同时反演时间分数阶方程的灌注系数和初始温度

Simultaneous Inversion of Perfusion Coefficient and Initial Temperature of Time Fractional Equation

研究了一类利用附加条件重构时间分数阶扩散方程中灌注系数和初始温度的反问题。文中的附加条件并非通常意义下的终端观测值,而是两个带有线性无关的权重函数的积分型观测数据。首先证明了一类弱形式的正问题解的存在唯一性。其次由于反问题是不适定的,利用Tikhonov正则化方法将原问题转化为变分问题,利用观测数据及先验估计,建立了相应的极小化严格凸泛函,给出了变分问题正则解的存在性、稳定性和收敛性。

A class of inverse problems of perfusion coefficient and initial temperature in time fractional diffusion equation reconstructed by additional conditions are studied.The additional condition in this paper is not the terminal observation value in the general sense,but two integral observation data with linearly independent weight functions.Firstly,the existence and uniqueness of solutions for a class of weak positive problems are proved.Secondly,because the inverse problem is ill posed,the Tikhonov regularization method is used to transform the original problem into a variational problem.Using the observation data and a priori estimation,the corresponding minimization strictly convex functional is established,and the existence,stability and convergence of the regular solution of the variational problem are given.