一个分数阶生长-抑制系统的参数反问题

AN INVERSE COEFFICIENT PROBLEM FOR A FRACTIONAL-ORDER ACTIVATOR-INHIBITOR SYSTEM

本文研究一个分数阶生长-抑制线性系统模型及其参数反问题.首先利用Laplace逆变换得到正问题解的唯一存在性.其次,考虑一个利用内点观测数据确定微分阶数与衰减率的反问题,应用极值原理在Laplace像空间中证明反演的唯一性.最后,基于正问题的有限差分解,应用同伦正则化算法进行数值反演.计算结果表明算法的收敛性及反问题的数值稳定性.

This article deals with a linear fractional-order activator-inhibitor model and a related inverse coefficient problem.The unique existence of the solution to the forward problem is obtained based on the Laplace inverse transform.An inverse problem of determining the fractional order and the attenuation rate is investigated using the measurements at one interior point,and the uniqueness of the inverse problem is proved in the mapping space of Laplace transform by the maximum principle.An implicit finite difference scheme is established to give the numerical solution of the forward problem,and numerical inversions with noisy data are performed by the homotopy regularization algorithm.The inversion solutions approximate to the exact solution demonstrate the algorithm's convergence and the numerical stability of the inverse problem.