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<p>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,&#xa0;2] with an integral boundary condition. The discontinuity is expected to happen at a point <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(d \in (0,2)\)</EquationSource> </InlineEquation>, while a tiny positive parameter adjusts the leading coefficient. The solution displays boundary layers at <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(x = 0\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(x = 2\)</EquationSource> </InlineEquation>, and there may be interior layers at <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(x = 1\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(x = d\)</EquationSource> </InlineEquation>, and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(x = 1 + d\)</EquationSource> </InlineEquation>, depending on the location of <i>d</i>. 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.</p>

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

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  • 发表日期：

    2026年2月21日

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  • Sekar Elango,
  • Mohammad Izadi,
  • Murugesan Manigandan

摘要

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 \(d \in (0,2)\) , 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.