Fractional telegraph equations

Abstract

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 D 2 u+2 aDu+ Au=0 was used, where D= d/ dt, and it was shown that, as t tends to infinity, u is approximated by v, where 2 aDv+ Av=0; here A=− d 2/ dx 2 on L 2( R) , or A can be a more general nonnegative selfadjoint operator. In this paper the concern is with the fractional telegraph equation E 2 u+2 aEu+ Au=0, where E= D γ and 0< γ<1; after solving this equation it is shown that u is approximated by v, where 2 aEv+ Av=0.