Backward Diffusion-Wave Problem: Stability, Regularization, and Approximation

Abstract

Our aim is the development and analysis of the numerical schemes for approximately solving the backward diffusion-wave problem, which involves a fractional derivative in time with order $\alpha\in(1,2)$. From terminal observations at two time levels, i.e., $u(T_1)$ and $u(T_2)$, we simultaneously recover two initial data $u(0)$ and $u_t(0)$ and hence the solution $u(t)$ for all $t > 0$. First, existence, uniqueness, and Lipschitz stability of the backward diffusion-wave problem were established under some conditions about $T_1$ and $T_2$. Moreover, for noisy data, we propose a quasi-boundary value scheme to regularize the “mildly" ill-posed problem, and show the convergence of the regularized solution. Next, to numerically solve the regularized problem, a fully discrete scheme is proposed by applying the finite element method in space and convolution quadrature in time. We establish error bounds of discrete solutions in both cases of smooth and nonsmooth data. Those estimates are very useful in practice since they indicate the ways to choose discretization parameters and regularization parameter, according to the noise level. Intensive numerical experiments show the accuracy of the numerical scheme and support our theoretical results.