A two-stage LGSM to identify time-dependent heat source through an internal measurement of temperature

We consider an inverse problem for estimating an unknown time-dependent heat source H(t)H(t) in a heat conduction equation Tt(x,t)=Txx(x,t)+H(t)Tt(x,t)=Txx(x,t)+H(t), with the aid of an extra measurement of temperature at an internal point. The Lie-group shooting method (LGSM) was used in the solution of this inverse problem; however, when the data are acquired at an internal point we require to develop a two-stage Lie-group shooting method (TSLGSM) to solve it. This novel approach is examined through numerical examples to convince that it is a rather accurate and efficient method, whose estimation error is small even for the identification of discontinuous and oscillatory heat source under large noise.