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Classical Layer-Resolving Scheme for a System of Two Singularly Perturbed Time-Dependent Problems with Discontinuous Source Terms and Spatial Delay

Author

Listed:
  • Joseph Paramasivam Mathiyazhagan

    (PG & Research Department of Mathematics, Bishop Heber College, Bharathidasan University, Tiruchirappalli 620017, Tamil Nadu, India)

  • Ramiya Bharathi Karuppusamy

    (PG & Research Department of Mathematics, Bishop Heber College, Bharathidasan University, Tiruchirappalli 620017, Tamil Nadu, India)

  • George E. Chatzarakis

    (Department of Electrical and Electronic Engineering Educators, School of Pedagogical & Technological Education (ASPETE), 15122 Marousi, Greece)

  • S. L. Panetsos

    (Department of Electrical and Electronic Engineering Educators, School of Pedagogical & Technological Education (ASPETE), 15122 Marousi, Greece)

Abstract

In this paper, a classical layer-resolving finite difference scheme is formulated to solve a system of two singularly perturbed time-dependent initial value problems with discontinuity occurring at ( y , t ) in the source terms and Robin initial conditions. The delay term occurs in the spatial variable, and the leading term of the spatial derivative of each equation is multiplied by a distinct small positive perturbation parameter, inducing layer behaviors in the solution domain. Due to the presence of perturbation parameters, discontinuous source terms, and delay terms, initial and interior layers occur in the solution domain. In order to capture the abrupt change that occurs due to the behavior of these layers, the solution is further decomposed into smooth and singular components. Layer functions are also formulated in accordance with layer behavior. Analytical results and bounds of the solution and its components are derived. The formulation of a finite difference scheme involves discretization of temporal and spatial axes by uniform and piecewise uniform meshes, respectively. The formulated scheme achieves first-order convergence in both time and space. At last, to bolster the numerical scheme, example problems are computed to prove the efficacy and accuracy of our scheme.

Suggested Citation

  • Joseph Paramasivam Mathiyazhagan & Ramiya Bharathi Karuppusamy & George E. Chatzarakis & S. L. Panetsos, 2025. "Classical Layer-Resolving Scheme for a System of Two Singularly Perturbed Time-Dependent Problems with Discontinuous Source Terms and Spatial Delay," Mathematics, MDPI, vol. 13(3), pages 1-28, February.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:3:p:511-:d:1583033
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    References listed on IDEAS

    as
    1. R. Soundararajan & V. Subburayan & Patricia J. Y. Wong, 2023. "Streamline Diffusion Finite Element Method for Singularly Perturbed 1D-Parabolic Convection Diffusion Differential Equations with Line Discontinuous Source," Mathematics, MDPI, vol. 11(9), pages 1-17, April.
    2. V.Y. Glizer, 2003. "Asymptotic Analysis and Solution of a Finite-Horizon H ∞ Control Problem for Singularly-Perturbed Linear Systems with Small State Delay," Journal of Optimization Theory and Applications, Springer, vol. 117(2), pages 295-325, May.
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