Effective Role of Cubic Splines for the Numerical Computations of Non-Linear Model of Viscoelastic Fluid

$ϖ$	Accurate solution	CNPS solution	\|error\| on CNPS	CPS solution	\|error\| on CPS
0	1	1	0.00E-00	1	0.00E-00
0.2	0.818730753	0.817688874	1.04E-03	0.817618382	1.11E-03
0.4	0.670320046	0.668633087	1.69E-03	0.668558559	1.76E-03
0.6	0.548811636	0.547122919	1.69E-03	0.547064978	1.75E-03
0.8	0.449328964	0.448284237	1.04E-03	0.448246958	1.08E-03
1	0.367879441	0.367879441	0.00E-00	0.367879441	0.00E-00

Tab. 2.

Comparison of accurate, CNPS along with CPS for problem 3.1 at $h = \frac{1}{10}$

$ϖ$	Accurate solution	CNPS solution	\|error\| on CNPS	CPS solution	\|error\| on CPS	[6]
0	1	1	0.00E-00	1	0.00E-00	0.00E-00
0.1	0.904837418	0.904837481	6.27E-08	0.904812814	2.46E-05	3.82E-06
0.2	0.818730753	0.818730875	1.22E-07	0.818688854	4.19E-05	1.36E-05
0.3	0.740818221	0.740818391	1.70E-07	0.740765438	5.28E-05	2.49E-05
0.4	0.670320046	0.670320248	2.02E-07	0.670262025	5.80E-05	3.31E-05
0.5	0.60653066	0.606530874	2.14E-07	0.606472401	5.83E-05	3.50E-05
0.6	0.548811636	0.548811841	2.05E-07	0.548757603	5.40E-05	3.01E-05
0.7	0.496585304	0.496585478	1.74E-07	0.496539521	4.58E-05	2.03E-05
0.8	0.449328964	0.44932909	1.26E-07	0.449295105	3.39E-05	9.57E-06
0.9	0.40656966	0.406569726	6.61E-08	0.406551127	1.85E-05	2.11E-06
1	0.367879441	0.367879441	0.00E-00	0.367879441	0.00E-00	0.00E-00

Comparison of AEs of CNPS and CPS with [6] of problem 3.1 at $h = \frac{1}{10}$

Graphically representation of Accurate solution, CNPS outcome, CPS outcome and their Absolute Errors for problem 3.1 at $h = \frac{1}{10}$

Tab. 3.

Comparison of accurate, CNPS along with CPS for problem 3.2 at $h = \frac{1}{5}$

$ϖ$	Accurate solution	CNPS solution	\|error\| on CNPS	CPS solution	\|error\| on CPS
0	1	1	0.00E-00	1	0.00E-00
0.2	0.654984602	0.654548343	4.36E-04	0.653822489	1.16E-03
0.4	0.402192028	0.401485267	7.07E-04	0.400509549	1.68E-03
0.6	0.219524654	0.218816219	7.08E-04	0.217928602	1.60E-03
0.8	0.089865793	0.089426899	4.39E-04	0.088878968	9.87E-04
1	0	0	0.00E-00	0	0.00E-00

Tab. 4.

Comparison of accurate, CNPS along with CPS for problem 3.2 at $h = \frac{1}{10}$

$ϖ$	Accurate solution	CNPS solution	\|error\| on CNPS	CPS solution	\|error\| on CPS	[6]
0	1	1	0.00E-00	1	0.00E-00	0.00E-00
0.1	0.814353676	0.814349741	0.814016517	3.37E-04	3.94E-06	1.91E-05
0.2	0.654984602	0.654977145	0.654367593	6.17E-04	7.46E-06	6.81E-05
0.3	0.518572754	0.518562529	0.517747628	8.25E-04	1.02E-05	1.24E-04
0.4	0.402192028	0.402180054	0.401241892	9.50E-04	1.20E-05	1.64E-04
0.5	0.30326533	0.303252787	0.302279968	9.85E-04	1.25E-05	1.72E-04
0.6	0.219524654	0.219512766	0.218594693	9.30E-04	1.19E-05	1.46E-04
0.7	0.148975591	0.148965506	0.148186139	7.89E-04	1.01E-05	9.70E-05
0.8	0.089865793	0.089858479	0.089290229	5.76E-04	7.31E-06	4.49E-05
0.9	0.040656966	0.040653124	0.040351558	3.05E-04	3.84E-06	9.44E-06
1	0	0	0	0.00E-00	0.00E-00	0.00E-00

Comparison of AEs of CNPS and CPS with [6] of problem 3.2 at $h = \frac{1}{10}$

Graphically representation of Accurate solution, CNPS outcome, CPS outcome and their Absolute Errors for problem 3.2 at $h = \frac{1}{10}$

Tab. 5.

