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

• Published in: International journal of heat and mass transfer Vol. 83; pp. 665 - 678
• Main Authors: Liu, Chein-Shan, Chang, Chih-Wen
• Format: Journal Article
• Language: English
• Published: 01.04.2015
• Subjects: Conduction; Dirichlet problem; Heat conduction; Heat transfer; Mathematical analysis; Mathematical models; Nonlinearity; Shooting
• ISSN: 0017-9310
• DOI: 10.1016/j.ijheatmasstransfer.2014.12.057

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 super(-87), and the relative noise level is in the order of 10 super(0), of which the numerical solution still has an accuracy in the order of 10 super(2). These results are quite remarkable in the computations of GBHCP.