Abstract
For a singularly perturbed weakly nonlinear second-order differential equation, we construct a sequence converging to the Cauchy problem solution. This is an asymptotical sequence because the deviation (in the sense of the norm of the space of continuous functions) of its nth element from the solution to the problem is proportional to the (n + 1)th power of the perturbation parameter. Such a sequence can be used to justify the asymptotics obtained by using boundary functions.
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A. N. Tikhonov, A. B. Vasil’eva, and A. G. Sveshnikov, Differential Equations, Course of Higher Mathematics and Mathematical Physics (Nauka, Moscow, 1998) [in Russian].
N. D. Kopachevskii and V. P. Smolich, Introduction to Asymptotic Methods, Special Course of Lectures (Tavrich. Nats. Univ., Simferopol’, 2009) [in Russian].
Yu. P. Boglaev, A. V. Zhdanov, and V. G. Stel’makh, “Uniform approximations to the solutions of certain singularly perturbed nonlinear equations,” Differ. Uravn. 14, 395–406 (1978).
A. B. Vasil’eva and V. F. Butuzov, Asymptotic Expansions of Solutions to Singularly Perturbed Equations (Nauka, Moscow, 1973) [in Russian].
A. B. Vasil’eva and V. F. Butuzov, Asymptotic Methods in the Theory of Singular Perturbations. Current Problems in Applied and Computational Mathematics (Vyssh. Shkola, Moscow, 1990) [in Russian].
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Original Russian Text © E.E. Bukzhalev, 2017, published in Vestnik Moskovskogo Universiteta, Seriya 15: Vychislitel’naya Matematika i Kibernetika, 2017, No. 3, pp. 10–17.
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Bukzhalev, E.E. The Cauchy problem for singularly perturbed weakly nonlinear second-order differential equations: An iterative method. MoscowUniv.Comput.Math.Cybern. 41, 113–121 (2017). https://doi.org/10.3103/S0278641917030037
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DOI: https://doi.org/10.3103/S0278641917030037