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Iteration method of approximate solution of the Cauchy problem for a singularly perturbed weakly nonlinear differential equation of an arbitrary order. (English) Zbl 1424.34062
Summary: 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 difference of its \(n\)th element from the solution of the problem is proportional to the \((n+1)\)th power of the perturbation parameter.
MSC:
34A45 Theoretical approximation of solutions to ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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