The Nonlocal Problem for a Hyperbolic Equation with a Parabolic Degeneracy

Abstract

In this paper, the nonlocal problem for a second-order partial differential equation is considered in the characteristic domain. The considered expression represents an equation of two independent variables x and y. It is a hyperbolic-type equation in the half-plane y > 0 with a parabolic degeneracy at y = 0. The line of the parabolic degeneracy y = 0 represents the cusp locus of characteristic curves. The novelty of the formulation of the problem consists in the fact that the boundary condition contains a linear combination of operators $D_{{0x}}^{\alpha }$ and $D_{{x1}}^{\alpha }$. For α > 0, these operators are fractional differentiation operators of order α, while for α < 0 they coincide with the Riemann–Liouville fractional integration operator of order α. For various orders of the operators included in the boundary condition, the unique solvability of the formulated problem is proven. The properties of the operators of fractional integro-differentiation and the properties of the Gaussian hypergeometric function are widely used in the proof. The solution of the problem is given in the explicit form.