Uniqueness of determining the variable fractional order in variable-order time-fractional diffusion equations

Abstract

Variable-order time-fractional diffusion equations provide very competitive modeling capabilities of challenging phenomena including anomalously subdiffusive transport of solutes in heterogeneous porous media and memory effect as constant-order time-fractional diffusion equations do, while eliminating the nonphysical singularity of the solutions near the initial time of the latter. Moreover, variable-order time-fractional diffusion equations themselves occur in many applications.We study the initial-boundary value problem of variable-order time-fractional diffusion equations and prove the uniqueness of determining the variable order in the initial-boundary value problem, from the observations of its solution on a sufficiently small open spatial interval over a sufficiently small time interval.