Weighted Reproducing Kernel Collocation Method and Error Analysis for Inverse Cauchy Problems

In this work, the weighted reproducing kernel collocation method (weighted RKCM) is introduced to solve the inverse Cauchy problems governed by both homogeneous and inhomogeneous second-order linear partial differential equations. As the inverse Cauchy problem is known for the incomplete boundary conditions, how to numerically obtain an accurate solution to the problem is a challenging task. We first show that the weighted RKCM for solving the inverse Cauchy problems considered is formulated in the least-squares sense. Then, we provide the corresponding error analysis to show how the errors in the domain and on the boundary can be balanced with proper weights. The numerical examples demonstrate that the weighted discrete systems improve the accuracy of solutions and exhibit optimal convergence rates in comparison with those obtained by the traditional direct collocation method. It is shown that neither implementation of regularization nor implementation of iteration is needed to reach the desired accuracy. Further, the locality of reproducing kernel approximation gets rid of the ill-conditioned system.