Iteration method of approximate solution of the Cauchy problem for a singularly perturbed weakly nonlinear differential equation of an arbitrary order

Abstract

We construct an iteration sequence converging (in the uniform norm in the space of continuous functions) tothe solution of the Cauchy problem for a singularly perturbed weakly nonlinear differential equation of an arbitrary order(the weak nonlinearity means the presence of a small parameter in the nonlinear term). The sequence thus constructedis also asymptotic in the sense that the departure of its nth element from the solution of the problem is proportional tothe (n + 1)th power of the perturbation parameter.