A Four-Parameter non-local Problem for a Fractional Wave Equation

Abstract

In this paper, we investigate a time-fractional diffusion-wave equation, where the classical second-order time derivative is replaced by a fractional derivative of order $\rho \in (1,2)$. We consider a class of non-local boundary value problems involving four real parameters $\alpha_1$, $\alpha_2$, $\beta_1$, and $\beta_2$ under general conditions.

Using the Fourier method in an abstract Hilbert space setting, we derive necessary and sufficient conditions for the well-posedness of the problem. We prove that the solution exists and is unique if certain algebraic conditions on the parameters are satisfied. In cases where these conditions fail, we describe the structure of the solution and show that uniqueness may be lost. For such cases, we also formulate orthogonality conditions that guarantee existence.