ldots,n-1;\quad n\geqslant 2, \tag{4} \]
corresponding to two systems of values of the constants \(k_1,\ldots,k_{n-1},c\). Also indicated (Sec. 4) is a formula expressing the solution of the Cauchy problem for equation (3) with \(n=1\) in terms of the solution of problem (3)—(4) with \(n>1\). In Sec. 5 some applications of the formulas obtained are given.
2.1. Equation (3) can be written in the form
\[ \left[t^{-n}\delta\bigl(\delta+n(b_1-1)\bigr)\cdots \bigl(\delta+n(b_{n-1}-1)\bigr)+c\right]u=Xu, \tag{3'} \]
where \(\delta=tD,\ D=\partial/\partial t\), and the numbers
\[ n(b_i-1)=\rho_i,\qquad i=1,\ldots,n-1, \tag{4'} \]
are found from \(k_1,\ldots,k_{n-1}\) from the system
\[ \sum_{i_1,\ldots,i_s=1}^{n-1} \rho_{i_1}\cdots\rho_{i_s} + a_1^{(n+1-s)} \sum_{i_1,\ldots,i_{s-1}=1}^{n-1} \rho_{i_1}\cdots\rho_{i_{s-1}} +\cdots+ a_{s-1}^{(n-1)} \sum_{i_1=1}^{n-1}\rho_{i_1} + a_s^{(n)} = k_s,\qquad s=1,\ldots,n-1, \tag{5} \]
whose coefficients are determined by the representation
\[ \delta^p = t^pD^p +a_1^{(p)}t^{p-1}D^{p-1} +\cdots+ a_{p-2}^{(p)}t^2D^2 +tD,\qquad p=1,\ldots,n. \tag{6} \]
The solution \(\widetilde u(t,x_1,\ldots,x_m;k_1,\ldots,k_{n-1},c;f)\) of problem (3)—(4) as a function of the parameters \(b_1,\ldots,b_{n-1},c\) will be denoted by \(u(t,x_1,\ldots,x_m;b_1,\ldots,b_{n-1},c;f)\) (more briefly: \(u(t,x;b_1,\ldots,b_{n-1},c;f)\)).
2.2. Suppose that for the values under consideration of the parameters \(b_1,\ldots,b_{n-1},c\) and of the given function \(f(x)\) there exists a solution of problem \((3')\)—(4). The following relations hold:
\[ u(t,x;b_1+\beta_1,\ldots,b_{n-1}+\beta_{n-1},c+\gamma;f)= \tag{7} \]
\[ = A_n \int_0^1\cdots\int_0^1 K(t,\xi;\widetilde b,\widetilde\beta,\gamma)\, u\!\left(t\sqrt[n]{\xi_1\cdots\xi_{n-1}},x;\, b_1,\ldots,b_{n-1},c;f\right) \,d\xi_1\cdots d\xi_{n-1}, \]
\[ K(t,\xi;b,\beta,\gamma)= \]
\[ ={}_0F_{n-1}\left(\beta_1,\ldots,\beta_{n-1};-\left(\frac{t}{n}\right)^n(1-\xi_1)\ldots(1-\xi_{n-1})\gamma\right) \prod_{i=1}^{n-1}\xi_i^{b_i-1}(1-\xi_i)^{\beta_i-1}, \]
\[ A_n^{-1}=\prod_{i=1}^{n-1}B(b_i,\beta_i);\qquad {}_0F_q(\mu_1,\ldots,\mu_q;t)= \]
\[ =\sum_{s=0}^{\infty} \frac{\Gamma(\mu_1+1)\ldots\Gamma(\mu_q+1)} {\Gamma(\mu_1+s+1)\ldots\Gamma(\mu_q+s+1)} \frac{t^s}{s!}; \]
\[ \operatorname{Re}(b_i+\beta_i)>\operatorname{Re}b_i>0^* \qquad (i=1,\ldots,n-1); \]
\[ u(t,x;b_1+\beta_1,\ldots,b_l+\beta_l,b_{l+1},\ldots,b_{n-1},c;f)= \tag{7¹} \]
\[ =A_{l+1}\int_0^1\ldots\int_0^1 \prod_{i=1}^{l}\xi_i^{b_i-1}(1-\xi_i)^{\beta_i-1} u\left(t\sqrt[n]{\xi_1\ldots\xi_l},x;b_1,\ldots,b_{n-1},c;f\right) \times \]
\[ \times d\xi_1\ldots d\xi_l,\qquad l=1,\ldots,n-1; \tag{8} \]
\[ \operatorname{Re}(b_i+\beta_i)>\operatorname{Re}b_i>0,\qquad i=1,\ldots,l; \tag{8¹} \]
\[ u(t,x;b_1,\ldots,b_{n-1},c;f)= \tag{9} \]
\[ =Bt^{\,n(1-b_{n-1})} \left\{ \prod_{i=1}^{n-1} \left[ \left(\frac{\partial}{\partial t^n}\right)^{N_i} t^{\,n(b_i-b_{i-1}+N_i)} \right] \right\} u(t,x;b_1+N_1,\ldots,b_{n-1}+N_{n-1},c;f); \]
\[ B^{-1}=\prod_{i=1}^{n-1}(b_i,N_i),\qquad (b,N)=b(b+1)\ldots(b+N-1);\quad N_i\ge 0,\ \text{integers}; \]
\[ b_0=1;\qquad b_i\ne0,-1,\ldots,-(N_i-1)\quad (i=1,\ldots,n-1); \tag{9¹} \]
here it is assumed that the linear operator \(X\), acting with respect to the variables \(x\), is commutable with integration with respect to the variables \(\xi\) (independent of \(x\)) in the right-hand sides of formulas (7), (8), and, respectively, with differentiation with respect to \(t\) in (9).
