Monitoring Dynamical Behavior and Optical Solutions of Space-Time Fractional Order Double-Chain Deoxyribonucleic Acid Model Considering the Atangana’s Conformable Derivative

Document Type : Research Paper


1 Department of Physics and Engineering Mathematics, Zagazig University, Faculty of Engineering, 44519, Egypt

2 Department of Mathematics, Saint Xavier University, Chicago, IL 60655, United States

3 Department of Basic Science, Faculty of Engineering, Delta University for Science and Technology, 11152, Gamasa, Egypt


DNA, or deoxyribonucleic acid, is found in every single cell and is the cell's primary information storage medium. DNA stores all an organism's genetic information, including the instructions it needs to grow, divide, and live. DNA is made up of four different building blocks called nucleotide bases: adenine (A), thymine (T), cytosine (C), and guanine (G). The genome is sequenced in vitro utilizing encoding strategies such as labelling one bond pair as 0 and the other as 1 to store digital information. In this study, the fractional differential order of double-chain DNA dynamical system was investigated, considering Atangana’s conformable fractional derivative. The conformable sub-equation method was applied to the system.  The analysis resulted in some interesting new exact solutions of the model. One-soliton kink solution, multiple-soliton solutions, and periodic-wave solutions are the three broad categories that may be used to describe the results. In order to better understanding the solutions found, we have visually investigated a few of them. Both solitary and anti-solitary waves of the DNA strands are seen, attesting to the nonlinear dynamics of the system. The gathered data might be used to conduct application evaluations and draw further scientific findings.


Main Subjects

Publisher’s Note Shahid Chamran University of Ahvaz remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

