  We consider the problem of determining a pair of functions $(u,f)$ satisfying
the heat equation $u_t -\Delta u =\varphi(t)f (x,y)$, where $(x,y)\in
\Omega=(0,1)\times (0,1)$ and the function $\varphi$ is given. The problem is
ill-posed. Under a slight condition on $\varphi$, we show that the solution is
determined uniquely from some boundary data and the initial temperature. Using
the interpolation method and the truncated Fourier series, we construct a
regularized solution of the source term $f$ from non-smooth data. The error
estimate and numerical experiments are given.
