Zafar Projected Differential Transform and Laplace Projected Differential Transform methods as exact solution methods for Klein-Gordon equations

Abstract

In this study, we introduced two innovative numerical methodologies tailored for the resolution of linear and nonlinear Klein-Gordon equations. The Zafar Projected Differential Transform Method (ZPDTM) and the Laplace Projected Differential Transform Method (LPDTM) are devised to address the inherent complexities of these equations encountered in diverse scientific and engineering realms. Rigorous mathematical analysis herein demonstrated the efficacy and efficiency of ZPDTM and LPDTM in obtaining precise and approximate solutions for Klein-Gordon equations. Seamlessly integrating the Zafar transform method with the projected differential transform method, ZPDTM offers a novel approach that significantly broadens the scope of applicability compared to traditional numerical techniques. Likewise, LPDTM, a fusion of robust techniques from the Laplace transform method and the projected differential transform method, presents a powerful methodology for tackling nonlinear differential equations encountered in various domains. Our study showcased the reliability and user-friendliness of ZPDTM and LPDTM, underscoring their potential for extensive utilization across diverse scientific and engineering disciplines. Furthermore, we highlighted the computational efficiency of these methodologies, which significantly alleviate the computational burdens relative to traditional methods while maintaining high numerical precision. By presenting a comprehensive overview of the application of ZPDTM and LPDTM, this study represents a significant advancement in numerical techniques for Klein-Gordon equations.