On an inverse problem: Recovery of non-smooth solutions to backward heat equation

We have recently developed two quasi-reversibility techniques in combination with Euler and Crank–Nicolson schemes and applied successfully to solve for smooth solutions of backward heat equation. In this paper, we test the viability of using these techniques to recover non-smooth solutions of backward heat equation. In particular, we numerically integrate the backward heat equation with smooth initial data up to a time of singularity (corners and discontinuities) formation. Using three examples, it is shown that the numerical solutions are very good smooth approximations to these singular exact solutions. The errors are shown using pseudo-L- and U-curves and compared where available with existing works. The limitations of these methods in terms of time of simulation and accuracy with emphasis on the precise set of numerical parameters suitable for producing smooth approximations are discussed. This paper also provides an opportunity to gain some insight into developing more sophisticated filtering techniques that can produce the fine-scale features (singularities) of the final solutions. Techniques are general and can be applied to many problems of scientific and technological interests.