Shilpa Malge
Department of Applied Sciences, Symbiosis Institute of Technology, Pune Campus, Symbiosis International (Deemed University), Lavale, 412115, Pune, Maharashtra, India.

Ram Kishun Lodhi
Department of Applied Sciences, Symbiosis Institute of Technology, Pune Campus, Symbiosis International (Deemed University), Lavale, 412115, Pune, Maharashtra, India.

DOI https://doi.org/10.33889/IJMEMS.2025.10.5.068

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Abstract

Various physical phenomena give rise to singularly perturbed differential equations with mixed shifts. Due to multiple parameters, singularly perturbed mixed delay boundary value problems are challenging to solve. This article considers a singularly perturbed differential-difference equation with delay and advance. To deal with the complexity of these equations, a non-polynomial spline numerical approach is adopted. For discretization, we have used a uniform mesh with equal spacing. Various theoretical results like stability and convergence are discussed. Numerical examples are solved to support the method and check the validity of the findings. The numerical order of convergence is determined and presented in the tables, along with the comparison of the results with the other existing methods. The comparison shows that the error is significantly less than the available solution in the literature. Also, the numerical order of convergence is determined to be equal to two. Graphs are drawn to observe the behavior of the solution for different values of parameters.

Keywords- Singular perturbation, Differential-difference equations, Boundary value problems, Mixed delay, Non-polynomial spline, Boundary layer.

Citation

Malge, S., & Lodhi, R. K (2025). Non-Polynomial Spline for Singularly Perturbed Differential-Difference Equation with Mixed Shifts and Layer Behavior. International Journal of Mathematical, Engineering and Management Sciences, 10(5), 1447-1462. https://doi.org/10.33889/IJMEMS.2025.10.5.068.