Galerkin Finite Element Approximation of a Stochastic Semilinear Fractional Wave Equation Driven by Fractionally Integrated Additive Noise

Abstract

We investigate the application of the Galerkin finite element method to approximate a stochastic semilinear space–time fractional wave equation. The equation is driven by integrated additive noise, and the time fractional order α∈(1,2). The existence of a unique solution of the problem is proved by using the Banach fixed point theorem, and the spatial and temporal regularities of the solution are established. The noise is approximated with the piecewise constant function in time in order to obtain a stochastic regularized semilinear space–time wave equation which is then approximated using the Galerkin finite element method. The optimal error estimates are proved based on the various smoothing properties of the Mittag–Leffler functions. Numerical examples are provided to demonstrate the consistency between the theoretical findings and the obtained numerical results.

Keywords

• stochastic fractional wave equation
• integrated additive noise
• Caputo derivative
• finite element method
• optimal error estimates