Numerical estimates for the regularization of nonautonomous ill-posed problems

The regularization of ill-posed problems has become a useful tool in studying initial value problems that do not adhere to certain desired properties such as continuous dependence of solutions on initial data. Because direct computation of the solution becomes difficult in this situation, many authors have alternatively approximated the solution by the solution of a closely-defined well-posed problem. In this paper, we demonstrate this process of regularization for the backward heat equation with a time-dependent diffusion coefficient, among other nonautonomous ill-posed problems. In the process, we provide two different approximate well-posed models and numerically compare convergence rates of their solutions to a known solution of the original ill-posed problem.