Semi Analytical Solutions of Transitional Korteweg-de Vries Equations

Differential equations (DEs) are essential tools in various scientific and engineering fields, used to model phenomena like population growth, heat transfer, and radioactive decay. However, solving DEs can be challenging. In the 19th century, perturbation techniques were developed to approximate solutions for nonlinear DEs, but they require a small parameter and are limited if such a parameter is absent. To overcome these limitations, the Optimal Homotopy Analysis Method (OHAM), introduced in 1992, allows for more flexibility in choosing linear operators. However, computing the inverse of these operators can be computationally intensive. In order to address this issue, Liao developed the Method of Directly Defining the Inverse Mapping (MDDiM), which makes the process easier by allowing the inverse mapping to be specified directly. In this study, we demonstrate two possible forms of the time-dependent function in the transitional Korteweg-de Vries (t-KdV) equation using a similarity transformation and propose solving it using both OHAM and MDDiM. The nonlinear KdV equation is a significant mathematical model that describes the motion of long waves in shallow water under the influence of gravity and has broader applications in quantum mechanics. Therefore, we studied the application of OHAM and MDDiM for t-KdV equations. The results of both methods are validated by comparing their solutions and minimizing the squared residual error.