The Modified Homogeneous Balance Methods for solving Korteweg–DeVries equations

This paper considers the analytic methods for finding solutions of the Korteweg–Devries (KdV) equation, a nonlinear evolution-type partial differential equation (PDE), that describes the propagation of solitary wave waters in shallow channels. In obtaining the solutions of the KdV equation, a Homogeneous Balance Method (HBM) and its improvements were observed by different researchers across the globe as an effective method and also, revealed some of the properties of the KdV equation such as convergence of solution. These Modified Homogeneous Balance Methods (MHBMs) encompass the first-order Riccati equations in usage. Surprisingly, an MHBM via a special form of a first-order Riccati equation of a single polynomial term, an MHBM via a second-order Riccati equation, and finally, an MHBM with the embodiment of a combination of a first-order Riccati equation and a second-order Riccati which have not been observed in the literature are considered in this paper. Unlike the solutions yielded by the MHBM via the first-order Riccati equation, the solutions yielded by the MHBM via the second-order Riccati equation are characterized by hyperbolic function. Unquestionably, the MHBM which incorporates the first-order Riccati equation of a single polynomial term observed to be of the highest speed of convergence of solution in any functional space as compared to the HBM and its improvements.