Solution to the fractional order Euler–Poisson–Darboux equation

Interest in fractional order equations, both ordinary and partial, has been steadily growing in recent decades. This is due to the need to model processes in which the current state significantly depends on the previous states of the process, i.e. the so-called systems with “residual” memory. The paper considers the Cauchy problem for a one-dimensional, homogeneous Euler–Poisson–Darboux equation with a differential operator of fractional order in time, which is a left-sided Bessel operator of fractional order. At the same time, the usual second-order differential operator is used for the spatial variable. The connection between the Meyer and Laplace transformation obtained using the Poisson transformation, which is a special case of the relation with the Obreshkov transformation, is shown. A theorem is proved that determines the conditions for the existence of a solution to the problem under consideration. When proving the theorem of the existence of a solution, the Meyer transform was used. In this case, the solution of the problem is presented explicitly through the generalized Green's function. The Green function constructed to solve the problem under consideration is defined by means of the generalized hypergeometric Fox $H$-function.