Numerical approach for differential-difference equations having layer behaviour with small or large delay using non-polynomial spline

Abstract

A numerical approach is suggested for the layer behaviour differential-difference equations with small and large delays in the differentiated term. Using the non-polynomial spline, the numerical scheme is derived. The discretization equation is constructed using the first order derivative continuity at non-polynomial spline internal mesh points. A fitting parameter is introduced into the scheme with the help of the singular perturbation theory to minimize the error in the solution. The maximum errors in the solution are tabulated to verify the competence of the numerical method relative to the other methods in literature. We also focused on the impact of large delays on the layer behaviour or oscillatory behaviour of solutions using a special mesh with and without fitting parameter in the proposed scheme.Graphs show the effect of the fitting parameter on the solution layer.

Fitted Parameter Exponential Spline Method for Singularly Perturbed Delay Differential Equations with a Large Delay

“…The solution's layer behaviour is preserved, and precise results are obtained when compared to the perturbation parameter; the shift parameter is smaller [22,23]. However, if δðεÞ is of order OðεÞ, the solution's layer behaviour is no longer well-preserved, and oscillations becomes visible.…”

“…With the motivation of the numerical scheme derived in the manuscript titled "Numerical Approach for Differential-Difference Equations having Layer Behaviour with Small or Large Delay using Non-Polynomial Spline," https://assets .researchsquare.com/files/rs-261904/v1_covered.pdf (Ref. [23]), we have derived a scheme for the problem having large delay in this manuscript. We thank the reviewers for their constructive comments/suggestions to strengthen the manuscript.…”

“…Kadalbajoo and his colleagues began widespread numerical work using finite differentiation techniques as well as the B-spline technique with fitted mesh [19][20][21][22]. The researchers in [23] suggested a numerical approach via nonpolynomial spline for the solution of differential-difference equations with small and large delay.…”

Singular Perturbations and Large Time Delays Through Accelerated Spline-Based Compression Technique

“…Kumar and Rao [24] considered a stabilized central difference method by modifying the error terms for the boundary value problem (BVP) of singularly perturbed differential equations with a large delay. The work in [25] suggested a non polynomial spline method for solving this type of problems. An almost first order convergent finite difference scheme by using piecewise Shishkin type mesh is presented in [26] and an exponentially fitted finite difference method is suggested in [27] to tackle the problem.…”

Numerical Simulation of Singularly Perturbed Delay Differential Equations With Large Delay Using an Exponential Spline

“…The authors in [9] used a nonpolynomial spline to develop a numerical solution for a DDEs having layer with a small and large delay in the differentiated terms. Using domain decomposition, the authors of [10] suggested a mixed difference technique to solve DDEs with mixed shifts.…”