On the numerical solution of the Cauchy problem for the Klein–Gordon equation using the Landweber method based on integral equations

This work addresses the numerical solution of the Cauchy problem for the Klein-Gordon equation in a two-dimensional doubly connected domain. Due to the ill-posed nature of the problem, the Landweber iterative regularization method is employed. Each iteration involves solving two well-posed mixed Dirichlet–Neumann boundary value problems. For each of these problems, the boundary integral equation method is applied using potential theory, resulting in a system of second-kind integral equations. This system is solved using the Nystrom method after parameterization and with appropriate handling of the kernel singularities. Numerical experiments for the Cauchy problem are provided to demonstrate the effectiveness and accuracy of the proposed approach.