Differentiation to Fractional Orders and the Fractional Telegraph Equation

Using methods of differential and integral calculus, is presented and discussed the calculation of a fractional Green's function, associated with the unidimensional case of the so-called general fractional telegraph equation with a spatial variable. This is a fractional partial differential equation with constant coefficients. The equation is solved by means of the juxtaposition of transforms, i.e., is introduced the Laplace transform in the time variable and the Fourier transform in the spatial variable. Several particular cases are discussed in terms of the parameters. Some known results are recovered. As a by-product of our main result, we obtain two new relations involving the two parameters Mittag-Leffler function.