Subdiffusion equation with Caputo fractional derivative with respect to another function

We show an application of a subdiffusion equation with 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 other type of diffusion may occur. The process can be interpreted as "ordinary" subdiffusion with fixed subdiffusion parameter (subdiffusion exponent) \(\alpha\) in which time scale is changed by the function \(g\). As example, we consider the transition from "ordinary" subdiffusion to ultraslow diffusion. The function \(g\) generates the additional aging process superimposed on the "standard" aging generated by "ordinary" subdiffusion. The aging process is analyzed using coefficient of relative aging of \(g\)--subdiffusion with respect to "ordinary" subdiffusion. The method of solving the \(g\)-subdiffusion equation is also presented.