Homotopy solution for nonlinear differential equations in wave propagation problems

In this paper, we derive a general series solution for nonlinear differential equations based on an alternate form of the homotopy analysis method. The conventional approach begins with a zero-order deformation equation, which includes an auxiliary operator for mapping of an initial approximation to the exact solution and an auxiliary parameter to ensure convergence of the series solution. We express the general series solution directly from the zero-order deformation equation in terms of the Bell polynomial and introduce a new dimension to the convergence characteristics through a second auxiliary parameter. Convergence theorems are provided to assure mapping to the correct solution in the new homotopy defined by two auxiliary parameters. Implementation of the general solution is demonstrated with the periodic long-wave problem governed by the Korteweg de Vries equation and the propagation of high-frequency waves in a relaxing medium given by the Vakhnenko equation. Comparison of the present and exact solutions confirms the effectiveness and validity of the proposed approach. The use of two auxiliary parameters substantially improves the convergence region and rate and provides series solutions to highly nonlinear equations with fewer terms.