Numerical approximation for a class of singularly perturbed delay differential equations with boundary and interior layer(s)

Abstract

This paper is devoted to the study of singularly perturbed delay differential equations with or without a turning point. The solution of the considered class of problems may exhibit boundary or interior layer(s) due to the presence of the perturbation parameter, the turning point, and the delay term. Some a priori estimates are derived on the solution and its derivatives. To solve the problem numerically, a finite difference scheme on piecewise uniform Shishkin mesh along with interpolation to tackle the delay term is proposed. The solution is decomposed into regular and singular components to establish parameter uniform error estimate. It is shown that the proposed scheme converges to the solution of the continuous problem uniformly with respect to the singular perturbation parameter. The numerical experiments corroborate the theoretical findings.

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Correspondence to Kapil K. Sharma.

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Rai, P., Sharma, K.K. Numerical approximation for a class of singularly perturbed delay differential equations with boundary and interior layer(s). Numer Algor 85, 305–328 (2020). https://doi.org/10.1007/s11075-019-00815-6

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Keywords

  • Delay differential equations
  • Singular perturbation
  • Turning point
  • Interior layer
  • Interpolation
  • Shishkin mesh

Mathematics Subject Classification (2010)

  • 34K26
  • 34K28
  • 65L10
  • 65L12
  • 65L20