This work is devoted to an inverse problem of identifying a source term depending on both spatial and time variables in a parabolic equation from single Cauchy data on a part of the boundary. A Crank–Nicolson Galerkin method is applied to the least squares functional with a quadratic stabilizing penalty term. The convergence of finite dimensional regularized approximations to the sought source as measurement noise levels and mesh sizes approach zero with an appropriate regularization parameter is proved. Moreover, under a suitable source condition, an error bound and a corresponding convergence rate are proved. Finally, several numerical experiments are presented to illustrate the theoretical findings.

这项工作致力于解决反问题，即根据部分边界上的单个柯西数据，根据抛物线方程中的空间和时间变量来识别源项。Crank–Nicolson Galerkin方法应用于具有二次稳定罚分项的最小二乘泛函。证明了在测量噪声水平和网格尺寸接近零的情况下，使用适当的正则化参数，有限维正则近似收敛于所寻找的源。此外，在适当的信源条件下，证明了误差范围和相应的收敛速度。最后，提出了几个数值实验来说明理论发现。