Higher order Homotopy-Perturbation via start-system

One of the problems in iterations is to determine the best initial value and to ensure the convergence of the iterations. So, the objective is to evaluate the effectiveness of higher order approximated solutions of Homotopy Taylor-perturbation using start-system to overcome the problems. Successive approximation procedures using start-system technique are applied to the Classical Newton-Raphson, the Newton-perturbation, the Higher Order Taylor-perturbation and the new higher order Homotopy Taylor-perturbation (HHTP). The results are compared and evaluated. Numerical examples are given to illustrate and support the suggested algorithms. Results show that HHTP offers as an alternative and effective way in solving nonlinear equations.