Construction of the Riemann–Hadamard Function for the Three-Dimensional Bianchi Equation

Abstract

In this paper, we state the Darboux problem and give the definition of the Riemann–Hadamard function for a third-order equation with dominant partial derivative (the Bianchi equation). Basing on the possibility of explicitly representing the Riemann function for a certain class of Bianchi equations of the third order, which are equivalent with respect to function, we establish sufficient conditions for coefficients of the Bianchi equation that provide the construction of the Riemann–Hadamard function in terms of hypergeometric functions.