Advanced Mathematical Approaches for Solving Fractional-Order Korteweg-de Vries Equations

In this work, we investigate the use of two new methods, residual power series natural transform method (RPSNTM) and new iteration natural transform method (NINTM), to tackle the fractional-order Korteweg-de Vries (KdV) equation. Many wave effects in fluid dynamics, plasma physics and traffic flow are described with the help of the fractional-order KdV equation. Tradi-tional techniques used to solve equations generally are not adapted to deal with the complications caused by fractional derivatives. The RPSNTM is presented as a valuable method to approximate how the solution will change with time by converting the problem into a step-by-step series. The NINTM is also used to make the solution more effective and reliable gradually. Applying both approaches offers a strong way to find approximate analytical answers to the KdV fractional-orderequation. Numerical data is used to reveal that the presented methods work better and faster than existing methods in terms of precision and speed of convergence. These results create new opportunities to apply fractional calculus in analyzing nonlinear waves.preparation.