Constructing accurate polynomial approximations for nonlinear differential initial value problems

This paper deals with the construction of approximate polynomial solutions, with a prefixed accuracy, of initial value problems for nonlinear ordinary differential equations. By approximating the right-hand side of the equation by an appropriate two-variables Chebyshev polynomial, then applying Frobenius method, and finally truncating this application, an approximate polynomial solution is constructed. Recent results of Chen et al. (2003) [B. Chen, R. García-Bolós, L. Jódar, M.D. Roselló, The truncation error of the two-variable Chebyshev series expansions, Comput. Math. Appl. 45 (2003) 1647–1653] and Chen et al. (2005) [B. Chen, R. García-Bolós, L. Jódar, M.D. Roselló, Chebyshev polynomial approximations for nonlinear differential initial value problems, Nonlinear Anal. 63 (5–7) (2005) e629–e637] are significantly improved in two directions by extending the existence domain of the approximation and by reducing the degree of the truncation polynomial. Several examples are given.