The solution y(x) of an IVP for a linear ODE with analytic coefficient functions is represented as a power series. A high–order Taylor polynomial is used for an approximate numerical solution. The Taylor remainder series is rigorously estimated by some geometric series. The method has been implemented and tested on a computer. Guaranteed enclosures are achieved by taking into account all roundo...

The present paper deals with the homotopy perturbation method. The question of whether the homotopy perturbation method is simply the conventional Taylor series expansion is examined. It is proven that under particular choices of the auxiliary parameters the homotopy perturbation method is indeed the Taylor series expansion of the sought solution of nonlinear equations.

A simple mathematical model of steady state  oxygen distribution subject to diffusive transport and non- linear uptake in a retinal cylinder has been developed. The approximate analytical solution to a reaction- diffusion equation are obtained by using series expansions. The computational results for the scaled variables are presented through graphs. The effect of the important parameters (1) d...

For a holomorphic function f in the open unit disc D and ζ ∈ , S n ( ) denotes -th partial sum of Taylor development at . Given an increasing sequence positive integers μ = we consider classes U functions such that subsequences sums { : 1 2 … } respectively approximate all polynomials uniformly on compact sets K ⊂ z C | ≥ with connected complement. We show these two universal series coincide if...