Generalized Finite Difference Method for Solving Stochastic Diffusion Equations

Abstract

Stochastic diffusion equations are crucial for modeling a range of physical phenomena influenced by uncertainties. We introduce the generalized finite difference method for solving these equations. Then, we examine its consistency, stability and convergence in mean-square, showing that the proposed method preserves stability and demonstrates favorable convergence characteristics under suitable assumptions. In order to validate the methodology, we present numerical results in one-, two-, and three-dimensional space domains.

Figures (8)

• Figure 1: Central node with its star.
• Figure 2: One dimension grid of nodes.
• Figure 3: Plot of the mean solution and the analytical solution for the stochastic diffusion equation in one dimension.
• Figure 4: Plot of the mean solution for the stochastic diffusion equation for one dimension with $\rho=0.01$ and $\mu=0.1$.
• Figure 5: Irregular clouds of points.

Theorems & Definitions (9)

• Definition 1
• Theorem 1
• proof
• Definition 2
• Theorem 2
• proof
• Definition 3
• Theorem 3
• proof