反常传热传质分数阶移动边界模型中多参数数值反演算法的研究

Fractional calculus has been widely used to describe the anomalous transport phenomena in complex systems. In general, the anomalous heat and mass transfer model with time-fractional derivative provides a more adequate and accurate description in capturing the memory and hereditary properties inherited by the transport process in heterogeneous porous media. From the point of stochastic view, fractional derivatives in time model sub-diffusion, related to long power-law waiting times between particle jumps. Specifically, in the anomalous sub-diffusive systems, the mean square displacement of the particles is no longer linear in time but highly related to the fractional exponent. What is essential is how to identify the fractional exponent and other physical parameters in the anomalous heat and mass transfer model from the experimental data, so as to establishing a convincing model that gives the best fitting for the measurement data. This project aims at establishing feasible anomalous transfer model with moving boundary based on physical law and presenting a fast and effective approach to get the estimation of fractional exponent and other physical parameters. That is, based on the convincible and feasible numerical approximation for the anomalous heat and mass transfer model with moving boundary, applying iterative regularization method to realize estimation of multi-parameters of interest. This inverse problem of parameters estimation is ill-posed, highly nonlinear and has a complex computational requirement, which leads to the necessity of seeking effective and efficient numerical algorithm for identifying parameters of the fractional heat and mass transfer model. This study will enrich the fractional calculus theory and has applicable value in anomalous transfer process. Application of the inverse problem to study fractional heat and mass transfer behavior provides a fresh idea and new approach for the research on anomalous transfer problem.

分数阶微积分理论被广泛用来描述复杂系统中的反常传输现象。事实上，时间分数阶反常传热传质模型比经典模型更能刻画奇异介质输运过程中所固有的记忆和遗传效应。分数导数所表征的输运模型中，粒子跳跃的等待时间满足幂律，粒子的平均平方位移与时间变化之间不再是简单的线性关系，而与分数指数紧密相关。一个自然而重要的问题是如何从观测数据中反演出刻画“记忆性”的分数阶指数和其他重要的物理性参数，从而建立与数据相吻合的反常传热传质模型。该课题旨在导出合理的反常传热传质移动边界模型，进而建立分数阶指数和其他物性参数数值反演的有效快速求解途径，即基于观测数据结合正问题数值算法和正则化迭代算法实现参数的估计。分数阶移动边界模型具有重要且广泛的应用背景，而参数反演问题的不适定性、高度非线性和计算复杂度等特点构成了该项目的难点。应用反问题方法研究复杂系统中的反常传输过程，从新的角度揭示分数阶微积分能够更准确的描述实际问题。

分数阶微积分算子所刻画的输运模型能够准确的描述奇异介质传输过程中的历史记忆效应和空间长程相关性。近年来，反常传输模型方面的研究不断发展和完善。本项目着眼于分数阶移动边界传输模型中的参数数值反演问题的相关研究，主要工作为：(1) 基于分数阶Fourier热导率构建了复合介质中带有分数阶导数边界的反常热传导模型，建立了正问题的有限差分格式，基于文献中的实验观测数据，利用Levenberg-Marquardt 算法实现边界热流和分数阶指数的同时反演；研究了一类非线性空间分数阶Schrödinger方程的Fourier谱方法，结合Bayesian算法实现了重要物理参数的数值估计；(2) 考虑了一类基于时空分数阶导数模型的欧式看涨期权的快速有限差分方法定价的研究；研究了基于空间分数阶导数模型的股票抵押问题，结合罚函数方法和FFT技术实现快速定价和最优执行边界的计算；(3) 探讨了一般时空分数阶偏微分方程的无网格方法，导出了径向基函数的时间/空间分数阶导数的逼近，并将该方法推广到2维非结构区域上分数阶导数模型的计算。.项目研究中发现相关课题进展的难点体现在：(1) 分数阶微积分模型正问题及反问题正则性方面的数学理论支撑不足；(2) 带有分数阶导数的移动边界传输模型中，一些重要守恒物理量需要重新建立框架。总体而言，基于分数阶微积分理论的反常传输理论具有巨大的潜力和挑战性，其中不乏很多开放性问题。