Piecewise homotopy methods for nonlinear ordinary differential equations

Piecewise homotopy perturbation methods are developed for the solution of nonlinear ordinary differential equations. These methods are based on the introduction of an artificial or book-keeping parameter and the expansion of the solution in a power series of this parameter and provide analytical solutions in open intervals which are smooth everywhere. Three piecewise-adaptive homotopy perturbation methods based on the use of either a fixed number of approximants and a variable step size, a variable number of approximants and a fixed step size or a variable number of approximants and a variable step size, are presented and applied to eight nonlinear ordinary differential equations. It is shown that piecewise-adaptive homotopy perturbation methods predict essentially the same solutions as MATLAB's variable-step, variable-order solvers and variable-order transition matrix techniques provided that five-term approximations of the decomposition method are applied to both the displacement and the velocity. It is also shown that piecewise homotopy perturbation techniques that use three-term approximations to both the displacement and the velocity provide essentially the same results as those obtained with a second-order accurate time-linearization technique when the same step is employed in both schemes.