Fractional telegrapher’s equation with generalized memory kernels: characteristic crossover anomalous dynamics

Abstract We consider a generalization of the fractional telegrapher’s equation by introducing memory kernels, which characterize the effects of the environment on the particle movement. By using the Fourier-Laplace transform method, we find the exact solution of the equation. We analyze the conditions under which the solution is nonnegative by using the theory of complete monotone and Bernstein functions. For a specific forms of the memory kernels we also show the nonnegativity of the solution by employing the subordination approach. The asymptotic behavior of the mean squared displacements of the fundamental solutions, in the short and long time limit, are analyzed by using Tauberian theorems. We show that the particle shows different characteristic crossover anomalous dynamics which might be useful to describe anomalous diffusive processes in complex media.