On the existence of solutions to the fractional derivative equations d(alpha)u/dt(alpha)+Au = f, of relevance to diffusion in complex systems

Fractional derivative equations account for relaxation and diffusion processes in a large variety of condensed matter systems. For instance, diffusion of position probability density displayed by a random walker in complex systems - such as glassy materials - is often modeled by fractional derivative partial differential equations (e.g. [1]). This paper deals with the existence of solutions to the general fractional derivative equation d(alpha)u/dt(alpha) + Au = f for 0 < alpha< 1, with A a self-adjoint operator. The results are proved using the von Neumann Dixmier theorem [2].