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

• 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.

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

 Citation: • • 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 $\bigg{[} \alpha = 2,\,a_0 = 6,\,\beta = 3,\,\delta = 4,\,\sigma = 1,\,\omega = 5,\, x \in [-1,5],\, t\in [-2,9] \bigg{]}$

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 $\bigg{[} \alpha = 2,\,a_0 = 6,\,\beta = 3,\,\delta = 4,\,\sigma = 1,\,\omega = 5,\, x \in [-1,5],\, t\in [-2,9] \bigg{]}$

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 $\bigg{[} \alpha = 2,\,a_0 = 6,\,\beta = 3,\,\delta = 4,\,\sigma = 1,\,\omega = 5,\, x \in [-1,5],\, t\in [-2,9] \bigg{]}$

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 $\bigg{[}\delta = 4,\,\omega = 5,\, x \in [-1,5],\, t\in [-2,9] \bigg{]}$

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

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

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

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

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

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$\times 10^{-16}$

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

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$\times 10^{-16}$
• Figures(8)

Tables(4)

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