The truncation method for a two-dimensional nonhomogeneous backward heat problem

We consider the backward heat problemut-uxx-uyy=f(x,y,t),(x,y,t)∈Ω×(0,T),u(x,y,T)=g(x,y),(x,y)∈Ω,with the homogeneous Dirichlet condition on the rectangle Ω = (0, π) × (0, π), where the data f and g   are given approximately. The problem is severely ill-posed. Using the truncation method for Fourier series we propose a simple regularized solution which not only works on a very weak condition on the exact data but also attains, due to the smoothness of the exact solution, explicit error estimates which include the approximation (ln(ϵ-1))3/2ϵ in H2(Ω). Some numerical examples are given to illuminate the effect of our method.