Uniqueness of orders and parameters in multi-term time-fractional diffusion equations by short-time behavior

Abstract

As the most significant difference from parabolic equations, long-time or short-time behavior of solutions to time-fractional evolution equations is dominated by the fractional orders, whose unique determination has been frequently investigated in literature. Unlike all the existing results, in this article we prove the uniqueness of orders and parameters (up to a multiplier for the latter) only by principal terms of asymptotic expansions of solutions near t=0 at a single point. Moreover, we discover special conditions on unknown initial values or source terms for the coincidence of observation data. As a byproduct, we can even conclude the uniqueness for initial values or source terms by the same data. The proof relies on the asymptotic expansion after taking the Laplace transform and the completeness of generalized eigenfunctions.

Initial-boundary value problems for coupled systems of time-fractional diffusion equations

“…Therefore, similar inverse problems have been considered intensively in the case of a single equation (see e.g. [13,16,23,26,27,30] and the survey [25]). Here we attempt to extend the result from a single equation to a weakly coupled system.…”

New Generalized Jacobi Galerkin Operational Matrices of Derivatives: An Algorithm for Solving Multi-Term Variable-Order Time-Fractional Diffusion-Wave Equations

“…The significance and difficulty of proving the existence, unicity, and dependency of parameters should be emphasized. The significance of determination of ν and ν j in (1) is evident in theory and practice due to the strong relationship between them and the heterogeneity and associated physical features of media [50]. For instance, in the situation of a single term case, Cheng et al [51] first demonstrated the uniqueness of ν using the boundary condition data that were provided.…”

Cauchy–Dirichlet Problem to Semilinear Multi-Term Fractional Differential Equations

“…Exact solutions of linear multi-term fractional diffusion equations with q i being positive constants on bounded domains are searched employing eigenfunction expansions in [5,26]. Solvability along with a maximum principle and the longtime asymptotic behavior of the solution for the initial-boundary value problems to the linear subdiffusion equation with (1.1) are studied in [20,21,24] (see also references therein). Finally, we quote [12], where existence and nonexistence of the mild solutions to the Cauchy problem for semilinear subdiffusion equation with the operator (1.1) are discussed.…”