弱耦合分数阶扩散方程组反系数问题的条件稳定性及数值方法研究

The studies of fractional diffusion equation, which have a wide range of practical applications in many fields of applied science, arise from modeling a class of process with temporal and spatial non-local dynamic behaviors. Hence, systematic studies on related forward and inverse problems play important roles in both theoretical and practical applications. In recent years, there exist abundant results on the studies of the forward problem of fractional differential equation; however, studies on the inverse problem of fractional differential equation are still in their infancy, which mainly concentrate in certain regularization theory and numerical methods of the inverse problem for an individual fractional differential equation. Until now there are no reports about the studies on the inverse problem of time fractional diffusion system which can efficiently describe the complex phenomena of multiphase systems, however, the involved inverse problem is very important in practical applications such as the prevention and treatment of multicomponent pollutant diffusion in underground environments, textile material design based on the moisture and heat transfer in textiles, etc. In order to develop theory and numerical methods of inverse problem for fractional differential equation, provide some new understandings for anomalous diffusion phenomena, as well as provide some guidance on relevant practical issues, we will mainly discuss the conditional stability for identifying the coefficients of a weakly-coupled fractional diffusion system by Carleman estimate of corresponding time-fractional diffusion system with half order, and the stable numerical reconstruction by utilizing the duality theory for time fractional diffusion equations followed by proposing duality methods for solving the inverse problem numerically.

分数阶扩散方程的研究源自对一类具有时空非局部动力学特性过程的建模，在应用科学诸多领域都有广泛应用，因此与之相关正反问题的研究尤为重要且具有理论和应用前景。近年来，分数阶扩散方程正问题研究已有丰富结果，而其反问题研究正处于起步阶段，主要集中在单个方程反问题的正则化理论及数值方法。对可以有效刻画复杂多相耦合反常扩散现象的时间分数阶扩散方程组反问题的研究尚无相关文献报道，而这类问题在实际中具有重要的应用价值，例如地下环境中多组分污染物扩散的防治，基于纺织材料中湿、热传递的材料设计等实际问题都需要解决此类反问题。因此本课题拟利用卡尔曼估计研究弱耦合时间1/2阶扩散方程组反系数问题的条件稳定性，从而在理论和数值上保证正则化解的有效收敛；并基于时间分数阶扩散方程对偶理论，提出数值求解反系数问题的对偶方法，实现反问题稳定的数值反演，探索分数阶扩散方程反问题数值求解的新途径，并对相关实际问题产生指导意义