Uniquely identifying the variable order of time-fractional partial differential equations on general multi-dimensional domains

We proved the unique determination of the variable order in a two-scale mobile-immobile variable-order time-fractional partial differential equation with a variable diffusivity tensor imposed on a general multi-dimensional domain, with the observations of the unknown solutions on any arbitrarily small spatial domain over a sufficiently small time interval. The proved theorem provides a guidance where the measurements should be performed and ensures that with these observations the uniqueness of the identification is theoretically guaranteed.

Automated summary

The uniqueness of the determination of the variable order in a time-fractional partial differential equation with a variable diffusivity tensor on a multi-dimensional domain was proven by observing the unknown solutions in small spatial domains over a small time interval. This theorem guides where measurements should be made and ensures theoretical guarantee of identification uniqueness with these observations.