Narrow Results

The azimuthal limitation of the Parabolic Equation (PE) approximation in solving the three-dimensional (3D) wave equation in cylindrical coordinate has been studied in this paper. Typically, 3D problems are dealt with an N × 2D approximation, which treats a 3D field as a fan-like composition of many 2D vertical slices (r - z plane in cylindrical coordinate) ignoring the θ-coupling terms. To deal with problems possessing 3D effects, the θ-coupling terms have to be considered in PE approximation. Nevertheless, the azimuthal limitation in the 3D PE approximation is not defined as well as the vertical angle limitation. Hence, the theoretical derivation estimating the azimuthal limitation is put forth in this work. Numerical results of a modified benchmark problem are also presented to validate the arguments and the wide angle version of the 3D PE model, FOR3D.

Conservation laws of various systems have been studied for decades due to their unparalleled importance in unraveling systems’ intricacies without having to go into microscopic details of the physical process involved. Their association with symmetries has not only had a stupendous impact in the formulation of the fundamental laws of physics, but also open doors to further explorations and unifications of others. In this study, we present the Lie symmetries and nonlinearly self-adjoint classifications of the wave equation on Bianchi I spacetime. For different forms of the metric potentials, generalized higher order non-trivial conserved vectors are constructed. Some exact invariant solutions are also exhibited.