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)u(x,y,T)=g(x,y), the initial data u(x,y,0)u(x,y,0) of the temperature function u(x,y,t),(x,y)∈I=(0,π)×(0,π),t∈[0,T]u(x,y,t),(x,y)∈I=(0,π)×(0,π),t∈[0,T] satisfying the following systemut-uxx-uyy=f(x,y,t),(x,y,t)∈I×(0,T),u(0,y,t)=u(π,y,t)=u(x,0,t)=u(x,π,t)=0(x,y,t)∈I×(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.