On the Cauchy Problem for the Iterated Generalized Two-axially Symmetric Equation of Hyperbolic Type

Abstract

The Cauchy problem for an iterated generalized two-axially symmetric equation of hyperbolic type is investigated. Unlike traditional methods, the generalized Erdelyi–Kober operator of fractional order is applied to solve the problem. The application of this operator allows us to reduce the equations with the lowest term and with the singular Bessel operator acting in one of the variables to an equation without the Bessel operator acting on this variable and without the lowest term. The proposed approach makes it possible to construct an explicit formula for solving the formulated problem. These formulas express solutions of the Cauchy problem in terms of the sum of the initial functions under the action of powers of the Bessel operator. They allow us to directly see the character of the dependence of the solution on the initial functions.