Lie-group differential algebraic equations method to recover heat source in a Cauchy problem with analytic continuation data

The present study recovers a time-dependent heat source H(t)H(t) in ut(x,t)=uxx(x,t)+H(t)ut(x,t)=uxx(x,t)+H(t), under measured initial heat flux, and Cauchy boundary conditions. The supplementary initial data are assumed to be analytic continuation ones being obtained by means of measurement, which are not given arbitrarily. We first transform the above problem into an inverse heat conduction problem without having the right-boundary value, and then, further transform it into another inverse heat source problem that we need to find F(t)F(t) in Tt(x,t)=Txx(x,t)-xF(t)/ℓTt(x,t)=Txx(x,t)-xF(t)/ℓ with measured initial and boundary conditions. By using the GL(n,R)GL(n,R) Lie-group differential algebraic equations (LGDAE) method to integrate the resultant ordinary differential equations with a priori bound |F(t)|⩽Fmax|F(t)|⩽Fmax, we can fast recover H(t)H(t) in real time. The accuracy and efficiency are assessed and confirmed by comparing the exact solutions with recovered results, where a large noise up to 10% or 20% is imposed on the input measured data.