APPLICATION OF GROUP CLASSIFICATION OF ONE CLASS OF QUASILINEAR EQUATIONS WITH RESPECT TO LIE ALGEBRAS OF DIMENSION NOT HIGHER THAN THREE IN THE RESEARCH OF CONSUMER BEHAVIOR

Abstract

This paper presents the group classification of a specific class of second-order quasilinear hyperbolic equations with two independent variables, focusing on non-decomposable, solvable, three-dimensional Lie algebras. This class of equations can be seen as a broad generalization of the nonlinear d’Alembert, Liouville, sine-Gordon, and Tzitzeica equations. The theoretical foundation of this work dates back to Sophus Lie, who pioneered the group analysis of differential equations, establishing that the principal classifying features of such equations are their symmetry properties. Accordingly, this research provides a complete description of the equations invariant under such algebras, along with their specific realizations.
The study demonstrates the direct applicability of this mathematical framework to the modeling of consumer behavior. The central premise is that the group symmetries and invariance found in the equations correspond to the relative stability of consumer behavior patterns, even amidst a changing marketing environment. By identifying these invariant characteristics, researchers can uncover fundamental trends that are not immediately apparent. The use of symmetries enables the simplification of complex consumer behavior models by reducing the number of variables, thereby enhancing their interpretability.
To perform the classification, the study employed modern symbolic computation tools, specifically Mathematica packages, to automate the process. The findings have practical significance for marketing research, offering applications in consumer profiling, campaign effectiveness evaluation, and the identification of key factors that influence purchasing behavior.