Fractional telegraph equations

We investigate several aspects of the fractional telegraph equations, in an effort to better understand the anomalous diffusion processes observed in blood flow experiments. In the earlier work Eckstein et al. [Electron. J. Differential Equations Conf. 03 (1999) 39–50], the telegraph equation D2u + 2aDu + Au = 0 was used, where D = d/dt, and it was shown that, as t tends to infinity, u is approximated by v, where 2aDv + Av = 0; here A = −d2/dx2 on L2(R), or A can be a more general nonnegative selfadjoint operator. In this paper the concern is with the fractional telegraph equation E2u+ 2aEu + Au = 0, where E = Dγ and 0<γ <1; after solving this equation it is shown that u is approximated by v, where 2aEv + Av = 0.  2002 Elsevier Science (USA). All rights reserved.