Nonlocal Boundary-Value Problem for a Parabolic-Hyperbolic Equation of the Second Kind

Abstract

In the work, a boundary value problem with non-local condition for the second kind degenerated parabolic-hyperbolic equation, was investigated. The solution of the considered problem was searched in the form of solution of the first boundary value problem in the parabolic domain and also as the solution of modified Cauchy’s problem in the hyperbolic domain. The special class $$R^{\lambda}_{00}$$ of regular solutions of modified Cauchy’s problem for hyperbolic type equation was introduced and the view of the solution simplified. Considered problem equivalently reduced (in the meaning uniqueness and existence of the solution) to Volterra’s integral equation of the second kind. Necessary conditions were found for providing continuity of the kernel and the independent term of the obtained integral equation. The identities involving hyperheometric functions were proved by using Laplace transform. In the work, the properties of Bessel functions, hyperheometric functions, Euler’s gamma function and Pohgammer’s symbol were used.