Computational and numerical simulations for the deoxyribonucleic acid (DNA) model

Raghda A. M. Attia; Dumitru Baleanu; Dianchen Lu; Mostafa M. A. Khater; El-Sayed Ahmed

doi:10.3934/dcdss.2021018

Article Contents

2021, Volume 14, Issue 10: 3459-3478. Doi: 10.3934/dcdss.2021018

This issue Previous Article On solutions of fractal fractional differential equations Next Article On a nonlocal problem involving the fractional

$ p(x,.) $

-Laplacian satisfying Cerami condition

Computational and numerical simulations for the deoxyribonucleic acid (DNA) model

1.
School of Management & Economics, Jiangsu University of Science and Technology, 212003 Zhenjiang, China
2.
Department of Basic Science, Higher Technological Institute 10^th of Ramadan City, El Sharqia, 44634, Egypt
3.
Institute of Space Sciences, Magurele-Bucharest, Romania
4.
Department of Mathematics, Faculty of Science, Jiangsu University, 212013, China
5.
Department of Mathematics, Obour High Institute For Engineering and Technology, 11828, Cairo, Egypt
6.
Mathematics Department, Faculty of Science, Taif University P.O.Box 11099, Taif 21944, Saudi Arabia

^* Corresponding author email: mostafa.khater2024@yahoo.com
^* Corresponding author email: mostafa.khater2024@yahoo.com

Received: October 2019

Revised: January 2020

Early access: March 2021

Published: October 2021

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Abstract

In this research paper, the modified Khater method, the Adomian decomposition method, and B-spline techniques (cubic, quintic, and septic) are applied to the deoxyribonucleic acid (DNA) model to get the analytical, semi-analytical, and numerical solutions. These solutions comprise much information about the dynamical behavior of the homogenous long elastic rods with a circular section. These rods constitute a pair of the polynucleotide rods of the DNA molecule which are plugged by an elastic diaphragm that demonstrates the hydrogen bond's role in this communication. The stability property is checked for some solutions to show more effective and powerful of obtained solutions. Based on the role of analytical and semi-analytical techniques in the motivation of the numerical techniques to be more accurate, the B-spline numerical techniques are applied by using the obtained exact solutions on the DNA model to show which one of them is more accurate than other, to explain more of the dynamic behavior of the homogenous long elastic rods, and to show the coincidence between the different types of obtained solutions. The obtained solutions verified with Maple 16 & Mathematica 12 by placing them back into the original equations. The performance of these methods shows the power and effectiveness of them for applying to many different forms of the nonlinear evolution equations with an integer and fractional order.

Keywords:

Deoxyribonucleic acid (DNA) model,
Analytical,
semi-analytical and approximate solutions,
Modified Khater method,
Adomian decomposition method,
B-spline techniques.

Mathematics Subject Classification: 34F05, 34F10, 37N25, 34D20.

Citation:

Full Text(HTML)

Figure 1. Periodic soliton wave of the longitudinal displacements of the top and down strands by using Eq. (2.6) in three(a), two(c)-dimensional and contour plot(b), when $ \bigg{[} \alpha = 2,\,a_0 = 6,\,\beta = 3,\,\delta = 4,\,\sigma = 1,\,\omega = 5,\, x \in [-1,5],\, t\in [-2,9] \bigg{]} $

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Figure 2. Periodic compacton traveling wave of the longitudinal displacements of the top and down strands by using Eq. (2.7) in three(a), two(c)-dimensional and contour plot(b), when $ \bigg{[} \alpha = 2,\,a_0 = 6,\,\beta = 3,\,\delta = 4,\,\sigma = 1,\,\omega = 5,\, x \in [-1,5],\, t\in [-2,9] \bigg{]} $

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Figure 3. Periodic kink traveling wave of the longitudinal displacements of the top and down strands by using Eq. (2.15) in three(a), two(c)-dimensional and contour plot(b), when $ \bigg{[} \alpha = 2,\,a_0 = 6,\,\beta = 3,\,\delta = 4,\,\sigma = 1,\,\omega = 5,\, x \in [-1,5],\, t\in [-2,9] \bigg{]} $

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Figure 4. Periodic cuspon traveling wave of the longitudinal displacements of the top and down strands by using Eq. (2.54) in three(a), two(c)-dimensional and contour plot(b), when $ \bigg{[}\delta = 4,\,\omega = 5,\, x \in [-1,5],\, t\in [-2,9] \bigg{]} $

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Figure 5. Two dimensional plot of exact and approximate solution in combined, separate, and radar plots of Eqs. (2.53), (2.57)

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Figure 6. Two dimensional plot of exact and numerical solution that obtained by cubic spline technique in combined, separate, and radar plots

