A non-local in time telegraph equation

In this work, we derive a non-local in time telegraph equation. Our model includes as particular cases the classical telegraph equation and the fractional in time telegraph equation among others. Further, we define the fundamental solution of the problem and we prove that it can be interpreted as a probability density function. Finally, using versions of the Karamata–Feller Tauberian theorem, we study the temporal behavior of the variance of the distribution process associated with the solution of the equation in large times as well as in short times.