3.1. It may be practically useful to reduce problem (3)—(4) with coefficients variable in \(t\) to the corresponding problem (3)—(4) with coefficients constant in \(t\) (i.e., to the case when \(k_1=\ldots=k_{n-1}=0\)). In this connection, we note that, since
\[ (\delta+\alpha)t^l w(t,x)=t^l(\delta+\alpha+l)w(t,x) \tag{*} \]
and \((\delta-(n-1))\ldots(\delta-1)\delta w=t^nD^n w\), equation \((3')\) for \(b_i=i/n\) \((i=1,\ldots,n-1)\) turns into \((D^n+c)u=Xu\). More generally: to the system of parameter values \(k_1,k_2=\ldots=k_{n-1}=0\) in (3) there corresponds the system of parameters \(b_1=(k_1+1)/n,\ b_2=2/n,\ldots,\ b_{n-1}=(n-1)/n\) in \((3')\).**
3.2. In order to express \(\widetilde u(t,x;k'_1,\ldots,k'_{n-1},c';f)\) in terms of \(\widetilde u(t,x;k_1,\ldots,k_{n-1},c;f)\), one should, from system (5), using the values \(k'_1,\ldots,k'_{n-1}\) and \(k_1,\ldots,k_{n-1}\), find the corresponding \(b'_1,\ldots,b'_{n-1}\) and \(b_1,\ldots,b_{n-1}\). In the cases \(\operatorname{Re}b'_i>\operatorname{Re}b_i>0\) \((i=1,\ldots,n-1)\), or \(b'_i=b_i-N_i\) \((i=1,\ldots,n-1)\), \(N_i\) are integers \(\ge0\), \(b'_i\ne0,-1,-2,\ldots\), we achieve the goal by applying (7) or, respectively, (9). In all other cases, but under the conditions \(\operatorname{Re}b_i>0\) and \(b'_i\ne0,-1,-2,\ldots\) \((i=1,\ldots,n-1)\), by successive use of (9) and (7), with appropriate \(N_i\) and \(\beta_i\), we obtain the required reduction.*** In particular, the formula of transition from \(u(c')\equiv u(t,x;b_1,\ldots,b_{n-1},c';f)\) to \(u(c)\) can be obtained successively—
* For uniqueness of the solution of problem \((3')\)—(4), it is necessary that \(\operatorname{Re} b_i\ge 1/n,\ i=1,\ldots,n-1\) (see items 3, 4).
** The parameters \(b_1,\ldots,b_{n-1}\) for arbitrarily prescribed \(k_1,\ldots,k_{n-1}\) are the roots of an algebraic equation of degree \((n-1)\), obtained directly from system (5); among them, even for real \(k_1,\ldots,k_{n-1}\), complex ones may occur.
*** If one of the parameters \(b_i\) (say \(b_1\)) is equal to \(-r\), \(r\) an integer \(\ge0\), then, using (9), with \(N_1=r-1\), we can reduce \(u\) for \(b_1=-r\) to the function \(u\) for \(b_1=-1\), and the latter (by a formula analogous to \((1_4)\) from (9)) can be brought to \(u\) for \(b_1>0\).
by applying formula (9) successively to \(u(c')\) with \(N_1=\cdots=N_{n-1}=1\), and formula (7) with \(\beta_1=\cdots=\beta_{n-1}=1,\ \gamma=c'-c\). The formula thus obtained, when written for \(b_i=i/n\) \((i=1,\ldots,n-1)\), establishes a connection between the solutions of the Cauchy problem (4) for the equations \((D^n+c)u=Xu\) and \(D^n u=Xu,\ n\geqslant 2\). For \(n=2\) it was obtained in \({}^{(12)}\); see also formula \((1_3)\) in \({}^{(9)}\); compare in this connection \({}^{(13)}\), since in the case under consideration \((k_1=\cdots=k_{n-1}=0)\) the singularity of the problem disappears.
