In this article, an invariantized finite difference scheme to find the solution of the heat equation, is developed. The scheme is based on a discrete symmetry transformation. A comparison of the results obtained by the proposed scheme and the Crank Nicolson method is carried out with reference to the exact solutions. It is found that the proposed invariantized scheme for the heat equation improves the efficiency and accuracy of the existing Crank Nicolson method.

In this paper, the Mei symmetries for the Lagrangians corresponding to the spherically and axially symmetric metrics are investigated. For this purpose, the Schwarzschild and Kerr black hole metrics are considered. Using the Mei symmetries criterion, we obtained four Mei symmetries for the Lagrangian of Schwarzschild and Kerr black hole metrics. The results reveal that, in the case of the Schwarzschild metric, the obtained Mei symmetries are a subset of the Lie point symmetries of equations of motion (geodesic equations), while in the case of the Kerr black hole metric, the Noether symmetry set is a subset of the Mei symmetry set and that Mei symmetries and the Lie point symmetries of the equations of motion are same.

The aim of this paper is to give the geometrical/physical interpretation of the conserved quantities corresponding to each Noether symmetry of the geodetic Lagrangian of plane symmetric space-times. For this purpose, we present a complete list of plane symmetric nonstatic space-times along with the generators of all Noether symmetries of the geodetic Lagrangian. Additionally, the structure constants of the associated Lie algebras, the Riemann curvature tensors, and the energy-momentum tensors are obtained for each case. It is worth mentioning that the list contains all classes of solutions that have been obtained earlier during the classification of plane symmetric space-times by isometries and homotheties.