Communications in Mathematical Sciences

Volume 22 (2024)

Number 2

Unified asymptotic analysis and numerical simulations of singularly perturbed linear differential equations under various nonlocal boundary effects

Pages: 394 – 434

DOI: https://dx.doi.org/10.4310/CMS.2024.v22.n2.a5

Authors

Xianjin Chen (Wu Wen-Tsun Key Laboratory of Mathematics, USTC, CAS & School of Mathematical Science, University of Science & Technology of China, Hefei, China)

Chiun-Chang Lee (Institute for Computational and Modeling Science, National Tsing Hua University, Hsinchu, Taiwan)

Masashi Mizuno (Department of Mathematics, College of Science and Technology, Nihon University, Chiyoda-Ku, Tokyo , Japan)

Abstract

While being concerned with a singularly perturbed linear differential equation subject to integral boundary conditions, the exact solutions, in general, cannot be specified, and the validity of the maximum principle is unassurable. Hence, a problem arises: how to identify the boundary asymptotics more precisely? We develop a rigorous asymptotic method involving recovered boundary data to tackle the problem. A key ingredient of the approach is to transform the “nonlocal” boundary conditions into “local” boundary conditions. Then, we perform an “$\varepsilon \log \varepsilon$-estimate” to obtain the refined boundary asymptotics of its solutions with respect to the singular perturbation parameter $\varepsilon$. Furthermore, for the inhomogeneous case, diversified asymptotic behaviors including uniform boundedness and asymptotic blow-up are obtained. Numerical simulations and validations are also presented to further support the corresponding theoretical results.

Keywords

singular perturbation, integral boundary condition, refined asymptotics, recovered boundary data, nonlocal boundary effect

2010 Mathematics Subject Classification

34B10, 34D15, 34E05, 34K26, 35J25

Received 15 September 2022

Received revised 22 June 2023

Accepted 9 July 2023

Published 1 February 2024