A robust septic hermite collocation technique for dirichlet boundary condition Heat conduction equation

k	SHCM-L	SHCM-C	[15]	[11]CHCM-L
0.01	7.1591 × 10⁻⁷	7.1591 × 10⁻⁷	4.2273 × 10⁻⁴	4.1333 × 10⁻⁷
0.005	1.7931 × 10⁻⁷	1.7932 × 10⁻⁷	1.0560 × 10⁻⁴	1.0353 × 10⁻⁷
0.0025	4.4851 × 10⁻⁸	4.4851 × 10⁻⁸	2.6395 × 10⁻⁵	2.5895 × 10⁻⁸

Table 2

A comparison of the computed error norm L₂ of the proposed scheme with those given in [10, 11] for M = 5, 10, 20 and k = 1/10⁶ at T = 1.

h	SHCM-L	SHCM-C	[10] (CN-I)	[10] (CN-II)	[11] CHCM-L
1/5	1.2153 × 10⁻¹⁴	1.0075 × 10⁻¹²	4.7696 × 10⁻⁶	8.5859 × 10⁻³	3.5716 × 10⁻⁸
1/10	1.9449 × 10⁻¹⁵	2.3323 × 10⁻¹⁴	7.7143 × 10⁻⁹	2.1412 × 10⁻³	2.2848 × 10⁻⁹
1/20	4.6215 × 10⁻¹⁵	7.4211 × 10⁻¹⁵	1.8820 × 10⁻¹¹	5.3498 × 10⁻⁴	1.4361 × 10⁻¹⁰
1/40	8.1433 × 10⁻¹⁵	9.6794 × 10⁻¹⁵

Table 3

A comparison of the computed error norm L₂ of the proposed scheme with those given in [11, 15] for M = 10, 20, 40 and k = 1/10⁶ at T = 0.1.

h	SHCM-L	[15]	[15]	[11] CHCM-L
		θ = 0.1	θ = 0.2
1/10	6.2482 × 10⁻¹³	1.6534 × 10⁻⁴	2.4968 × 10⁻⁴	1.6957 × 10⁻⁶
1/20	3.3280 × 10⁻¹²	4.3906 × 10⁻⁷	7.8152 × 10⁻⁷	1.0426 × 10⁻⁷
1/40	5.8595 × 10⁻¹²	8.6154 × 10⁻¹⁰	8.0111 × 10⁻¹⁰	6.4916 × 10⁻⁹
		θ = 0.3	θ = 0.4
1/10	6.2482 × 10⁻¹³	3.2357 × 10⁻⁴	3.6276 × 10⁻⁴	1.6957 × 10⁻⁶
1/20	3.3280 × 10⁻¹²	1.0791 × 10⁻⁶	1.2556 × 10⁻⁶	1.0426 × 10⁻⁷
1/40	5.8595 × 10⁻¹²	9.2171 × 10⁻¹⁰	1.1166 × 10⁻⁹	6.4916 × 10⁻⁹
		θ = 0.5	θ = 0.6
1/10	6.2482 × 10⁻¹³	3.5225 × 10⁻⁴	2.8316 × 10⁻⁴	1.6957 × 10⁻⁶
1/20	3.3280 × 10⁻¹²	1.2494 × 10⁻⁶	1.0109 × 10⁻⁶	1.0426 × 10⁻⁷
1/40	5.8595 × 10⁻¹²	1.2167 × 10⁻⁹	1.1107 × 10⁻⁹	6.4916 × 10⁻⁹
		θ = 0.7	θ = 0.8
1/10	6.2482 × 10⁻¹³	1.5954 × 10⁻⁴	1.9640 × 10⁻⁴	1.6957 × 10⁻⁶
1/20	3.3280 × 10⁻¹²	5.4419 × 10⁻⁷	9.1503 × 10⁻⁷	1.0426 × 10⁻⁷
1/40	5.8595 × 10⁻¹²	7.4804 × 10⁻¹⁰	8.6751 × 10⁻¹⁰	6.4916 × 10⁻⁹
		θ = 0.9	θ = 0.95
1/10	6.2482 × 10⁻¹³	4.5197 × 10⁻⁴	6.1645 × 10⁻⁴	1.6957 × 10⁻⁶
1/20	3.3280 × 10⁻¹²	2.0820 × 10⁻⁶	2.8818 × 10⁻⁶	1.0426 × 10⁻⁷
1/40	5.8595 × 10⁻¹²	2.4668 × 10⁻⁹	3.6858 × 10⁻⁹	6.4916 × 10⁻⁹

Table 4

A comparison of the computed error norm L_∞ of the proposed scheme with those given in [7, 11, 12] for various values of h = k at T = 1.

