The authors consider the following fractional telegraph equation \[ (t(\partial/(\partial t)))^{\nu}(t(\partial/(\partial t)))^{\nu}u+2\lambda (t(\partial/(\partial t)))^{\nu}u=c^2((\partial^2 u)/(\partial x^2)) \] where \((t(\partial/(\partial t)))^{\nu}\) stands for the Caputo-like modification of the Hadamard derivative of order \(\nu\), and the generalized telegraph equation \[ D_{t}^{2 \gamma,\delta}u (x,t)+2\lambda D_{t}^{\gamma,\delta}u (x,t)=c^2 D_{0x}^{\alpha}u (x,t)-\omega u (x,t)+F(x,t),x \in R,t\geq 0 \] where \(D_{t}^{\gamma,\delta}\) is the Hilfer fractional derivative and \(D_{0x}^{\alpha}\) is the Riesz-Feller derivative. They found explicit Fourier transforms of their solutions and gave some interesting probabilistic interpretations.