The continuous-time random walk (CTRW) model is a useful tool for the description of diffusion in nonequilibrium systems, which is broadly applied in nature and life sciences, e.g., from biophysics to geosciences. In particular, the integro-differential equations for diffusion and diffusion-advection are derived asymptotically from the decoupled CTRW model and a generalized Chapmann–Kolmogorov equation, with generic waiting time probability density function (PDF) and external force. The advantage of the integro-differential equations is that they can be used to investigate the entire diffusion process i.e., covering initial-, intermediate- and long-time ranges of the process. Therefore, this method can distinguish the evolution detail for a system having the same behavior in the long-time limit but with different initial- and intermediate-time behaviors. An integro-differential equation for diffusion-advection is also presented for the description of the subdiffusive and superdiffusive regime. Moreover, the methods of solving the integro-differential equations are developed, and the analytic solutions for PDFs are obtained for the cases of force-free and linear force.
Continuous time random walk and diffusion with generalized fractional Poisson process
