Power Series Extender Method for the Solution of Nonlinear Differential Equations

We propose a power series extender method to obtain approximate solutions of nonlinear differential equations. In order to assess the benefits of this proposal, three nonlinear problems of different kind are solved and compared against the power series solution obtained using an approximative method. The problems are homogeneous Lane-Emden equation of index, governing equation of a burning iron particle, and an explicit differential-algebraic equation related to battery model simulations. The results show that PSEM generates highly accurate handy approximations requiring only a few steps. The main advantage of PSEM is to extend the domain of convergence of the power series solutions of approximative methods as Taylor series method, homotopy perturbation method, homotopy analysis method, variational iteration method, differential transform method, and Adomian decomposition method, among many others. From the application of PSEM, it results in handy easy computable expressions that extend the domain of convergence of high order power series solutions.