带有分数阶热流条件的时间分数阶热波方程及其参数估计问题

摘要

研究了Caputo导数定义下带有分数阶热流条件的一维时间分数阶热波方程及其参数估计问题。首先，对正问题给出了解析解；其次，基于参数敏感性分析，利用最小二乘算法同时对分数阶阶数α和热松弛时间τ进行参数估计；最后对不同的热流分布函数所构成的两个初边值问题，分别进行参数估计仿真实验，分析温度真实值和估计值的拟合程度。实验结果表明，最小二乘算法在求解时间分数阶热波方程的两参数估计问题中是有效的。本文为分数阶热波模型的参数估计提供了一种有效的方法。%An inversion problem of estimating parameters for a one-dimensional time fractional thermal wave equation with fractional heat flux conditions and Caputo fractional derivatives is investigated. To begin with, the analytical solution of the direct problem is obtained. Then, based on the parameter sensitivity analysis, the least-squares method is used to estimate both the fractional orderαand the relaxation timeτ simultaneously. Finally, two different heat flux distributions are given as different boundary conditions to perform the simulation experiments, respectively. By analyzing the degree of fitting curves, results show that the least-squares method performs well in parameter estimation for this fractional thermal wave equation. This study provides an effective method of estimating the parameters of fractional thermal wave equations.