In the present study a transient inverse geometry heat conduction problem (shape identification problem) is solved using the Conjugate Gradient Method (CGM) and Boundary Element Method (BEM)-based inverse algorithm to estimate the unknown irregular boundary shape. Results obtained by using the conjugate gradient method to solve this inverse moving boundary problems are justified based on the numerical experiments. It is concluded that the accurate configuration can be estimated by the conjugate gradient method except for the initial and final time. The reason and improvement of this singularity are addressed. Finally the effects of the measurement errors on the inverse solutions are discussed.