Abstract The theory and practice of inverse problem of partial differential equations play an important role in science and engineering applications. This paper deals with the numerical solutions of the inverse problem to simultaneously identify the right-hand and lowest coefficient that dependent on time only in a two dimensional parabolic equation. Backward difference in time and finite element procedure in space are constructed for solving this kind of inverse problem. Based on a special decomposition, the algorithm transforms the original problem into three standard elliptic problems at the new time level. Numerical examples are given to demonstrate the ability of the proposed computational algorithm for solving such coefficient inverse problem.