A Lie-group shooting method for reconstructing a past time-dependent heat source

We consider an inverse problem for the reconstruction of a past unknown time-dependent heat source H(t), t < tf, in a heat conduction equation Tt(x,t) = Txx(x,t) + H(t) with the aid of an extra measurement of temperature gradient on the left-boundary, where a final time condition is measured at the present terminal time tf. This inverse problem is quite difficult to be solved numerically owing to a twofold ill-posedness, as a combination of the backward heat conduction problem and the inverse heat source identification problem, which is abbreviated as inverse heat source/backward heat conduction problem (IHSBHCP). The new method proposed here, namely the Lie-group shooting method (LGSM), is examined through the tests by several numerical examples. Although the recovery of an unknown heat source is carried out under a presently measured temperature at a final time and under a large measurement noise both imposed on the final time data and all boundary data, the LGSM still works effectively and accurately. The accuracy in the reconstruction of H(t) is almost uneffected by different levels of noise.