h = k	SHCM-L	SHCM-C	[7]	[12] (CN)	[12] (CBVM)	[11] CHCM-L
0.2	5.1578 × 10⁻⁵	5.1552 × 10⁻⁵	1.4145 × 10⁻¹	1.1 × 10⁻¹	2.8 × 10⁻²	5.8498 × 10⁻⁵
0.1	3.1586 × 10⁻⁵	3.1587 × 10⁻⁵	3.7195 × 10⁻²	3.0 × 10⁻²	3.8 × 10⁻³	3.1584 × 10⁻⁵
0.05	9.7106 × 10⁻⁶	9.7107 × 10⁻⁶	8.4588 × 10⁻³	6.9 × 10⁻³	2.7 × 10⁻⁴	9.7065 × 10⁻⁶
0.025	2.5489 × 10⁻⁶	2.5489 × 10⁻⁶	2.0698 × 10⁻³	1.7 × 10⁻³	1.3 × 10⁻⁵	2.5485 × 10⁻⁶
0.0125	6.4490 × 10⁻⁷	6.4490 × 10⁻⁷	5.1473 × 10⁻⁴	4.2 × 10⁻⁴	5.1 × 10⁻⁷	6.4488 × 10⁻⁷
0.00625	1.6171 × 10⁻⁷	1.6171 × 10⁻⁷		1.1 × 10⁻⁴	3.6 × 10⁻⁸	1.6171 × 10⁻⁷
0.01	4.1333 × 10⁻⁷	4.1333 × 10⁻⁷				4.1332 × 10⁻⁷
0.005	1.0353 × 10⁻⁷	1.0353 × 10⁻⁷				1.0353 × 10⁻⁷
0.0025	2.5895 × 10⁻⁸	2.5895 × 10⁻⁸				2.5895 × 10⁻⁸
0.002	1.6574 × 10⁻⁸	1.6574 × 10⁻⁸				1.6574 × 10⁻⁸
0.001	4.1437 × 10⁻⁹	4.1437 × 10⁻⁹				4.1437 × 10⁻⁹

Table 5

A comparison of the computed error norms L₂ and L_∞ of the proposed scheme with those given in [9, 11] for h = 1/16, k = 0.01 at various T.

t_final	L₂			L_∞
t_final	SHCM-L	[11] CHCM-L	[9]	SHCM-L	[11] CHCM-L	[9]
0.1	5.1774 × 10⁻⁴	2.9917 × 10⁻⁴	4.86 × 10⁻³	2.9891 × 10⁻⁴	2.9891 × 10⁻⁴	6.79 × 10⁻³
0.3	2.1558 × 10⁻⁴	1.2457 × 10⁻⁴	8.87 × 10⁻⁵	1.2447 × 10⁻⁴	1.2447 × 10⁻⁴	3.76 × 10⁻⁴
0.5	4.9872 × 10⁻⁵	2.8818 × 10⁻⁵	1.73 × 10⁻³	2.8793 × 10⁻⁵	2.8793 × 10⁻⁵	2.44 × 10⁻⁴
0.7	9.6911 × 10⁻⁶	5.5999 × 10⁻⁶	2.04 × 10⁻⁴	5.5950 × 10⁻⁶	5.5953 × 10⁻⁶	3.17 × 10⁻⁴
0.9	1.7294 × 10⁻⁶	9.9934 × 10⁻⁷	2.14 × 10⁻³	9.9847 × 10⁻⁷	9.9856 × 10⁻⁷	3.14 × 10⁻³
1.0	7.1591 × 10⁻⁷	4.1368 × 10⁻⁷	2.15 × 10⁻³	4.1332 × 10⁻⁷	4.1338 × 10⁻⁷	3.32 × 10⁻³

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DOI: https://doi.org/10.2478/ijmce-2025-0019 | Journal eISSN: 2956-7068

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Language: English

Page range: 253 - 266

Submitted on: Feb 23, 2024

Accepted on: May 11, 2024

Published on: Sep 19, 2024

Published by: Harran University

In partnership with: Paradigm Publishing Services

Publication frequency: 2 issues per year

Keywords:

1D Heat equation,

approximate solutions,

septic hermite collocation method,

finite elements method,

stability analysis

MSC:

41A15,

65M60,

65M12,

65M15,

35K05

Related subjects:

Computer sciences,

Computer sciences, other,

Engineering,

Introductions and overviews,

Physics,

© 2024 Selçuk Kutluay, Nuri Murat Yağmurlu, Ali Sercan Karakaş, published by Harran University
This work is licensed under the Creative Commons Attribution 4.0 License.

Volume 3 (2025): Issue 2 (December 2025)