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

Abstract

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) = 𝜑(x) and the spectrum of the fractional Laplacian operator (−Δ) s∕2 (in a bounded domain X ⊂ R d , 0 < s < 2). First, we show the existence and uniqueness of a mild solution in L p (0, T; L 2 (Ω, V)) ∩ C((0, T]; L 2 (Ω, 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 𝜑 and the source 𝑓 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.

On recovering the source term for the heat equation with memory term

“…-If m = 0 the problem (1.1) is called classical parabolic equation. This problem has been studied a lot in [9,11,10,2,18,20,12,22,21,5,17,19,14,13].…”