A new algorithm for direct and backward problems of heat conduction equation

Direct heat conduction problem (DHCP) and backward heat conduction problem (BHCP) are numerically solved by employing a new idea of fictitious time integration method (FTIM). The DHCP needs to consider the stability of numerical integration in the sense that the solution may be divergent for a specific time stepsize and specific spatial stepsize. The BHCP is renowned as strongly ill-posed because the solution does not continuously depend on the given data. In this paper, we transform the original parabolic equation into another parabolic type evolution equation by introducing a fictitious time variable, and adding a fictitious viscous damping coefficient to enhance the stability of numerical integration of the discretized equations by employing a group preserving scheme. When 10 numerical examples are amenable, we find that the FTIM is applicable to both the DHCP and BHCP. Even under seriously noisy initial or final data, the FTIM is also robust against disturbance. More interestingly, when we use the FTIM, we do not need to use different techniques to treat DHCP and BHCP as that usually employed in the conventional numerical methods. It means that the FTIM can unifiedly approach both the DHCP and BHCP, and the gap between direct problems and inverse problems can be smeared out.