  Two main aims of this paper are to develop a numerical method to solve an
inverse source problem for parabolic equations and apply it to solve a
nonlinear coefficient inverse problem. The inverse source problem in this paper
is the problem to reconstruct a source term from external observations. Our
method to solve this inverse source problem consists of two stages. We first
establish an equation of the derivative of the solution to the parabolic
equation with respect to the time variable. Then, in the second stage, we solve
this equation by the quasi-reversibility method. The inverse source problem
considered in this paper is the linearization of a nonlinear coefficient
inverse problem. Hence, iteratively solving the inverse source problem provides
the numerical solution to that coefficient inverse problem. Numerical results
for the inverse source problem under consideration and the corresponding
nonlinear coefficient inverse problem are presented.
