A backward parabolic equation with a time‐dependent coefficient: Regularization and error estimates

We consider the problem of determining the temperature $u(x,t)$ , for $(x,t)\in [0,\pi ]\times [0,T)$ in the parabolic equation with a time‐dependent coefficient. This problem is severely ill‐posed, i.e., the solution (if it exists) does not depend continuously on the given data. In this paper, we use a modified method for regularizing the problem and derive an optimal stability estimation. A numerical experiment is presented for illustrating the estimate.