Well-posedness and strong convergence analysis of stochastic space-time fractional wave problems driven by fractional Brownian sheet

Abstract

This paper is interesting in studying well-posedness and strong convergence analysis for stochastic space-time fractional wave equations driven by fractional Brownian sheet. We first discuss well-posedness of the original stochastic space-time fractional wave equation in terms of standard Picard's iteration argument and give regularity and Hölder continuity analysis of its mild solution. Afterwards, we introduce a Wong-Zakai approximation to fractional Brownian sheet using spectral basis so that a regularized stochastic space-time fractional wave equation is derived, and then the corresponding regularity, Hölder continuity, and error estimate of its regularized solution are also investigated. Further, we use spectral Galerkin method and backward Euler convolution quadrature scheme to discretize the regularized equation in space and time, respectively, and then we provide rigorously strong convergence analysis for the solution of discrete scheme. Finally, numerical examples are carried out to verify our theoretical convergence results.