Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation

We consider a one-dimensional fractional diffusion equation: partial derivative(alpha)(t) u(x, t) = partial derivative/partial derivative x (p(x) partial derivative u/partial derivative x (x, t)), 0 < x < l, where 0 < alpha < 1 and partial derivative(alpha)(t) denotes the Caputo derivative in time of order alpha. We attach the homogeneous Neumann boundary condition at x = 0, l and the initial value given by the Dirac delta function. We prove that alpha and p(x), 0 < x < l, are uniquely determined by data u(0, t), 0 < t < T. The uniqueness result is a theoretical background in experimentally determining the order alpha of many anomalous diffusion phenomena which are important, for example, in environmental engineering. The proof is based on the eigenfunction expansion of the weak solution to the initial value/boundary value problem and the Gel'fand-Levitan theory.