The Qualitative Features of the Numerical Integration Problems with a Boundary Layer by Nonlocal Transformations
https://doi.org/10.1134/S2304487X19060099
Abstract
The qualitative features of the numerical integration of two-point boundary-value problems of boundary-layer type by using nonlocal transformations are described. Such transformations, sometimes also called Sundman-type transformations, are defined by using an auxiliary differential equation and allow one to “stretch” the boundary-layer region (after which any adequate numerical methods with a fixed stepsize can be applied). Multiparameter nonlinear singularly perturbed boundary-value problems with a small parameter having exact solutions in elementary functions are presented, which can be used to test various numerical methods on non-uniform grids. Particular attention is paid to the study of the most difficult boundary-value problems for numerical analysis, which have non-monotonic solutions or degenerate solutions at the boundary of the boundary-layer. A comparison of numerical and exact solutions shows the high efficiency of the nonlocal transformation method for numerical integration of boundary-value problems with a boundary layer.
Keywords
ordinary differential equations with a small parameter,
singularly perturbed boundary-value problems,
boundary layer,
nonlocal transformations,
exact and numerical solutions
Supporting agencies: The work was carried out on the topic of the state assignment (state registration number AAAA-A17-117021310385-6) and with partial financial support from the Russian Foundation for Basic Research (project No. 18-29-10025)
About the Authors
A. D. Polyanin
Institute for Problems in Mechanics, Russian Academy of Sciences; Bauman State Technical University; National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)
Russian Federation
Russian Federation
119526
105005
115409
Moscow
I. K. Shingareva
University of Sonora
Mexico
Mexico
Sonora
Hermosillo
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Review
For citations:
Polyanin A.D., Shingareva I.K. The Qualitative Features of the Numerical Integration Problems with a Boundary Layer by Nonlocal Transformations. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2019;8(6):515-532. (In Russ.) https://doi.org/10.1134/S2304487X19060099
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