An LGSM to identify nonhomogeneous heat conductivity functions by an extra measurement of temperature

Considering here is an inverse problem for estimating the unknown nonhomogeneous heat conductivity function α(x)α(x) in Tt(x,t)=∂(α(x)Tx)/∂x+h(x,t)Tt(x,t)=∂(α(x)Tx)/∂x+h(x,t) with the aid of an extra measurement of temperatures at a final time, which may be disturbed by random noise. A Lie-group shooting method (LGSM) is developed from the one-step Lie-group elements obtained by a spatial-discretization of heat conduction equation and by using the central difference or forward difference for α′(x)α′(x) in spatial domain. The heat conductivities are available by directly solving linear equations. The new methods have twofold advantages in that no a priori information of heat conductivity is required and no iterations in the calculation process are needed. The accuracy and robustness of present methods are confirmed by comparing estimated results with exact solutions. The LGSM is stable and accurate, although the estimations are carried out under a large measurement noise.