Finding unknown heat source in a nonlinear Cauchy problem by the Lie-group differential algebraic equations method

We consider an inverse heat source problem of a nonlinear heat conduction equation, for recovering an unknown space-dependent heat source under the Cauchy type boundary conditions. With the aid of measured initial temperature and initial heat flux, which are disturbanced by random noise causing measurement error, we develop a Lie-group differential algebraic equations (LGDAE) method to solve the resultant differential algebraic equations. The Lie-group numerical method has a stabilizing effect to retain the solution on the associated manifold, which thus naturally has a regularization effect to overcome the ill-posed property of the nonlinear inverse heat source problem. As a consequence, we can quickly recover the unknown heat source under noisy input data only through a few iterations. The initial data used in the recovery of heat source are assumed to be the analytic continuation ones which are not given arbitrarily. Certainly, the measured initial data belong to this type data.