On a terminal value problem for stochastic space-time fractional wave equations

This work is to investigate terminal value problem for a stochastic time fractional wave equation, driven by a cylindrical Wiener process on a Hilbert space. A representation of the solution is obtained by basing on the terminal value data u(T, x) = phi(x) and the spectrum of the fractional Laplacian operator (-Delta)(s/2) (in a bounded domain X subset of R-d, 0 < s < 2). First, we show the existence and uniqueness of a mild solution in Lp(0, T; L-2(Omega, V))boolean AND C((0, T]; L-2(Omega, L-2(X))), for a suitable sub-space V of L-2(X). A limitation of this result is the lack of time continuity at t = 0. Second, we study the inverse problem (IP) of recovering u(0, x) when the terminal value data.. and the source.. are given. We give an explanation why the time continuity of the solution at t = 0 could not derived. The main reason comes from unboundedness of a solution operator, so the problem (IP) is then ill-posed, that is, recovery u(0, x) cannot be obtained in general. Hence, we propose a truncation regularization method with a suitable choice of the regularization parameter. Finally, we present a numerical example to demonstrate our proposed method.

Automated summary

This work investigates the terminal value problem for a stochastic time fractional wave equation. The existence and uniqueness of a mild solution is shown, although time continuity is lacking at t = 0. The inverse problem of recovering the initial value is studied, and it is found that the problem is ill-posed. A truncation regularization method is proposed as a solution.