Solving the Euler–Poisson–Darboux Equation of Fractional Order

• Published in: Siberian mathematical journal 2023-05, Vol.64 (3), p.707-719
• Main Authors: Dzarakhohov, A. V., Shishkina, E. L.
• Format: Article
• Language: English
• Subjects: Cauchy problems; Existence theorems; Green's functions; Laplace transforms; Mathematics; Mathematics and Statistics; Operators (mathematics); Partial differential equations
• ISSN: 0037-4466; 1573-9260
• DOI: 10.1134/S0037446623030187

Interest in fractional ordinary and partial differential equations has been steadily increasing in the recent decades. This is due to the necessity of modeling the processes whose current state depends significantly on the previous ones, i.e., the so-called systems with residual memory. We consider the Cauchy problem for the one-dimensional, homogeneous Euler–Poisson–Darboux equation with a differential operator of fractional order in time being the left-sided fractional Bessel operator. At the same time, we use the ordinary differential operator in the space variable of the second order. We reveal the connection between the Meyer and Laplace transform which is obtained by the Poisson transform and presents a special case of the relation with the Obreshkov transformation. We prove the theorem that yields the conditions of the existence of a solution to the problem by using the Meyer transform. In this case, a solution to the problem is represented explicitly in terms of the generalized Green’s function that determines the generalized hypergeometric Fox -function.