A Numerical Approach to Time-Space Fractional Stochastic Diffusion Equation

The primary objective of this paper is to propose a Crank-Nicolson scheme tailored to analyze a time-space fractional stochastic diffusion equation subject to additive noise. Using the Caputo definition for time fractional derivatives and the shifted Grunwald-Letnikov definition for space fractional derivatives, a fractional order Crank-Nicolson finite difference scheme is devised to obtain a numerical solution. Considerable attention has been paid to examining the stability and convergence properties of this fractional order Crank-Nicolson method. To validate the effectiveness of the developed method, a comprehensive investigation using a test problem is conducted to assess its accuracy. The study harnesses the power of Python software to perform numerical simulations, demonstrating the efficiency and effectiveness of the proposed scheme. These simulations indicate that the numerical method developed effectively provides approximate solutions for the time-space fractional stochastic diffusion equation, especially under the influence of additive noise. Furthermore, the results gleaned from surface plots of the numerical solution for the test problem unveil a notable trend: as the intensity of the noise increases, disturbances in the surface of the mean solution escalate, underscoring the impact of noise on the system’s behavior. By offering a thorough analysis and validation of the proposed scheme, this research contributes valuable insights to the broader fields of applied mathematics and computational sciences.