复合介质中分数阶反常热传导模型多参数数值反演算法的研究

In last decades, fractional calculus has been widely used to describe the anomalous transport phenomena in complex systems. In general, the anomalous diffusion 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 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. An naturally question of great importance is how to identify the fractional exponent and other thermal parameters in the anomalous thermal diffusion model from the experimental data, so as to establishing an convincing model that gives the best fitting for the measurement data. 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 anomalous thermal diffusion model in multilayered media. This project aims at presenting a fast and effective approach to get the estimation of fractional exponent and other thermal parameters. That is, based on the numerical solution for the anomalous heat conduction model in composite media, applying iterative regularization method to realize estimation of multi-parameters of interest. This study will enrich the fractional calculus theory and has applicable value in anomalous heat conduction. Application of the inverse problem to study thermal sub-diffusion behavior provides a fresh idea and new approach for the research on anomalous heat conduction problem.

近年来，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，时间分数阶反常热传导模型比经典模型更能刻画奇异介质传导过程中所固有的记忆和遗传效应。分数阶导数所表征的次热扩散模型中，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，而与分数阶指数紧密相关。一个自然而重要的问题是如何从实验数据中反演出刻画“记忆性”的分数阶指数和其他重要的热物性参数，从而建立与数据相吻合的反常热传导模型。该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导致了借助于有效快速数值算法实现分数阶热扩散模型中参数估计研究的必要性。该课题旨在研究复合介质中反常热传导模型分数阶指数和其他热物性参数数值反演的有效快速求解途径，即基于实验测量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。应用反问题方法研究次热扩散，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。