A new shooting method for quasi-boundary regularization of backward heat conduction problems

Abstract A quasi-boundary regularization leads to a two-point boundary value problem of the backward heat conduction 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 shooting method 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 shooting method is stable to recover the initial temperature very well.

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