Classifications of different dimensional partial differential equations and their invariant solutions via symmetry reductions and optimal systems

The study analyzes the (۱+۲)-dimensional modified Breaking Soliton equation using classical symmetries and explores the (۱+۱)-dimensional heat equation and modified Boussinesq equation through Lie symmetry analysis and Lie subalgebras. Classical symmetries are derived from the solutions of nonlinear partial differential equations, and optimal systems are constructed using commutator relationships and adjoint representations. The research presents new invariant solutions and their graphical analyses, which are valuable for applied sciences and numerical simulations. The solutions explain phenomena such as circular membrane vibrations, heat conduction, and electromagnetic waves. Wave, contour, and patch contour solutions are used in sound, light, weather forecasting, medical imaging, and materials science. This paper provides a comprehensive analysis of the generalized single and double reduction methods, highlighting the significance of inherited symmetries at each stage of the reduction process.

کلیدواژه ها:

Classifications ، Symmetry reductions ، Optimal systems ، Invariant solutions