This study investigates the application of the Elzaki Projected Differential Transform Method (EPDTM) to fractional-order nonlinear Korteweg-de Vries (KdV) equations, which describe various nonlinear wave phenomena in physics and engineering. The method effectively addresses both linear and nonlinear operators and fractional derivatives. Through two illustrative examples, the method accurately captures the dynamics of fractional-order wave systems and achieves results in excellent agreement with exact solutions. The findings demonstrate the method_s precision, fast convergence, and computational efficiency, underscoring EPDTM's potential as a robust tool for solving nonlinear partial differential equations with fractional dynamics. Keywords Fractional Korteweg-de Vries equations, Elzaki transform, Projected Differential Transform Method, Fractional calculus, Numerical simulation..

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In this study, we propose two innovative numerical methods for solving Klein- Gordon equations: the Zafar Projected Differential Transform Method (ZPDTM) and the Laplace Projected Differential Transform Method (LPDTM). By integrating the Zafar and Laplace transforms respectively with the Projected Differential Transform Method, these approaches offer improved computational efficiency and enhanced solution accuracy. The performance of both methods is demonstrated through their application to linear and nonlinear forms of the Klein-Gordon equation, showing strong agreement with exact solutions and reduced computational overhead. These results highlight the versatility and reliability of ZPDTM and LPDTM in addressing complex differential models encountered in physics and engineering. Keywords Zafar Transform; Laplace Transform; Projected Differential Transform; Klein- Gordon equations; Exact solution..

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