3.3. For \(\gamma=0\) formula (7) turns into formula (8), but with \(l=n-1\). Formula (9) is obtained by inverting (8) for integer \(\beta_i\). Applying (8) for \(l=1\), in view of Sec. 3.1, we obtain a formula expressing \(\tilde u(t,x;k_1,0,\ldots,0,c;f)\), \(k_1>0\), in terms of \(\tilde u(t,x;0,0,\ldots,0,c;f)\), noted (in a special case) in \({}^{(10)}\); see also \({}^{(11)}\). For \(n=2\) formulas (7)—(9) and (12) were obtained in \({}^{(9)}\).
3.4. Putting in equation \((3')\) \(u=t^l v\) and taking into account the equality (*) (Sec. 3.1) for \(l=(1-b_i)n\) \((i=1,\ldots,n-1)\), we find that, along with
\(u_0\equiv u(t,x;b_1,\ldots,b_{n-1},c;f)\), the equation \((3')\) is also satisfied by
\[ u_i\equiv t^{n(1-b_i)}u(t,x;b_1-b_i+1,\ldots,b_{i-1}-b_i+1,2-b_i,b_{i+1}-b_i+1,\ldots,\ldots,b_{n-1}-b_i+1,c;f)* \]
\((i=1,\ldots,n-1)\); hence it follows that if a solution of problem \((3')\)—(4) exists for arbitrary values of \(b_i\) \((i=1,\ldots,\ldots,n-1)\), then it is not unique if at least one of the parameters \(b_i\) \((i=1,\ldots,n-1)\), for example \(b_j\), is such that \(\operatorname{Re} b_j<1/n\), for in that case \(u_j\) is a solution of equation \((3')\) satisfying zero initial conditions.
3.5. Formula (8) can be represented in the form
\[ u(t,x;b_1+\beta_1,\ldots,b_l+\beta_l,b_{l+1},\ldots,b_{n-1},c;f)= \]
\[ =\int_0^1 K_l(\xi;b_1,\ldots,b_l;\beta_1,\ldots,\beta_l)\, u\bigl(t\sqrt[n]{\xi},x;b_1,\ldots,b_{n-1},c;f\bigr)\,d\xi \tag{8'} \]
\[ (l=1,\ldots,n-1) \]
with kernel obtained recurrently from
\[ K_1(\xi;b_1;\beta_1)= \frac{1}{B(b_1,\beta_1)}\xi^{b_1-1}(1-\xi)^{\beta_1-1} \]
by the formula
\[ K_p(\xi;b_1,\ldots,b_p;\beta_1,\ldots,\beta_p)= \]
\[ =\int_\xi^1 K_1(z;b_p;\beta_p)\, K_{p-1}\!\left(\frac{\xi}{z};b_1,\ldots,b_{p-1};\beta_1,\ldots,\beta_{p-1}\right) \frac{dz}{z}\quad (p=2,\ldots,n-1). \tag{10} \]
- Let \(w(t,x;f)\) be the solution of the problem
\[ \partial w/\partial t=Xw,\qquad w|_{t=0}=f(x), \tag{11} \]
and let \(u(t,x;b_1,\ldots,b_{n-1};f)\) be the solution of problem \((3')\)—(4) for \(c=0\). The formula
\[ w(t,x;f)=\int_0^\infty\cdots\int_0^\infty \prod_{i=1}^{n-1}\frac{e^{-\xi_i}\xi_i^{\,b_i-1}}{\Gamma(b_i)}\, u\bigl(n\sqrt[n]{\xi_1\cdots\xi_{n-1}}\,t,x;b_1,\ldots,b_{n-1};f\bigr)\, d\xi_1\cdots d\xi_{n-1}, \tag{12} \]
holds; here it is assumed that the linear operator \(X\) is interchangeable with integration with respect to \(\xi_1,\ldots,\xi_{n-1}\) in the right-hand side of (12).
Concerning formula (12), there is a statement analogous to that made in Sec. 3.5 with respect to formula (8).
- Some applications of the preceding results.
