<abstract xmlns="http://www.w3.org/1999/xhtml"><p>The purpose of this paper is to present a uniform finite difference method for numerical solution of a initial value problem for semilinear second order singularly perturbed delay differential equation. A numerical method is constructed for this problem which involves appropriate piecewise-uniform Shishkin mesh on each time subinterval. The method is shown to uniformly convergent with respect to the perturbation parameter. A numerical experiment illustrate in practice the result of convergence proved theoretically.</p></abstract>

The purpose of this paper is to present a uniform finite difference method for numerical solution of a initial value problem for semilinear second order singularly perturbed delay differential equation. A numerical method is constructed for this problem which involves appropriate piecewise-uniform Shishkin mesh on each time subinterval. The method is shown to uniformly convergent with respect to the perturbation parameter. A numerical experiment illustrate in practice the result of convergence proved theoretically.

A finite difference method on layer-adapted mesh for singularly perturbed delay differential equations

Department of Mathematics, Faculty of Science

Applied Mathematics and Nonlinear Sciences

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

The purpose of this paper is to present a uniform finite difference method for numerical solution of a initial value problem for semilinear second order...

ɛ	N = 64	N = 128	N = 256	N = 512	N = 1024
2⁻¹	0.0085116	0.0042832	0.0021485	0.0010760	0.0005384
2⁻¹	0.99	1.00	1.00	1.00
2⁻²	0.0140007	0.0070930	0.0035693	0.0017090	0.0008906
2⁻²	0.98	0.98	0.99	1.00
2⁻⁴	0.0241181	0.0143603	0.0083166	0.0047146	0.0026331
2⁻⁴	0.75	0.78	0.81	0.84
2⁻⁶	0.0230541	0.0137267	0.0079497	0.0045066	0.0025149
2⁻⁶	0.75	0.78	0.81	0.84
2⁻⁸	0.0227881	0.0135684	0.0078580	0.0044546	0.0024859
2⁻⁸	0.75	0.78	0.81	0.84
2⁻¹⁰	0.0227216	0.0135288	0.0078350	0.0044416	0.0024786
2⁻¹⁰	0.75	0.78	0.81	0.84
2⁻¹²	0.0227050	0.0135189	0.0078293	0.0044382	0.0024768
2⁻¹²	0.75	0.78	0.81	0.84
2⁻¹⁴	0.0227008	0.0135164	0.0078279	0.0044361	0.0024763
2⁻¹⁴	0.75	0.78	0.81	0.84
2⁻¹⁶	0.0226998	0.0135158	0.00782756	0.0044374	0.0024762
2⁻¹⁶	0.75	0.78	0.81	0.84

ɛ	N = 64	N = 128	N = 256	N = 512	N = 1024
2⁻¹	0.0079814	0.0040103	0.0020101	0.0010063	0.0005033
2⁻¹	0.98	0.99	1.00	1.00
2⁻²	0.0012138	0.0061392	0.0030875	0.0015482	0.0007775
2⁻²	0.98	0.99	1.00	1.00
2⁻⁴	0.0202600	0.0120754	0.0069885	0.0039609	0.0022110
2⁻⁴	0.75	0.79	0.81	0.84
2⁻⁶	0.0194243	0.0115426	0.0066802	0.0037862	0.0021126
2⁻⁶	0.75	0.78	0.82	0.84
2⁻⁸	0.0191427	0.0114094	0.0066031	0.0037425	0.0020882
2⁻⁸	0.75	0.78	0.82	0.84
2⁻¹⁰	0.0190868	0.0113761	0.0065838	0.0037315	0.0020821
2⁻¹⁰	0.75	0.78	0.82	0.84
2⁻¹²	0.0190729	0.0113678	0.0065790	0.0037288	0.0020806
2⁻¹²	0.75	0.78	0.82	0.84
2⁻¹⁴	0.0190696	0.0113657	0.0065778	0.0037281	0.0020668
2⁻¹⁴	0.75	0.78	0.82	0.84
2⁻¹⁶	0.0190685	0.0113652	0.0065754	0.0037280	0.0020801
2⁻¹⁶	0.75	0.78	0.82	0.84

A finite difference method on layer-adapted mesh for singularly perturbed delay differential equations

Published Online: Mar 31, 2020

Page range: 425 - 436

Received: Jun 11, 2019

Accepted: Aug 13, 2019

DOI: https://doi.org/10.2478/amns.2020.1.00040

Keywords
Delay differential equation, Singular perturbation, Finite difference scheme, Piecewise-uniform mesh, Error estimates

© 2019 Fevzi Erdogan et al., published by Sciendo

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

Maximum errors and rates of convergence on ωN,1

Maximum errors and rates of convergence on ωN,2

A finite difference method on layer-adapted mesh for singularly perturbed delay differential equations

Published Online: Mar 31, 2020

Page range: 425 - 436

Received: Jun 11, 2019

Accepted: Aug 13, 2019

DOI: https://doi.org/10.2478/amns.2020.1.00040

KeywordsDelay differential equation, Singular perturbation, Finite difference scheme, Piecewise-uniform mesh, Error estimates

© 2019 Fevzi Erdogan et al., published by Sciendo

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

Maximum errors and rates of convergence on ωN,1

Maximum errors and rates of convergence on ωN,2

Keywords
Delay differential equation, Singular perturbation, Finite difference scheme, Piecewise-uniform mesh, Error estimates