On Holomorphic Regularization of Nonlinear Singularly Perturbed Boundary Value Problems
DOI:
https://doi.org/10.24160/1993-6982-2018-4-152-156Keywords:
Tikhonov’s system, holomorphic integrals, pseudoholomorphic solutions, family of homomorphismsAbstract
The article considers a method for solving singularly perturbed boundary value problems based on holomorphic regularization of singular perturbations, which makes it possible to obtain solutions of these problems in the form of series in powers of a small parameter that converge in the usual sense (rather than asymptotically). The primary challenge encountered in solving nonlinear boundary problems is the need to prove that the solution of this problem exists on the whole interval. In this regard, the method of upper and lower solutions proposed by Chaplygin and Nagumo is the most well-elaborated one. It should be noted that each researcher uses its own approach to constructing barrier functions. In the majority of cases, efforts taken to find them are commensurable in complexity with finding the solution of the boundary value problem itself. In any case, this step is followed by constructing the solution in the form of asymptotically convergent series in powers of a small parameter. In the present study, the solution of a nonlinear singularly perturbed second-order equation with Dirichlet-type boundary conditions is found in the form of a series in powers of a small parameter converging in the usual sense.
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Для цитирования: Качалов В.И., Бесова М.И. О голоморфной регуляризации нелинейных сингулярно возмущенных краевых задач // Вестник МЭИ. 2018. № 4. С. 152—156. DOI: 10.24160/1993-6982-2018-4-152-156.
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For citation: Kachalov V.I., Besova M.I. On Holomorphic Regularization of Nonlinear Singularly Perturbed Boundary Value Problems. MPEI Vestnik. 2018;4:152—156. (in Russian). DOI: 10.24160/1993-6982-2018-4-152-156.