5.1. Let \(X=\nabla_n,\ \nabla_n\equiv \partial^n/\partial x_1^n+\cdots+\partial^n/\partial x_m^n\), and let \(f(x)\) be an analytic function \((n>2)\). Put
\[ s\{f\}=\frac{1}{n^m} \sum_{l_1,\ldots,l_m=1}^{n} f\bigl(x_1+t a_1 e^{2l_1\pi i/n},\ldots,x_m+t a_m e^{2l_m\pi i/n}\bigr), \tag{13} \]
* Thus we arrive, for equation \((3')\), at a generalization of one of the recurrence relations of Darboux—Weinstein \({}^{(1,2)}\); the second generalization has the form
\[ C_1 t^{1-n}D u(t,x;b_1,\ldots,b_{n-1},c;f) = u(t,x;b_1+1,\ldots,b_{n-1}+1,c;(X-c)f), \]
\[ C_1\equiv n^{\,n-1}b_1\ldots b_{n-1}. \]
where \(a_j=(\eta_{1,j}\ldots \eta_{n-1,j})^{2/n}\) \((j=1,\ldots,m)\), and \(\eta_{k,1},\ldots,\eta_{k,m}\) are the polar coordinates of a point of the sphere \(S_{k,m}\) \((k=1,\ldots,n-1)\) of radius 1 with center at the origin. For \(m>1\) consider the following averaging (14) (with weight) of the function \(f(x)\):
\[ V_n(t,x;f)=\frac{1}{\Omega} \int_{S^+_{1,m}}\cdots\int_{S^+_{n-1,m}} s\{f\}\prod_{j=1}^{n-1}(\eta_{j,1}\ldots \eta_{j,m})^{(2j-n)/n} \,ds_{1,m}\cdots ds_{n-1,m}; \tag{14} \]
\[ \Omega=2^{(1-n)(m-1)} \prod_{j=1}^{n-1} \left[\Gamma\left(\frac{j}{n}\right)\right]^m \left[\Gamma\left(\frac{mj}{n}\right)\right]^{-1}; \]
\(S^+_{k,m}\) is the region of the sphere \(S_{k,m}\) for which \(\eta_{k,j}\geqslant 0\) \((j=1,\ldots,m)\), and \(dS_{k,m}\) is the surface element of the sphere \(S_{k,m}\).
It can be shown that \(V_n(t,x;f)\) satisfies the equation
\[ t^{-n}\delta(\delta+m-n)\ldots(\delta+(n-1)m-n)V_n=\nabla_n V_n \tag{15} \]
and the initial conditions (4)\(^*\). Consequently,
\[ V_n(t,x;f)=u(t,x;m/n,\ldots,(n-1)m/n,0;f). \tag{16} \]
Thus, using (14), (16) for \(m>1\), we can, by formulas (7), (9) (see item 3.2), write the solution of problem (3)—(4), \(X=\nabla_n\), explicitly (in the class of analytic initial data) for arbitrary \(b_1,\ldots,b_{n-1},c\). Similarly also for \(m=1\); here it suffices to rely on the solution of problem (4) for the equation \(D^n u=\partial^n u/\partial x_1^n\), namely:
\[ u\left(t,x;\frac1n,\ldots,\frac{n-1}{n},0;f\right) = \frac1n[f(x+\varepsilon_1 t)+\ldots+f(x+\varepsilon_n t)], \tag{17} \]
where \(\varepsilon_1,\ldots,\varepsilon_n\) are the roots of the \(n\)-th degree of 1.
5.2. Using formula (12) and taking into account (16) (and, for \(m=1\), (17)), we can also write the solution of problem (11) for \(X=\nabla_n\) in explicit form (in the class of analytic initial data).
5.3. If we assume that the operator \(X\) is the operator of multiplication by the number \(\lambda\), then
\[ u(t,x;b_1,\ldots,b_{n-1},c;f) = {}_0F_{n-1}(b_1,\ldots,b_{n-1};(t/n)^n(\lambda-c))\,f(x) \]
and formula (7) turns into the “integral addition theorem” for the hypergeometric function \({}_0F_{n-1}(b_1,\ldots,b_{n-1};t)\), namely:
\[ {}_0F_{n-1}(b_1+\beta_1,\ldots,b_{n-1}+\beta_{n-1};t_1+t_2)= \]
\[ = A_n \int_0^1\cdots\int_0^1 {}_0F_{n-1}(b_1,\ldots,b_{n-1};\xi_1\ldots \xi_{n-1}t_1)\, {}_0F_{n-1}(\beta_1,\ldots,\beta_{n-1};(1-\xi_1)\ldots \]
\[ \ldots(1-\xi_{n-1})t_2) \prod_{i=1}^{n-1}\xi_i^{b_i-1}(1-\xi_i)^{\beta_i-1}\,d\xi_i^{**}. \tag{18} \]
Relation (18), as well as the relations obtained from formulas (8), (9), (12) under the assumption about the operator \(X\) made in item 5.3, being found independently, can be regarded as a heuristic source of formulas (7), (8), (9), (12); see in this connection \((^{15})\).
Received
22 VII 1964
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* For \(n=2\), (15) is the Darboux–Asgeirsson equation, and \(V_2(t,x;f)\) is the mean value of the function \(f(x)\) on the sphere of radius \(t\) with center at the point \(x\).
** For \(n=2\) this is the well-known Sonine formula for Bessel functions \((^{16})\).