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

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.