一类时间分数阶扩散方程中的源项反演解法

The Numerical Method for Reconstructing Source Term in a Time Fractional Diffusion Equation

考虑了一类具有Neumann边界的时间分数阶扩散方程源项反演问题.首先,从分离变量法出发将反问题归结为第1类Volterra积分方程,从而揭示出反问题的不适定性;其次,为了获得反问题的条件稳定性,通过分数阶数值微分将第1类Volterra积分方程转化为第2类Volterra积分方程,建立源项反问题的条件稳定性和误差估计;最后,引进磨光正则化,获得稳定的分数阶数值导数,将其代入求解第2类积分方程,从而稳定地重建出仅依赖时间变量的源项.数值实验结果验证了所得反演算法的有效性.

An inverse source problem in a time fractional diffusion equation with Neumann boundary is considered.Firstly,from the method of separation of variables for solving the direct problem,the inverse source problem is turned into a Volterra integral equation of the first kind,which reveals ill-posedness of the inverse problem.Secondly,for obtaining conditional stability of the inverse problem,the Volterra integral equation of the first kind is transformed into a second kind Volterra integral equation by using fractional derivative,then the conditional stability and error estimate are established.Lastly,from stable approximation of the fractional derivative computed by utilizing the mollification regularization,the time-dependent source term is reconstructed stably by solving the Volterra integral equation of the second kind.Results of numerical experiments verify the effectiveness of the inversion algorithm.