A highly accurate LGSM for severely ill-posed BHCP under a large noise on the final time data

Abstract

We consider a backward heat conduction problem (BHCP) in a slab, by subjecting to data at a final time, and some different type boundary conditions at two ends of the slab. The BHCP is known to be highly ill-posed. In order to numerically solve the BHCP we develop a new Lie-group shooting method (LGSM) in the spatial direction. It can retrieve very well the initial data with a high order accuracy. Several numerical examples of the BHCP demonstrate that the LGSM is applicable, even for those of strongly ill-posed ones with a large value of final time. Under the noisy final data the LGSM is robust against the disturbance. The new method is applicable for a severely ill-posed case with a final data very small in the order of 10 −43, and the noise level is in the order of 10 −1, of which the numerical solution still has an accuracy in the order of 10 −2. The results are rather significant in the computations of BHCP.