<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>This study deals with singularly perturbed Volterra integro-differential equations with delay. Based on the properties of the exact solution, a hybrid difference scheme with appropriate quadrature rules on a Shishkin-type mesh is constructed. By using the truncation error estimate techniques and a discrete analogue of Grönwall’s inequality it is proved that the hybrid finite difference scheme is almost second order accurate in the discrete maximum norm. Numerical experiments support these theoretical results and indicate that the estimates are sharp.</p>
</abstract>

This study deals with singularly perturbed Volterra integro-differential equations with delay. Based on the properties of the exact solution, a hybrid difference scheme with appropriate quadrature rules on a Shishkin-type mesh is constructed. By using the truncation error estimate techniques and a discrete analogue of Grönwall’s inequality it is proved that the hybrid finite difference scheme is almost second order accurate in the discrete maximum norm. Numerical experiments support these theoretical results and indicate that the estimates are sharp.

A second order numerical method for singularly perturbed Volterra integro-differential equations with delay

Faculty of Economics and Administrative Sciences, Department of Econometrics

International Journal of Mathematics and Computer in Engineering

This work is licensed under the Creative Commons Attribution 4.0 International License.

This study deals with singularly perturbed Volterra integro-differential equations with delay. Based on the properties of the exact solution, a hybrid...

Number of mesh points N

ɛ	N=64	N=128	N=256	N=512	N=1024	N=2048

2⁻⁰	5.6119e-5	1.4030e-5	3.5075e-6	8.7687e-7	2.1922e-7	5.4804e-8
2⁻⁰	2.572	2.477	2.409	2.358	2.319	-
2⁻⁴	1.9811e-3	4.9279e-4	1.2304e-4	3.0751e-5	7.6880e-6	1.9220e-6
2⁻⁴	2.581	2.479	2.410	2.358	2.319	-
2⁻⁸	2.0850e-3	7.0533e-4	2.3017e-4	7.2825e-5	2.2479e-5	6.7995e-6
2⁻⁸	2.011	2.001	2.000	2.000	2.000	-
2⁻¹²	2.0847e-3	7.0525e-4	2.3015e-4	7.2817e-5	2.2477e-5	6.7988e-6
2⁻¹²	2.011	2.001	2.000	2.000	2.000	-
2⁻¹⁶	2.0847e-3	7.0525e-4	2.3015e-4	7.2817e-5	2.2477e-5	6.7988e-6
2⁻¹⁶	2.011	2.001	2.000	2.000	2.000	-
2⁻²⁰	2.0847e-3	7.0525e-4	2.3015e-4	7.2817e-5	2.2477e-5	6.7988e-6
2⁻²⁰	2.011	2.001	2.000	2.000	2.000	-
2⁻²⁴	2.0847e-3	7.0525e-4	2.3015e-4	7.2817e-5	2.2477e-5	6.7988e-6
2⁻²⁴	2.011	2.001	2.000	2.000	2.000	-
2⁻²⁸	2.0847e-3	7.0525e-4	2.3015e-4	7.2817e-5	2.2477e-5	6.7988e-6
2⁻²⁸	2.011	2.001	2.000	2.000	2.000	-

e^N	2.0847e-3	7.0525e-4	2.3015e-4	7.2817e-5	2.2477e-5	6.7988e-6
r^N	2.011	2.001	2.000	2.000	2.000	-

A second order numerical method for singularly perturbed Volterra integro-differential equations with delay

Article Category: Original Study

Data publikacji: 31 paź 2023

Zakres stron: 85 - 96

Otrzymano: 07 cze 2023

Przyjęty: 07 sie 2023

DOI: https://doi.org/10.2478/ijmce-2024-0007

Słowa kluczowe
Singular perturbation, delay differential equation, integro-differential equation, finite difference scheme, Shishkin mesh

© 2024 Fevzi Erdoğan et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Error estimates and convergence rates for various N for Example

A second order numerical method for singularly perturbed Volterra integro-differential equations with delay

Article Category: Original Study

Data publikacji: 31 paź 2023

Zakres stron: 85 - 96

Otrzymano: 07 cze 2023

Przyjęty: 07 sie 2023

DOI: https://doi.org/10.2478/ijmce-2024-0007

Słowa kluczoweSingular perturbation, delay differential equation, integro-differential equation, finite difference scheme, Shishkin mesh

© 2024 Fevzi Erdoğan et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Error estimates and convergence rates for various N for Example

Słowa kluczowe
Singular perturbation, delay differential equation, integro-differential equation, finite difference scheme, Shishkin mesh