[1] Williams, K.L., "Gene mapping," Encyclopedia of bioinformatics and computational biology, S. Ranganathan, M. Gribskov, K. Nakai and C. Schönbach (Editors), Academic Press, Oxford, 2019.
[2] Zilberman, D., Gehring, M., Tran, R.K., Ballinger, T., Henikoff, S., Genome-wide analysis of arabidopsis thaliana DNA methylation uncovers an interdependence between methylation and transcription, Nature Genetics, 39, 2007, 61-69.
[3] Cevallos, Y., Nakano, T., Tello-Oquendo, L., Rushdi, A., Inca, D., Santillán, I., Shirazi, A.Z., Samaniego, N., A brief review on DNA storage, compression, and digitalization, Nano Communication Networks, 31, 2022, 100391.
[4] Douglas, C.M.W., Stemerding, D., Challenges for the European governance of synthetic biology for human health, Life Sciences, Society and Policy, 10, 2014, 6.
[5] Katz, L., Chen, Y.Y., Gonzalez, R., Peterson, T.C., Zhao, H., Baltz, R.H., Synthetic biology advances and applications in the biotechnology industry: A perspective, Journal of Industrial Microbiology and Biotechnology, 45, 2018, 449-461.
[6] Daniel, M., Vanitha, M., Bubble solitons in an inhomogeneous, helical DNA molecular chain with flexible strands, Physical Review E, 84, 2011, 031928.
[7] Hao, Y., Li, Q., Fan, C., Wang, F., Data storage based on DNA, Small Structures, 2, 2021, 2000046.
[8] Dong, Y., Sun, F., Ping, Z., Ouyang, Q., Qian, L., DNA storage: Research landscape and future prospects, National Science Review, 7, 2020, 1092-1107.
[9] Yao, S.-W., Mabrouk, S.M., Inc, M., Rashed, A.S., Analysis of double-chain deoxyribonucleic acid dynamical system in pandemic confrontation, Results in Physics, 42, 2022, 105966.
[10] Rashed, A.S., Kassem, M.M., Hidden symmetries and exact solutions of integro-differential Jaulent–Miodek evolution equation, Applied Mathematics and Computation, 247, 2014, 1141-1155.
[11] Mabrouk, S.M., Rashed, A.S., Analysis of (3 + 1)-dimensional Boiti – Leon –Manna–Pempinelli equation via Lax pair investigation and group transformation method, Computers & Mathematics with Applications, 74, 2017, 2546-2556.
[12] Kassem, M.M., Rashed, A.S., N-solitons and cuspon waves solutions of (2 + 1)-dimensional broer–kaup–kupershmidt equations via hidden symmetries of lie optimal system, Chinese Journal of Physics, 57, 2019, 90-104.
[13] Rashed, A.S., Analysis of (3+1)-dimensional unsteady gas flow using optimal system of lie symmetries, Mathematics and Computers in Simulation, 156, 2019, 327-346.
[14] Saleh, R., Rashed, A.S., New exact solutions of (3 + 1)dimensional generalized Kadomtsevpetviashvili equation using a combination of lie symmetry and singular manifold methods, Mathematical Methods in the Applied Sciences, 43, 2020, 2045-2055.
[15] Saleh, R., Rashed, A.S., Wazwaz, A.-M., Plasma-waves evolution and propagation modeled by sixth order Ramani and coupled Ramani equations using symmetry methods, Physica Scripta, 96, 2021, 085213.
[16] Mabrouk, S.M., Rashed, A.S., N-solitons, kink and periodic wave solutions for (3+1)-dimensional Hirota bilinear equation using three distinct techniques, Chinese Journal of Physics, 60, 2019, 48-60.
[17] Kaewta, S., Sirisubtawee, S., Sungnul, S., Application of the exp-function and generalized Kudryashov methods for obtaining new exact solutions of certain nonlinear conformable time partial integro-differential equations, Computation, 9, 2021, 52.
[18] Bekir, A., Zahran, E.H.M., Painlevé approach and its applications to get new exact solutions of three biological models instead of its numerical solutions, International Journal of Modern Physics B, 34, 2020, 2050270.
[19] Saleh, R., Kassem, M., Mabrouk, S.M., Exact solutions of nonlinear fractional order partial differential equations via singular manifold method, Chinese Journal of Physics, 61, 2019, 290-300.
[20] Saleh, R., Kassem, M., Mabrouk, S.M., Investigation of breaking dynamics for Riemann waves in shallow water, Chaos, Solitons & Fractals, 132, 2020, 109571.
[21] Saleh, R., Mabrouk, S.M., Wazwaz, A.M., The singular manifold method for a class of fractional-order diffusion equations, Waves in Random and Complex Media, 2022, DOI: 10.1080/17455030.2021.2017069.
[22] El-Ganaini, S., Comment on ″optical soliton to multi-core (coupling with all the neighbors) directional couplers and modulation instability″, The European Physical Journal Plus, 137, 2022, 483.
[23] Ma, W.-X., Zhang, Y., Tang, Y., Tu, J., Hirota bilinear equations with linear subspaces of solutions, Applied Mathematics and Computation, 218, 2012, 7174-7183.
[24] Wazwaz, A.-M., Structures of multiple soliton solutions of the generalized, asymmetric and modified Nizhnik–Novikov–Veselov equations, Applied Mathematics and Computation, 218, 2012, 11344-11349.
[25] Wazwaz, A.-M., A study on two extensions of the Bogoyavlenskii–Schieff equation, Communications in Nonlinear Science and Numerical Simulation, 17, 2012, 1500-1505.
