This dissertation considers the inverse heat conduction problems (IHCP) for both one dimensional and two dimensional space models. The problem is to reconstruct the surface heat flux and/or temperature histories from transient, measured temperatures inside solids. The stability properties of this classical inverse problem are discussed. The object is to approximate the IHCP by a stable well-posed problem and determine its solution as the reconstructed surface heat flux or temperature. Two new approaches are given for different geometries in one space dimension and in two space dimensions. The first of these methods is based on singular perturbation techniques and the second on a data filtering interpretation of the mollification method. For the first time, the theoretical analysis for the two dimensional IHCP is presented.; The main result of the thesis is that by combining our new methods with space marching procedures one can develop fully explicit and unconditionally stable finite differences schemes to approximately solve the IHCP. The mollification method presented here can be easily extended to numerically solve more complicated nonlinear IHCP, for both one and two dimensional cases.; Several numerical experiments support the stability and convergence properties predicted by the error analysis of the algorithms.
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