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,@p)x(0,@p),t@?[0,T] satisfying the following systemu"t-u"x"x-u"y"y=f(x,y,t),(x,y,t)@?Ix(0,T),u(0,y,t)=u(@p,y,t)=u(x,0,t)=u(x,@p,t)=0(x,y,t)@?Ix(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.

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