Determination of a two-dimensional heat source: Uniqueness, regularization and error estimate

Let QQ be a heat conduction body and let ϕ=ϕ(t)ϕ=ϕ(t) be given. We consider the problem of finding a two-dimensional heat source having the form ϕ(t)f(x,y)ϕ(t)f(x,y) in QQ. The problem is ill-posed. Assuming ∂Q∂Q is insulated and ϕ≢0ϕ≢0, we show that the heat source is defined uniquely by the temperature history on ∂Q∂Q and the temperature distribution in QQ at the initial time t=0t=0 and at the final time t=1t=1. Using the method of truncated integration and the Fourier transform, we construct regularized solutions and derive explicitly error estimate.