Abstract:Aquasi-boundaryregularization leads to a two-point boundary value problem of the backwardheatconduction equation. The ill-posed problem is analyzed by using the semi-discretization numerical schemes. Then the resulting ordinary differential equations in the discretized space are numerically integrated towards the time direction by the Lie-group shootingmethod to find the unknown initial conditions. The key point is based on the erection of a one-step Lie group element G(T) and the formation of a generalized mid-point Lie group element G(r). Then, by imposing G(T)=G(r) we can search for the missing initial conditions through a minimum discrepancy of the targets in terms of the weighting factor r∈(0,1). Several numerical examples were worked out to persuade that this novel approach has good efficiency and accuracy. Although the final temperature is almost undetectable and/or is disturbed by large noise, the Lie group shootingmethod is stable to recover the initial temperature very well.