A methodological approach to solving the Korteweg–de Vries equation in its various forms

The Korteweg-de Vries (KdV) equation, an evolution-type nonlinear partial differential equation (PDE), describes the propagation of solitary water waves as observed in the literature. Finding the exact solution to such an equation usually involves a robust analytical method. The exact solution of the KdV serves as a benchmark for which numerical solutions to the same equation are compared with and estimate the error margins, as well as the level of accuracy of these numerical solutions by the numerical methods. Although several analytic methods have been used to obtain the exact solution to the KdV equation but no comparison of these analytic methods has been observed in the literature to establish the properties of the analytic methods, especially in the convergence of the solution in terms of the computational time. Thus, the undeniable fact about the property of the method for finding the solution to the KdV equation establishes the fact and the ease of usage of the analytic method. To provide such information on the exact solution to the KdV equation, four analytical methods: the Exp-Function Method (E-FM), the Self-Seminar Solution Method (S-SSM), the Laplace Decomposition Method (LDM), and the Sumudu-Generalized Laplace Transform Decomposition Method (SGLDM) are applied. Based on the established property of the analytic method for finding the exact solution to the KdV equation, it was revealed that the E-FM takes 8ms computational time, which has the fastest rate of convergence of solution to the KdV equation, followed by the S-SSM (10ms), then the LDM (25ms), and finally, the SGLDM (30ms). This implies that, in obtaining the solution to the KdV equation, the E-FM requires the fewest solution procedures as compared with the S-SSM, the LDM, and the SGLDM.