Application of Homogeneous Balance Method in Solving Non-linear Partial Differential Equations: A Systematic Analysis

Abstract

This research paper presents a comprehensive analysis of the Homogeneous Balance Method (HBM) as an effective technique for obtaining exact solutions to non-linear partial differential equations (PDEs). Through rigorous mathematical analysis and practical applications, we demonstrate the method's efficacy across various classes of non-linear PDEs. The study encompasses theoretical foundations, methodological implementations, and comparative analyses with other solution techniques. Our findings indicate that HBM offers significant advantages in terms of computational efficiency and solution accuracy, particularly for certain classes of non-linear PDEs. The research also explores practical applications in wave propagation, fluid dynamics, and quantum mechanics, providing valuable insights into the method's versatility and limitations.