Method for obtaining corrective power-series solutions to algebraic and transcendental systems
Abstract
A method for obtaining approximate analytical solutions of algebraic and transcendental systems is described. Power-series expansions of the roots are constructed directly from the system of interest. It is assumed that the system has arisen from a physical problem that has associated with it a suitably small parameter, herein denoted α. It is also assumed that there is no difficulty in identifying which roots are of physical interest, and that these roots possess a Maclaurin-series expansion in α. As in numerical continuation methods, this method requires the availability of the solution of the system when α = 0. The roots of the α = 0 system appear as the zeroth terms of the expansions. The terms in the expansions beyond the zeroth are viewed as "corrective" in the sense that their inclusion modifies the α = 0 solution to produce the roots when α≠ 0. The primary application of the method is to physical problems in which a solution is available when some parameter is zero, and it is desired to know how the roots change when the parameter departs from zero by a relatively small amount. Thus, the method is analogous to perturbation techniques. Several illustrative examples are given.
