TITLE:
Uniform Error Estimates for Singularly Perturbed Cauchy Problems via Green’s Functions and Spectral Decomposition
AUTHORS:
António Camunga Tchikoko, David Benje Mucenje, Euclides Faustino da Costa Fernando
KEYWORDS:
Singular Perturbations, Initial Layer, Error Estimates, Uniform Convergence, Green’s Function
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.10,
October
12,
2026
ABSTRACT: We study a singularly perturbed Cauchy problem for a second order linear ordinary differential equation with constant coefficients, in which a small parameter
ε>0
multiplies the highest derivative. Because the reduced first order equation cannot generally satisfy both initial conditions, a layer of width
O(
ε
)
develops near the initial point. An exact Green’s function representation, combined with the separation of the characteristic roots into slow and fast modes, yields an
O(
ε
)
error bound for the solution in the uniform norm on the full interval. For the derivative, we isolate an exponentially localized term proportional to the initial-data mismatch and bound the remaining contribution by
O(
ε
)
. When the initial slope is compatible with the reduced equation, the order one fast component is absent and convergence holds in
C
1
with order
O(
ε
)
. A closed form example proves that the uniform solution rate and the order one derivative discrepancy at the initial point are optimal. We also state the additional kernel and spectral assumptions needed for analogous results for variable coefficients, higher order equations, and systems.