Diffusion equations with Markovian switching: Well-posedness, numerical generation and parameter inference

Complex systems in real world often experience abrupt changes or jumps that cannot be accurately captured by traditional partial differential equation modeling methods. However, the Markov switching model can effectively describe these abrupt system changes. To address this issue, it is essential to consider adopting the Markov switching model. In this paper, we propose a novel diffusion equation model with Markovian switching to represent the state jump phenomena of complex systems. We then establish the well-posedness property of this model and provide a numerical method with non-uniform grids to accurately simulate the solution of this mixture model with time-varying parameters. Moreover, a discrete sparse Bayesian learning algorithm are presented to estimate the diffusion equation’s parameters from spatiotemporal data with noise. Finally, we conduct several numerical simulation experiments to validate the effectiveness and precision of the proposed method. The results demonstrate that the Markov switching model is more adaptable than the traditional models for spatiotemporal data and can more accurately simulate the state jump phenomena of real-world systems.