Singular Cauchy Problem Research Articles

Some singular cauchy problems

Existence and uniqueness theorems for some generalizedEuler-Poisson-Darboux equations are proved and growth and convexity properties of the solutions are studied for multiply subharmonic initial values.

The Solution and Huygens' Principle for a Singular Cauchy Problem

The Solution and Huygens' Principle for a Singular Cauchy Problem

On the singular cauchy problem for a generalization of the Euler-Poisson-Darboux equation in two space variables

The present paper contains an existence and uniqueness theorem for the singularCauchy problem for the non-homogeneousEuler-Poisson-Darboux equation: $$\begin{gathered} u_{xx} + u_{yy} - u_{tt} - \frac{k}{t}u_t = f(x,y,t),t > 0,k > 0, \hfill u(x,y,0) = u_t (x,y,0) = 0. \hfill \end{gathered}$$ The solution of this problem is used to prove an existence and uniqueness theorem for the following singularCauchy problem: $$\begin{gathered} u_{xx} + u_{yy} - u_{tt} - \frac{k}{t}u_t - h(x,y,t)u = 0,t > 0,k > 0, \hfill u(x,y,0) = g(x,y),u_t (x,y,0) = 0, \hfill \end{gathered}$$ by the method of successive approximations.