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

Joseph Paramasivam Mathiyazhagan, Ramiya Bharathi Karuppusamy, George E. Chatzarakis () and S. L. Panetsos
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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

Mathematics, 2025, vol. 13, issue 3, 1-28

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.

Keywords: finite difference scheme; singularly perturbed problems; discontinuous source terms; spatial delay; Robin initial conditions; interior layers (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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