Numerical Method for Solving an Inverse Boundary Problem with Unknown Initial Conditions for Parabolic PDE Using Discrete Regularization

Abstract

We consider an inverse boundary values problem for parabolic PDE with unknown initial conditions. In this problem both Dirichlet and Neumann boundary conditions are given on a part of the boundary and it is required to determine the corresponding function on the remaining part of the boundary. To solve this problem, the numerical method based on finite difference schemes and regularization technique is proposed. The computing scheme involves solving the equation for each spatial step that allows to obtain the numerical solution in internal points of the domain and on the boundary. We prove a conditional stability of the method. The reliability and the efficiency of the method were confirmed by computational results.