On the time-fractional Cattaneo equation of distributed order

The present work revisits the time-fractional Cattaneo equation (TFCE) by considering a generic class of time-fractional derivatives of distributed order, alpha is an element of (0, 1]. At first, we shed light on two (main) forms of TFCE; one (natural generalization) uses the Caputo fractional derivative and the other (Compte-Metzler generalization) is defined in the Riemann-Liouville sense, and the equivalence condition between their solutions is provided. Some special cases of these two forms are addressed. Hence, different versions of distributed-order TFCE are proposed. The fundamental solution of the two general forms of distributed-order TFCE are derived using integral transform technique. The non-negativity of solutions of TFCE and its distributed-order versions is examined using the "special-type functions" technique; Gorenflo, Luchko and Stojanovic (2013). Two classes of order distribution are considered: discrete-order and continuous-order distributions, and some random choices of non-negative order densities affirm graphically the non-negativity of the general forms of distributed-order TFCE. The double-order (as an example on the discrete-order), and exponentially exp(lambda alpha) and sinusoidally sin(pi mu alpha) distributed-order (as an example on the continuous-order) are chosen. Finally, connections between some models of distributed-order TFCE and continuous time random walk theory are established, and the corresponding mean-squared displacements are discussed. (C) 2018 Elsevier B.V. All rights reserved.