UNIQUENESS FOR INVERSE PROBLEM OF DETERMINING FRACTIONAL ORDERS FOR TIME-FRACTIONAL ADVECTION-DIFFUSION EQUATIONS

We consider initial boundary value problems of time-fractional advection-diffusion equations with the zero Dirichlet boundary value partial derivative(alpha)(t) u(x,t) = -Au(x, t), where -A = Sigma(d)(i, j=1) partial derivative(i)(a(ij)(x)partial derivative(j)) + Sigma(d)(j=1) b(j)(x)partial derivative(j) + c(x). We establish the uniqueness for an inverse problem of determining an order alpha of fractional derivatives by data u(x(0), t) for 0 < t < T at one point x(0) in a spatial domain Omega. The uniqueness holds even under assumption that Omega and A are unknown, provided that the initial value does not change signs and is not identically zero. The proof is based on the eigenfunction expansions of finitely dimensional approximating solutions, a decay estimate and the asymptotic expansions of the Mittag-Leffler functions for large time.