Particular solutions of the multidimensional singular ultrahyperbolic equation generalizing the telegraph and Helmholtz equations

Abstract

This article deals with the construction of particular solutions for a second-order multidimensional singular partial differential equation, which generalizes the famous telegraph and Helmholtz equations. The constructed particular solutions are expressed in terms of the multiple confluent hypergeometric function, which is analogous to the multiple Lauricella function and the famous Bessel function. A limit correlation theorem for the multiple confluent hypergeometric function is proved, and a system of partial differential equations associated with the confluent function is derived. Thanks to the proven properties of the multiple confluent hypergeometric function. The particular solutions of the multidimensional partial differential equation with the singular coefficients are written in explicit forms and it is determined that these solutions have a singularity at the vertex of a multidimensional cone.

Keywords

• particular solution
• Lauricella function
• multiple confluent hypergeometric function
• a limit correlation theorem
• a system of the partial differential equations