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Figure 7. Two dimensional plot of exact and numerical solution that obtained by quintic spline technique in combined, separate, and radar plots

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Figure 8. Two dimensional plot of exact and numerical solution that obtained by septic spline technique in combined, separate, and radar plots

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Table 1. Analytical and semi-analytical solution of Eq. (2.1) at different point $ \bigg{[}0.1\leq\xi\leq1\bigg{]} $

Value of	Values of	Values of	Absolute Value
$ \xi $	Approximate Solutions	Exact Solutions	of Error
0.1	0.0832639583	0.0832639583	3.85979 $ \times 10^{-12} $
0.2	0.1661133254	0.1661133244	9.8317$ \times 10^{-10} $
0.3	0.2481417478	0.2481417227	2.50479$ \times 10^{-8} $
0.4	0.3289591155	0.3289588670	2.48485$ \times 10^{-7} $
0.5	0.4081992403	0.4081977707	1.46967$ \times 10^{-6} $
0.6	0.4855272861	0.4855210208	6.26536$ \times 10^{-6} $
0.7	0.5606472102	0.5606259072	0.0000213030
0.8	0.6333096392	0.6332482704	0.0000613687
0.9	0.7033207446	0.7031650088	0.0001557360
1	0.7705527947	0.7701952621	0.0003575330

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Table 2. Analytical, numerical, and absolute values of error of obtained solutions of Eq. (2.1)

Value of	Values of	Values of	Absolute Values
$ \xi $	Approximate Solutions	Exact Solutions	of Error
0	0.0000000000	0.0000000000	0.0000000000
0.1	-0.0832541541	-0.0832639583	9.8042$ \times 10^{-6} $
0.2	-0.1660944493	-0.1661133244	0.0000188750
0.3	-0.2481152092	-0.2481417227	0.0000265135
0.4	-0.3289267807	-0.3289588670	0.0000320863
0.5	-0.4081627183	-0.4081977707	0.0000350524
0.6	-0.4854860388	-0.4855210208	0.0000349819
0.7	-0.5605943403	-0.5606259072	0.0000315669
0.8	-0.6332236466	-0.6332482704	0.0000246238
0.9	-0.7031509210	-0.7031650088	0.0000140878
1	-0.7701952621	-0.7701952621	1.11022$ \times 10^{-16} $

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Table 3. Analytical, numerical, absolute values of error of obtained solution of Eq. (2.1)

Value of $ \xi $	Approximate	Exact	Absolute Error
0	3.46945$ \times 10^{-18} $	0.0000000000	3.46945$ \times 10^{-18} $
0.1	-0.0832639540	-0.0832639583	4.22639$ \times 10^{-9} $
0.2	-0.1661133141	-0.1661133244	1.02421$ \times 10^{-8} $
0.3	-0.2481417081	-0.2481417227	1.46466$ \times 10^{-8} $
0.4	-0.3289588492	-0.3289588670	1.78158$ \times 10^{-8} $
0.5	-0.4081977514	-0.4081977707	1.92278$ \times 10^{-8} $
0.6	-0.4855210020	-0.4855210208	1.87683$ \times 10^{-8} $
0.7	-0.5606258910	-0.5606259072	1.62749$ \times 10^{-8} $
0.8	-0.6332482583	-0.6332482704	1.20805$ \times 10^{-8} $
0.9	-0.7031650035	-0.7031650088	5.25365$ \times 10^{-9} $
1	-0.7701952621	-0.7701952621	0.0000000000

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Table 4. Analytical, numerical, absolute values of error of obtained solution of Eq. (2.1)

Value of	Values of	Values of	Absolute value
$ \xi $	Approximate Solution	Exact solution	of Error
0	0.0000000000	0.0000000000	0
0.1	-0.0832592356	-0.083263958	4.72267$ \times 10^{-6} $
0.2	-0.1661049996	-0.1661133244	8.32474$ \times 10^{-6} $
0.3	-0.2481357283	-0.2481417227	5.99441$ \times 10^{-6} $
0.4	-0.3289523831	-0.3289588670	6.48396$ \times 10^{-6} $
0.5	-0.4081921510	-0.4081977707	5.61967$ \times 10^{-6} $
0.6	-0.4855155144	-0.4855210208	5.50631$ \times 10^{-6} $
0.7	-0.5606213388	-0.5606259072	4.56842$ \times 10^{-6} $
0.8	-0.6332432277	-0.6332482704	5.04269$ \times 10^{-6} $
0.9	-0.7031623445	-0.7031650088	2.66428$ \times 10^{-6} $
1	-0.7701952621	-0.7701952621	1.11022$ \times 10^{-16} $

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