Comparison of accurate, CNPS along with CPS for problem 3.3 at $h = \frac{1}{10}$

$ϖ$	Accurate solution	CNPS solution	\|error\| on CNPS	CPS solution	\|error\| on CPS	[6]
0	1	1	1	0.00E-00	0.00E-00	0.00E-00
0.1	1.105170918	1.105171098	1.105120542	5.04E-05	1.80E-07	3.82E-06
0.2	1.221402758	1.221403102	1.221310719	9.20E-05	3.43E-07	1.35E-05
0.3	1.349858808	1.349859281	1.349734356	1.24E-04	4.73E-07	2.64E-05
0.4	1.491824698	1.491825254	1.49167782	1.47E-04	5.56E-07	3.85E-05
0.5	1.648721271	1.648721853	1.648562907	1.58E-04	5.82E-07	4.64E-05
0.6	1.8221188	1.82211935	1.821961083	1.58E-04	5.50E-07	4.70E-05
0.7	2.013752707	2.01375317	2.01360923	1.43E-04	4.62E-07	3.94E-05
0.8	2.225540928	2.225541259	2.225427036	1.14E-04	3.31E-07	2.48E-05
0.9	2.459603111	2.459603282	2.45953623	6.69E-05	1.70E-07	8.50E-06
1	2.718281828	2.71828	2.71828	0.00E-00	0.00E-00	0.00E-00

Tab. 6.

Comparison of accurate, CNPS along with CPS for problem 3.3 at $h = \frac{1}{5}$

$ϖ$	Accurate solution	CNPS solution	\|error\| on CNPS	CPS solution	\|error\| on CPS	[6]
0	1	1	0.00E-00	1	0.00E-00	0.00E-00
0.2	1.221402758	1.221403377	6.19E-07	1.22103274	3.70E-04	5.42E-05
0.4	1.491824698	1.491825707	1.01E-06	1.491234204	5.90E-04	1.53E-04
0.6	1.8221188	1.822119714	9.13E-07	1.821484771	6.34E-04	1.95E-04
0.8	2.225540928	2.225541357	4.28E-07	2.225083141	4.58E-04	1.04E-04
1	2.718281828	2.71828	0.00E-00	2.71828	0.00E-00	0.00E-00

Comparison of AEs of CNPS and CPS with [6] of problem 3.3 at $h = \frac{1}{10}$

Graphically representation of Accurate solution, CNPS outcome, CPS outcome and their Absolute Errors for problem 3.3 at $h = \frac{1}{10}$

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DOI: https://doi.org/10.2478/ama-2025-0021 | Journal eISSN: 2300-5319 | Journal ISSN: 1898-4088

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Language: English

Page range: 177 - 188

Submitted on: Jan 5, 2024

Accepted on: Jun 25, 2024

Published on: Jun 6, 2025

Published by: Bialystok University of Technology

In partnership with: Paradigm Publishing Services

Keywords:

finite difference method,

central difference,

boundary value problems,

Related subjects:

Mechanical engineering,

Mechanics,

Bioengineering and biomedical engineering,

Biomechanics,

Civil engineering,

Environmental engineering

© 2025 Aasma Khalid, Rubab Batool, Kashif Ali Abro, Akmal Rehan, published by Bialystok University of Technology
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 19 (2025): Issue 2 (June 2025)

Effective Role of Cubic Splines for the Numerical Computations of Non-Linear Model of Viscoelastic Fluid

Figures & Tables

Tab.1.

Tab. 2.

Fig. 1.

Fig. 2.

Tab. 3.

Tab. 4.

Fig. 3.

Fig. 4.

Tab. 5.

Tab. 6.

Fig. 5.

Fig. 6.

Paradigm

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