|Fractional derivatives can be used to model time delays in a diffusion process. When the order of the fractional derivative is distributed over the unit interval, it is useful for modeling a mixture of delay sources. In some special cases distributed order derivative can be used to model ultra-slow diffusion. In the fist part of the thesis, we extend the results of Baeumer and Meerschaert  in the single order fractional derivative case to distributed order fractional derivative case. In particular, we develop the strong analytic solutions of distributed order fractional Cauchy problems.
In this thesis, we also study the asymptotic behavior of the trace of the semigroup of a killed relativistic α-stable process in any bounded R−smooth boundary open set. More precisely, we establish two-term estimates of the trace with an error bound of e^(2mt)t^(2−d)/α. When m = 0, our result reduces to the result established by Ban ̃uelos and Kulczycki for stable processes given in .