Principal Algebra, Invariant Solutions and Representations for Optimal Systems of the Burgers–Huxley Equation

This manuscript is committed to the investigation of search reduction methodologies, identification and analysis of invariant solutions, and formulation of algebraic representations about the optimal system. These endeavors are conducted within the contextual framework of Lie symmetries, specifically applied to elucidate the complexities inherent in the Burgers–Huxley Equation. A comprehensive analysis is carried out to achieve a thorough classification of the symmetry group for a generalized form of the Burgers–Huxley Equation, leading to the derivation of its principal algebra and the optimal system, as well as the exploration of its reductions, possible invariant solutions, and algebraic representations. These findings offer valuable insights into the underlying mathematical structures and properties of the Burgers–Huxley Equation. The classification and identification of the principal algebra contribute to advancing our understanding of the applications of this differential equation and establishing a solid foundation for future research endeavors.