二维复杂区域时间分数阶扩散方程源项识别数值方法研究

The environmental pollution problem is always one of the international focus issues. The study on source identification of diffusion equation is necessary to prevent and control the environmental pollution. In recent years, studies have found that anomalous diffusion phenomena can be well described by time-fractional diffusion equations and the source term identification problem has been widely concerned. In the practical application, the considered domain of concerned problem is usually multiply connected region with irregular physical boundaries. The finite difference method is difficult to tackle this kind of problem, and also using the classical finite element method may cause expensive numerical computation and low efficiency. In term of the advantages of the radial basis function method for complex physical boundary problems, this topic intends to provide numerical methods to identify the source terms of two dimensional time-fractional diffusion equations on three different domains, a simply connected region with irregular boundaries，a multiply connected region with regular and with irregular boundaries. This program will theoretically investigate the existence and uniqueness of inversion solution with different additional data, then further analyze the stability and convergence of numerical methods and estimate the corresponding error. Finally, numerical identification methods can be proposed by using the radial basis function method, improved regularization method and the optimization algorithms containing particle swarm optimization algorithm and genetic algorithm. The study is to provide theoretical foundation for pollution prediction and control.

环境污染问题一直是目前国内外关注的焦点问题之一，研究污染物扩散方程源项识别问题对预防和控制环境污染具有重要的实际意义。近年来，研究发现时间分数阶扩散方程能够较好地描述污染物的反常扩散现象，因而其源项识别问题也引起了广泛关注。然而，实际应用中问题的影响区域是多连通的且物理边界不规则，有限差分法难以处理此类问题，而有限元法需划分很多次单元，计算量大且效率较低。鉴于径向基函数法求解复杂物理边界问题的优势，本课题拟研究求解二维复杂区域即单连通区域不规则边界、多连通区域规则或不规则边界条件下的时间分数阶扩散方程源项识别的数值方法。首先分析不同附加数据下问题解的存在唯一性以及数值识别方法的稳定性和收敛性，并给出相应的误差估计。最后采用局部径向基函数法并结合改进正则化方法，通过粒子群算法以及遗传算法等最优化理论，给出源项识别的数值方法，为污染预测和治理提供了理论依据。

研究污染物扩散方程源项识别问题对预防环境污染和环境治理具有重要的实际意义。本项目针对二维区域复杂物理边界情形，研究时间分数阶扩散方程模型在不同附加数据下的污染源的数值反演问题。研究的关键问题是正则解的唯一性分析以及数值算法的稳定性和收敛性分析。针对复杂边界问题，本项目引入了无网格算法，利用局部径向基函数，解决了空间离散问题，其目的是为了提高计算效率和降低计算机存储量。无网格算法可以有效地解决边界的无规则以及有洞区域的布点问题，且局部径向基函数法可以充分利用局部信息减少计算量。针对连续源问题，本项目解决了仅含时间变量和仅含空间变量的点源问题，面源问题比较复杂，有待进一步研究。针对识别问题，主要采用正则化方法，其关键技术是稳定泛函的选取以及正则化参数的选择。在数值实现过程中，主要考虑了后验策略，结合自适应方式对正则化参数进行修正和选取。针对问题解的唯一性理论分析，还需要进一步研究和探索，目前尚未给出统一性的结论。本项目所研究的科学问题是关于环境污染预警和治理的基础应用性研究问题，研究成果能够为防治环境污染策略提供一定的理论依据和提供必要的防治策略。