Spatial boundary problem with the Dirichlet–Neumann condition for a singular elliptic equation

Abstract

The present work devoted to the finding explicit solution of a boundary problem with the Dirichlet-Neumann condition for elliptic equation with singular coefficients in a quarter of ball. For this aim the method of Green's function have been used. Since, found Green's function contains a hypergeometric function of Appell, we had to deal with decomposition formulas, formulas of differentiation and some adjacent relations for this hypergeometric function in order to get explicit solution of the formulated problem.

Holmgren problem for Helmholtz equation with the three singular coefficients

“…were constructed, respectively, in [14], [10] and [6], where α, β, γ and λ are real numbers. For equations (2) and (3), the Dirichlet, Neumann and Holmgren problems were solved in some parts of the ball [13,18,24].…”

The Dirichlet problem for elliptic equation with several singular coefficients

“…were constructed, respectively, in [24] and [15]. For equations (1.1) and (1.2), the Dirichlet, Neumann and Holmgren problems [19,31,25] were solved in some parts of the ball.…”

The Dirichlet problem for elliptic equation with several singular coefficients

“…x u x + 2β y u y + 2γ z u z = 0, 0 < 2α, 2β, 2γ < 1 (2) were constructed, respectively, in [21] and [16]. For equations (1) and (2), the Dirichlet, Neumann and Holmgren problems [20,26,32] were solved in some parts of the ball.…”