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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 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. [ABSTRACT FROM AUTHOR]

This research work is devoted to investigations of the existence and uniqueness of the solution of a non-local boundary value problem with discontinuous matching condition for the loaded equation. Considering parabolic-hyperbolic type equations involves the Caputo fractional derivative and loaded part joins in Riemann-Liouville integrals. The uniqueness of a solution is proved by the method of integral energy and the existence is proved by the method of integral equations. [ABSTRACT FROM AUTHOR]

This manuscript deals with a new set of sufficient conditions for the existence of solutions for a class of impulsive fractional neutral stochastic integro-differential systems (IFNSIDS) with nonlocal conditions (NLCs) and state-dependent delay (SDD) in Hilbert spaces. An example is provided to illustrate the obtained theory. [ABSTRACT FROM AUTHOR]