Runge-Kutta Finite Element Method for the Fractional Stochastic Wave Equation

This paper presents the development and application of the Runge-Kutta Finite Element Method (RK-FEM) to solve fractional stochastic wave equations. Fractional differential equations (FDEs) play a significant role in modelling complex systems with memory and hereditary properties, while the inclusion of stochastic components accounts for randomness inherent in physical systems. The fractional stochastic wave equation represents a natural extension of classical wave equations, incorporating both fractional time derivatives and stochastic processes to model phenomena such as anomalous diffusion and noise-driven wave propagation. We propose a hybrid numerical scheme that combines the high accuracy of the Runge-Kutta Method or temporal discretization with the flexibility of the Finite Element Method (FEM) for spatial discretization. The Caputo fractional derivative is used to describe the time-fractional component of the equation. A white noise-driven stochastic term is incorporated into the system to account for randomness. We analyze the stability and convergence properties of the RK-FEM scheme and demonstrate its effectiveness through numerical simulations. The results illustrate that the proposed method provides accurate and stable solutions for fractional stochastic wave equations, making it a robust tool for investigating wave phenomena in complex and uncertain environments.