Nonlinear problems with unknown initial temperature and without final temperature, solved by the GL(N, R) shooting method

We consider a nonlinear heat conduction equation for recovering unknown initial temperature under Dirichlet or Neumann boundary conditions. This problem is a generalized backward heat conduction problem (GBHCP), which not necessarily subjects to data at a final time. The GBHCP is known to be highly ill-posed, for which we develop a novel GL(N, R) shooting method (GLSM) in the spatial direction. It can retrieve very well the initial data with a high order accuracy. Several numerical examples of the GBHCP demonstrate that the GLSM is applicable, even for those of strongly ill-posed ones with large values of final time. Under the noisy final data the GLSM is robust against the disturbance. The new method is applicable for a case with a final data very small in the order of 10−87, and the relative noise level is in the order of 100, of which the numerical solution still has an accuracy in the order of 10−2. These results are quite remarkable in the computations of GBHCP.