The iterative solution of non-linear ordinary differential equations in Chebyshev series

Abstract

The Newton iteration formula is applied to the solution of non-linear ordinary differential equations. For the first-order equation y′ = f(x,y) successive application of Newton's rule yield the formula y′_i = f(x,y_i−1) + (y_i − y_{i − 1})f_y(x,y_i−1). All functions which occur in the formula are represented by their Chebyshev series, and the analytic operations involved are performed by arithmetic operations on the coefficients of these series. This iterative method is of particular value when solving boundary-value problems since the more usual step-by-step methods are less powerful in these cases. Several examples are given to illustrate the effectiveness of the method.