Solution of the Cauchy problem for the four-dimensional hyperbolic equation with Bessel operator

The modified Cauchy problem is investigated for a four-dimensional second order equation of hyperbolic type with spectral parameter and with the Bessel operator. The equation contains a singular differential Bessel operator on all variables. To solve the formulated problem, a generalized Erdélyi–Kober fractional order operator is applied. To solve the problem, a generalized Erdélyi–Kober fractional order operator is applied. A formula is obtained for calculating the high order derivatives of the generalized operator Erdélyi–Kober, that is then used in the study of the problem. We also consider the confluent hypergeometric function of four variables, which generalizes the Humbert function; some properties of this function are proved. Taking into account the proven properties of the Erdélyi–Kober operator and the confluent hypergeometric function, the solution of the modified Cauchy problem is presented in a compact integral form that generalizes the Kirchhoff formula. The obtained formula allows us to see directly the nature of the dependence of the solution on the initial functions and, in particular, to establish the smoothness conditions for the classical solution. The paper also contains a brief historical introduction to differential equations with Bessel operators.