Recovering the Fractional Order and an Initial Value Simultaneously in a Time-Fractional Diffusion-Wave Equation

For a one-dimensional time-fractional diffusion-wave equation, we investigate joint recovery of its fractional order and the initial velocity from measurements taken at one interior point. Uniqueness is proved through analytic continuation, Laplace transforms, an integral mean value theorem, and Mittag-Leffler function properties. The ill-posed reconstruction is regularized in a Bayesian setting using the iterative regularizing ensemble Kalman method. The posterior measure is shown to exist and to depend stably on the data; its numerical approximation converges under an assumed error bound for the forward solver. The resulting method is assessed in three numerical tests, which illustrate its effectiveness as well as its stability.