In this study, a new approach called the reversed fixed point iteration method (RFPIM) is implemented to solve a class of fundamental partial differential equations, namely viscous and inviscid Burgers equations. The results confirm that the current method has the potential to effectively solve problems governed by nonlinear PDEs representing the phenomena of advection and diffusion. Derivations reveal that even in challenging situations such as advection dominance, the RFPIM is highly capable of capturing the natural behaviour of the problem under study. Another important finding, depending on the considered problem, is that the RFPIM can allow the use of relatively large time steps. This feature is of great importance in reducing the storage burden and CPU time for the computational society. The observations reveal that the current approach leads to a significant improvement in the understanding of inverse problems and inverse optimal control problems.