Perturbation Substitution Method for Ordinary Differential Equations

Abstract

A new perturbation method is proposed. In addition to the perturbation series expansion of the dependent variable, the independent variable is also expanded as arbitrary functions. The arbitrary function expansions gives more flexibility in choosing the specific forms of the functions so that secular terms, small-divisor terms, blow-up terms which limit the validity of expansions can be eliminated. The method has the capability to produce a number of solutions ranging from regular perturbation solutions to even exact solutions if available. Several linear and nonlinear ordinary differential equation problems are treated with the new method. A boundary layer type problem is treated as well. The link between the new method and the other perturbation methods are outlined in the examples considered. The advantage of the new method is that it inherits more arbitrariness in the expansions so that many different approximate solutions including the regular solutions and even exact solutions can be constructed. The disadvantage is that the selection of the independent expansion functions is not straightforward and the solutions depend on the specific choices.