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. §1. Introduction. Recently there are many anomalous diffusion phenomena observed which show different aspects from the classical diffusion. For example, Adams and Gelhar [1] pointed that field data in the saturated zone of a highly heterogeneous aquifer are not well simulated by the classical advection-diffusion equation which is based on
In this paper, based on the conditional stability estimate for
ill-posed inverse problems, we propose a new strategy for
a priori choice of regularizing parameters in Tikhonov's
regularization and we show that it can be applied to a wide
class of inverse problems. The convergence rate of the
regularized solutions is also proved.
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