Uniqueness in an Inverse Problem for One-dimensional Fractional Diffusion Equation

We consider a one-dimensional fractional diffusion equation: ∂α t u(x, t) = ∂ ∂x ( p(x) ∂u ∂x (x, t) ) , 0 < x < `, where 0 < α < 1 and ∂α t denotes the Caputo derivative in time of order α. We attach the homogeneous Neumann boundary condition at x = 0, ` and the initial value given by the Dirac delta function. We prove that α and p(x), 0 < x < `, are uniquely determined by data u(0, t), 0 < t < T . The uniqueness result is a theoretical background in experimentally determining the order α of many anomalous diffusion phenomena which are important for example in the 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. §