Contents Online
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
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
The research of C.-C. Lee was partially supported by MOST grants of Taiwan with numbers 108-2115-M-007-006-MY2 and 110-2115-M-007-003-MY2. The research of M. Mizuno was supported by JSPS KAKENHI grants with numbers JP18K13446 and JP22K03376.
Received 15 September 2022
Received revised 22 June 2023
Accepted 9 July 2023
Published 1 February 2024