The solution of the one-dimensional, linear, inverse, unsteady heat conduction problem (IHCP) in a slab geometry is analysed. The Initial temperature is known, together with a condition on an accessible part of the boundary of the body under investigation. Additional temperature measurements in time are taken with a sensor positioned at an arbitrary location within the solid material, and it is required to determine the temperature and the heat flux on the remaining part of the unspecified boundary. As the problem is improperly posed the direct method of solution cannot be used and hence the least squares, regularization and energy method have been introduced into the boundary element method (BEM) formulation. When noise is present in the measured data some of the numerical results obtained using the least squares method exhibit oscillatory behaviour, but these large oscillations are substantially reduced on the introduction of the minimal energy technique based on minimizing the kinetic energy functional subject to certain constraints. Furthermore, the numerical results obtained using this technique compare well with the results obtained using regularization procedures, showing a good stable estimation of the available test solutions. Further, the constraints, subject to which the minimization is performed, depend on a small parameter of which selection is more natural and easier to implement than the choice of the regularization parameter, which is always a difficult task when using the regularization procedures.