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Reconstruction of moving boundary for one-dimensional heat conduction equation by using the Lie-group shooting method

We consider an inverse problem for the recovery of an unknown time-dependent moving boundary ξ(t) in a one-dimensional heat conduction equation. By using the domain embedding technique, the inverse problem of moving boundary identification is transformed into a time-dependent parameter function identification problem of an advection–diffusion partial differential equation, where ξ(t) and play the role of unknown parameters. From a viewpoint of numerical differentiation, that appearing in the governing equation causes the identification of ξ(t) being rather difficult, because the differential itself is an ill-posed operator. In order to overcome this difficulty we let , instead of ξ(t), be the unknown variable, and ξ(t) is calculated from by an integration. Once using the Lie-group shooting method (LGSM) we can derive a quite simple system of linear algebraic equations to iteratively calculate and then ξ(t). It is demonstrated through numerical examples that the LGSM is accurate, efficient and stable; the maximum error of numerical solutions is in the order of 10−4–10−2, even a large noise in the level of 10−2 is imposed on the measured data.