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

Abstract

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. This paper deals with the existence of solutions to the general fractional derivative equation d^αu/dt^α+Au = f for 0 < α < 1, with A a self-adjoint operator. The results are proved using the von Neumann–Dixmier theorem.