一类非线性时间分数阶扩散方程反问题的变分型正则化

Variational Regularization of the Inverse Problem of a Class of Nonlinear Time-Fractional Diffusion Equations

考虑了一类二维非线性时间分数阶扩散方程,并从最终位置获取的测量数据来反演物质在u(0,y,t)处的物理信息.这个问题是严重不适定的,即问题的解并不连续依赖于测量数据,因此提出了变分型正则化方法来稳定求解该问题.给出了精确解与正则近似解之间的误差估计,数值算例验证了该方法的有效性.

The nonlinear time-fractional diffusion equations were considered in the 2D domain,and the physical information in initial state u(0,y,t)seriously ill-posed,that is,the solution to this problem does not continuously depend on the measured data.Therefore,a variational regularization method was proposed to construct the approximate solution to the problem,and the convergence error estimates of the exact and approximate solutions were obtained under the assumption of the priori bounds on the exact solutions.Finally,a numerical example was given to verify the effectiveness of the proposed method.