A Semi-Analytic Hybrid Approach for Solving the Buckmaster Equation: Application of the Elzaki Projected Differential Transform Method (EPDTM)

Abstract

The Buckmaster equation, a nonlinear partial differential equation (PDE) central to modeling the dynamics and deformation of flat fluid plates, presents significant analytical and computational challenges due to its inherent complexity. Traditional solution approaches predominantly rely on numerical methods, which, although effective, are often computationally intensive and face limitations in handling nonlinearity. In this study, we propose and apply the Elzaki projected differential transform method (EPDTM), a semi-analytic approach, to solve the Buckmaster equation. The EPDTM combines the strengths of the Elzaki transform and the projected differential transform method, offering a precise and computationally efficient framework to tackle such nonlinear equations. We present approximate solutions for two specific cases of the Buckmaster equation and generalize our analysis to its broader form. A detailed comparative analysis of the EPDTM results with exact solutions, using tables, 3D plots, and error graphs, demonstrates the negligible absolute errors achieved by the method. Convergence plots further validate the rapid alignment of the EPDTM solutions with the exact solutions, showcasing their accuracy and reliability. Compared with existing numerical methods, EPDTM significantly reduces computational demand while maintaining high precision, even when handling nonlinearity. The findings underscore the potential of the EPDTM as a robust and efficient tool for solving complex nonlinear PDEs such as the Buckmaster equation. This method provides an effective alternative to traditional numerical approaches and opens new opportunities for its application in broader mathematical modeling and scientific domains.