A non-local problem for mixed type equations involving the Caputo fractional derivative

This work is devoted to the study of a non-local problem for an equation of mixed type, which for 0<t<β is a subdiffusion equation with a fractional derivative in the Caputo sense and for -α<t<0 is a hyperbolic type equation with a classical derivative. The Laplace operator is involved in the elliptic part of the equations. A non-local condition has the form u(x,-α)-u(x,β)=φ(x), x∈Ω, where φ(x) is a given function. Using the Fourier method, conditions were found on the function φ(x) and on the right-hand side of the equation, under which a solution to the problem exists and it is unique.