An Adaptive Semi-Analytical Approach in Solving Nonlinear Korteweg-De Vries Equations

This paper introduces a novel method named the Adaptive Hybrid Reduced Differential Transform Method (AHRDTM) for solving Nonlinear Korteweg-De Vries Equations (NKdVEs). AHRDTM provides convergent semi-analytical solutions over long-time frames by generating subintervals of varying lengths, significantly reducing the number of time-steps and processing time needed for solutions, distinguishing it from the traditional multistep approach of RDTM. Notably, AHRDTM avoids the need for perturbation, linearization or discretization, enhancing its adaptability and reliability. The findings demonstrate that AHRDTM provides highly accurate and efficient solutions for NKdVEs. Additionally, the method is straightforward, significantly reduces the computational effort required to solve NKdVE problems and shows promise for application to a wide range of partial differential equations (PDEs). The efficacy of AHRDTM is illustrated through tables and graphical representations.