We discuss and derive the analytical solution for three basic problems of the so-called time-fractional telegraph equation. The Cauchy and Signaling problems are solved by means of juxtaposition of transforms of the Laplace and Fourier transforms in variable t and x, respectively. the appropriate structures and negative prosperities for their Green functions are provided. The boundary problem in a bounded space domain is also solved by the spatial Sine transform and temporal Laplace transform, whose solution is given in the form of a series.
F. Huang. "Analytical Solution for the Time-Fractional Telegraph Equation." J. Appl. Math. 2009 1 - 9, 2009. https://doi.org/10.1155/2009/890158