A new regularized method for two dimensional nonhomogeneous backward heat problem

We consider the problem of finding, from the final data u(x, y, T) =g(x,y), the initial data u(x, y, 0) of the temperature function u(x, y, t), (x, y) ∈ I = (0, π) x (0, π), t ∈ [0, T] satisfying the following system ut - uxx - uyy = f (x,y, t), (x,y, t) ∈ I x (0, T), u(0, y, t) = u(π, y, t) = u(x, 0, t) = u(x, π, t) = 0 (x, y, t) E I x (0, T). The problem is severely ill-posed. In this paper a simple and convenient new regularization method for solving this problem is considered. Meanwhile, some quite sharp error estimates between the approximate solution and exact solution are provided. A numerical example also shows that the method works effectively.