G-subdiffusion equation with Caputo fractional derivative with respect to another function as a universal model of anomalous transport: A Survey

The g-subdiffusion equation is a significant generalization of the ordinary fractional subdiffusion equation. Its main purpose is to describe diffusion processes occurring in media whose structure or properties evolve over time. This advanced modeling capability is achieved by incorporating the fractional time derivative of Caputo with respect to another function. We show an application of a subdiffusion equation with a Caputo fractional time derivative with respect to another function g to describe subdiffusion in a medium having a structure evolving over time. In this case, a continuous transition from subdiffusion to another type of diffusion may occur. The process can be interpreted as ordinary subdiffusion in which the time scale is changed by the function g. We will present several examples of complex diffusion processes in which, by choosing the appropriate function g, we will obtain a smooth transition between different types of anomalous diffusion. We will also show a method for solving this type of equation using the generalized Laplace transform.