What question did this study set out to answer?

The aim is to develop efficient numerical approaches for second-order delay differential equations with discontinuities and integral boundary conditions.

February 23, 2026Open Access

Efficient Numerical Approaches for Solving Singularly Perturbed Delay Differential Equations with Discontinuities and Integral Boundary Constraints

Key Points

The aim is to develop efficient numerical approaches for second-order delay differential equations with discontinuities and integral boundary conditions.
Applied a finite difference method on a segmented Shishkin mesh.
Investigated the behavior of solutions over the interval [0, 2].
Analyzed the effect of a small perturbation parameter on the leading coefficient.
Demonstrated first-order uniform convergence of the numerical scheme concerning the perturbation parameter.
Identified boundary layers at x = 0 and x = 2, with potential interior layers based on the discontinuity location.

Abstract

This research focuses on studying a specific problem involving a second-order delay differential equation that has a discontinuous source term over the interval 0, 2 with an integral boundary condition. The discontinuity is expected to happen at a point d (0, 2), while a tiny positive parameter adjusts the leading coefficient. The solution displays boundary layers at x = 0 and x = 2, and there may be interior layers at x = 1, x = d, and x = 1 + d, depending on the location of d. To tackle this issue, a finite difference method is implemented on a segmented Shishkin mesh. We establish first-order uniform convergence of the scheme with respect to the perturbation parameter, supported by numerical validation.

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