Summary
The present paper contains an existence and uniqueness theorem for the singularCauchy problem for the non-homogeneousEuler-Poisson-Darboux equation:
The solution of this problem is used to prove an existence and uniqueness theorem for the following singularCauchy problem:
by the method of successive approximations.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Bibliography
F. Tricomi,Sulle equazioni lineari alle derivate parziali di 2 o ordine, di tipo misto, « Atti della R. Accademia Nazionale dei Lincei », Serie Quinta, Mem. Cl. Sci. Fis. Mat. Nat.,14 (1923), p. 134.
R. Courant andD. Hilbert,Methoden der Mathematischen Physik,2, Berlin (1937), p. 402.
J. Kampé de Fériet,La fonction hypergéométrique, « Mémorial des Sci. Mat. »,85, Paris (1937).
M. Riesz,L'intégrale de Riemann-Liouville et le problème de Cauchy, « Acta Math. »,81 (1948), pp. 1–223.
M. N. Olevskii, Solution of the Dirichlet problem for the equation\(\Delta u + \frac{p}{{x_n }}\frac{{\partial u}}{{\partial x_n }} = \rho\) for a hemisperical region, (in Russian), « Doklady Akad. Nauk SSSR »,64 (1949), pp. 767–770.
L. Gårding,Linear hyperbolic partial differential equations with constant coefficients, « Acta Math. »,85 (1951), pp. 1–61.
P. Germain andR. Bader,Sur quelques problèmes aux limites, singuliers, pour une équation hyperbolique, « Comptes Rendus », Paris,231 (1950), pp. 268–270.
F. Bureau,Intégrales de Fourier et problème de Cauchy, « Annali di Matematica », series 4,32 (1951), pp. 205–233 (especially pp. 206–207).
J. Hadamard,Lectures on Cauchy's problem in linear partial differential equations, New York (1952).
M. B. Kapilevic,On an equation of mixed elliptic-hyperbolic type, (in Russian), « Mat. Sbornik », (new series),30 (1952), pp. 11–38.
A. Weinstein,On the wave equation and the equation of Euler-Poisson, « Proc. 5th Symp. in Applied Math. of American Math. Soc. », Carnegie Inst. of Tech. (1952). Published in New York (1954). See also « Comptes Rendus »,234 (1952), pp. 2584–2585.
J. B. Diaz andH. F. Weinberger,The solution of the singular initial value problem for the Euler-Poisson-Darboux equation, « Proc. American Math. Soc. »,4 (1953), pp. 703–715.
R. M. Davis, Abstract of a paper presented to the American Mathematical Society, meeting at New York (April 1954).
A. Weinstein,The singular solutions and the Cauchy problem for generalized Tricomi equations, « Communications on Pure and Applied Mathematics »,7 (1954), pp. 105–116.
J. B. Diaz andA. Weinstein,On the fundamental solutions of a singular Beltrami operator,R. von Mises Anniversary Volume, to appear 1954.
E. K. Blum,The solutions of the Euler-Poisson-Darboux equation for negative values of the parameter, « Duke Journ. of Math. », to appear 1954.
E. K. Blum,The Euler-Poisson-Darboux equation in the exceptional cases, « Proc. American Math. Soc. », to appear 1954.
Author information
Authors and Affiliations
Additional information
The research of this author was supported by the United States Air Force through the Office of Scientific Research.
Rights and permissions
About this article
Cite this article
Diaz, J.B., Ludford, G.S.S. On the singular cauchy problem for a generalization of the Euler-Poisson-Darboux equation in two space variables. Annali di Matematica 38, 33–50 (1955). https://doi.org/10.1007/BF02413513
Issue Date:
DOI: https://doi.org/10.1007/BF02413513