On the nonlocal problem for a hyperbolic equation with a parabolic degeneration

A. V. Tarasenko ; J. O. Yakovleva

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 6 (2022), pp. 60-66

Résumé

The nonlocal problem for the second-order partial differential equation is investigated in the characteristic region. The given equation is the equation of two independent variables $x$, $y$. The given equation is an equation of hyperbolic type in the half-plane $y>0$ with parabolic degeneracy at $y=0$. The line of parabolic degeneracy $y = 0$ represents the geometric locus of the cusp points of the characteristic curves. The novelty of the statement of the problem lies in the fact that the boundary condition contains a linear combination of the operators $D_{0x}^{\alpha}$ and $D_{x1}^{\alpha}$. These operators for $ \alpha>0 $ are fractional differentiation operators of order $ \alpha $, and for $ \alpha 0 $ they coincide with the Riemann-Liouville fractional integration operator of order $ \alpha $. The unique solvability of the posed problem is proved for various values of the orders of the operators included in the boundary condition. The properties of the operators of fractional integro-differentiation and the properties of the Gauss hypergeometric function are used in the proof. The solution of the problem is given in explicit form.

Détail

Keywords: boundary value problem, operator of fractional integration, operator of fractional differentiation, hypergeometric function.
Mots-clés : Euler-Darboux equation