In this article, we investigate a backward problem for identifying the initial value of the time-fractional telegraph equation: \(\begin{aligned} \varvec{\partial _{t}^{2\alpha }u(x,t) + 2a \partial _{t}^{\alpha }u(x,t) - \Delta {u(x,t)} = f(x,t), \quad x \in \Omega , \quad t \in (0, T),} \end{aligned}\) where \(\varvec{0< \alpha < \frac{1}{2}}\) and \(\varvec{-\Delta }\) is the negative Laplace operator. Our analysis relies on the result of V.A. Il’in (1960) on the convergence of Fourier series in domains \(\varvec{\Omega \subset \mathbb {R}^{N}}\) . This result enables us to construct a solution representation using the Fourier method and analyze its properties. By introducing the set \(\varvec{\omega (\alpha ,-\Delta )}\) , which consists of time values \(\varvec{T^{\alpha }}\) for which the eigenvalues of \(\varvec{-\Delta }\) satisfy certain resonance conditions, we establish the existence of the solution. Specifically: If \(\varvec{T^{\alpha } \notin \omega (\alpha ,-\Delta )}\) , the solution exists and is unique. If \(\varvec{T^{\alpha } \in \omega (\alpha ,-\Delta )}\) , the solution exists, but uniqueness fails due to resonance effects. The Fourier method serves as the primary analytical tool, enabling the generalization of our findings to fractional diffusion telegraph equations with more general elliptic operators. This work contributes to the theoretical understanding of backward problems in fractional calculus and their applications in mathematical physics.
