Review: Nonlinear Differential Equations by Applying the Method of Homogeneous Balance

Abstract

The Homogeneous Balance Method (HBM) for solving nonlinear differential equations is thoroughly reviewed in this paper, along with its theoretical formulation and analytical technique for solving and analyzing initial boundary value problems represented by fractional partial differential equations (FPDEs). Although it can be challenging to find exact or close analytical solutions for these equations, especially when dealing with nonlinear problems, FPDEs are useful in explaining complex scientific events involving heritable traits. In order to resolve nonlinear health issues (FPDE), the homogeneous balancing method (HBM) is studied and developed in a novel way. HB provides a straightforward method that uses a closed-form solution to solve both linear and nonlinear fractional models. A number of fractional calculus concepts are included in this work's HBM, including fractional derivative techniques, which include rational, trigonometric, and hyperbolic solutions as well as examples of nonlinear FPDEs that are solved and analytically examined to demonstrate how the method operates. The results of the extended version of the HBM are examined and compared with those documented in the literature to demonstrate the efficacy of the method. Overall, this study suggests a practical and effective analytical method for resolving nonlinear FPDEs by extending the HB technique with result analysis.