Three Regularization Methods For Backward Heat Conduction Problems With Convection Term

A temperature distribution at t < T is reconstructed from the temperature data u(x,T) = <φt(X) at t = T > 0 in a heat conduction body. This is referred to backward heat conduction problem (BHCP), it is also called the controllability problem. Mathematically BHCP is a classical severely ill-posed problem, i.e., a small perturbation in the input data can blow up the solution. Thus, it is very difficult to solve the problem numerically. However in practice, the data φt(x) is obtained by measurements and there must exist errors. Hence it needs special and effective regularization methods to solve the problem. Since the BHCP has extensive application background and important theoretic significance. In recent decades people have been probing new theories and algorithms for this problem.In this paper under some a-priori smoothness assumptions, we provide Fourier regulation method, revised Tikhonov regulation method and filtering regulation method for solving the problem systemtically. These are all new methods. Moreover we obtain order optimal error estimates for all the proposed methods.