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

Authors: ALEXEY ALIMOV, EVGENY BUKHZHALEV

Abstract: We construct an iteration sequence converging (in the uniform norm in the space of continuous functions) to the 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 constructed is also asymptotic in the sense that the departure of its $n$th element from the solution of the problem is proportional to the $(n+1)$th power of the perturbation parameter.

Keywords: Singular perturbations, Banach contraction principle, method of asymptotic iterations, Routh-Hurwitz stability criterion

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