Anomalous Diffusion on a Growing Domain

The ubiquity of subdiffusive transport in physical and biological systems has led to intensive efforts to provide robust theoretical models for this phenomena. Additionally many physical and biological phenomena occur on domains which evolve with time. We have derived a master equation, using a continuous time random walk, for particles diffusing on a domain that grows with time. Our method supports systems undergoing either standard diffusion and subdiffusion. This allows us to construct models that represent physical and biological systems which incorporate both diffusion and a domain that is growing. The resulting equations feature fractional derivatives. The implementation of the master equation is illustrated with a simple model of subdiffusing proteins in a growing membrane.