[26] Gepreel, K.A., Shehata, A.R., Rational jacobi elliptic solutions for nonlinear differential–difference lattice equations, Applied Mathematics Letters, 25, 2012, 1173-1178.
[27] Bhrawy, A.H., Abdelkawy, M.A., Biswas, A., Cnoidal and snoidal wave solutions to coupled nonlinear wave equations by the extended Jacobi’s elliptic function method, Communications in Nonlinear Science and Numerical Simulation, 18, 2013, 915-925.
[28] Zayed, E.M., Abdelaziz, M.A., Exact solutions for the nonlinear Schrödinger equation with variable coefficients using the generalized extended tanh-function, the sine–cosine and the exp-function methods, Applied Mathematics and Computation, 218, 2011, 2259-2268.
[29] Betchewe, G., Thomas, B.B., Victor, K.K., Crepin, K.T., Explicit series solutions to nonlinear evolution equations: The sine–cosine method, Applied Mathematics and Computation, 215, 2010, 4239-4247.
[30] Yaghobi Moghaddam, M., Asgari, A., Yazdani, H., Exact travelling wave solutions for the generalized nonlinear schrödinger (gnls) equation with a source by extended tanh–coth, sine–cosine and exp-function methods, Applied Mathematics and Computation, 210, 2009, 422-435.
[31] Taşcan, F., Bekir, A., Analytic solutions of the (2+1)-dimensional nonlinear evolution equations using the sine–cosine method, Applied Mathematics and Computation, 215, 2009, 3134-3139.
[32] Wazwaz, A.-M., The tanh–coth and the sine–cosine methods for kinks, solitons, and periodic solutions for the Pochhammer–Chree equations, Applied Mathematics and Computation, 195, 2008, 24-33.
[33] Wazwaz, A.-M., The sine-cosine and the tanh methods: Reliable tools for analytic treatment of nonlinear dispersive equations, Applied Mathematics and Computation, 173, 2006, 150-164.
[34] Wazwaz, A.-M., The tanh and the sine–cosine methods for a reliable treatment of the modified equal width equation and its variants, Communications in Nonlinear Science and Numerical Simulation, 11, 2006, 148-160.
[35] Wazwaz, A.-M., The tanh and the sine–cosine methods for compact and noncompact solutions of the nonlinear Klein–Gordon equation, Applied Mathematics and Computation, 167, 2005, 1179-1195.
[36] Wazwaz, A.-M., The tanh and the sine-cosine methods for the complex modified kdv and the generalized kdv equations, Computers & Mathematics with Applications, 49, 2005, 1101-1112.
[37] Wazwaz, A.-M., A sine-cosine method for handling nonlinear wave equations, Mathematical and Computer Modelling, 40, 2004, 499-508.
[38] Mohamed, N.A., Rashed, A.S., Melaibari, A., Sedighi, H.M., Eltaher, M.A., Effective numerical technique applied for Burgers' equation of (1+1)-, (2+1)-dimensional, and coupled forms, Mathematical Methods in the Applied Sciences, 44, 2021, 10135-10153.
[39] Sweilam, N.H., Al-Mekhlafi, S.M., Baleanu, D., Nonstandard finite difference method for solving complex-order fractional Burgers’ equations, Journal of Advanced Research, 25, 2020, 19-29.
[40] Osman, M.S., Baleanu, D., Adem, A.R., Hosseini, K., Mirzazadeh, M., Eslami, M., Double-wave solutions and lie symmetry analysis to the (2 + 1)-dimensional coupled Burgers equations, Chinese Journal of Physics, 63, 2020, 122-129.
[41] Majeed, A., Kamran, M., Iqbal, M.K., Baleanu, D., Solving time fractional Burgers’ and Fisher’s equations using cubic B-spline approximation method, Advances in Difference Equations, 2020, 2020, 175.
[42] Abdel-Gawad, H.I., Tantawy, M., Baleanu, D., Fractional kdv and Boussenisq-Burger's equations, reduction to pde and stability approaches, Mathematical Methods in the Applied Sciences, 43, 2020, 4125-4135.
[43] Seadawy, A.R., Bilal, M., Younis, M., Rizvi, S.T.R., Althobaiti, S., Makhlouf, M.M., Analytical mathematical approaches for the double-chain model of DNA by a novel computational technique, Chaos, Solitons & Fractals, 144, 2021, 110669.
[44] Alka, W., Goyal, A., Nagaraja Kumar, C., Nonlinear dynamics of DNA – Riccati generalized solitary wave solutions, Physics Letters A, 375, 2011, 480-483.
[45] Xian-Min, Q., Lou, S., Exact solutions of nonlinear dynamics equation in a new double-chain model of DNA*, Communications in Theoretical Physics, 39, 2003, 501.
[46] De-Xing, K., Sen-Yue, L.O.U., Jin, Z., Nonlinear dynamics in a new double chain-model of DNA, Communications in Theoretical Physics, 36, 2001, 737-742.
[47] Mabrouk, S.M., Explicit solutions of double-chain DNA dynamical system in (2+ 1)-dimensions, International Journal of Current Engineering and Technology, 9, 2019, 655-660.
[48] Ouyang, Z.-Y., Zheng, S., Travelling wave solutions of nonlinear dynamical equations in a double-chain model of DNA, Abstract and Applied Analysis, 2014, 2014, 317543.
[49] Khalil, R., Al Horani, M., Yousef, A., Sababheh, M., A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264, 2014, 65-70.
[50] Yépez-Martínez, H., Gómez-Aguilar, J.F., Fractional sub-equation method for Hirota–Satsuma-coupled KdV equation and coupled mkdv equation using the Atangana’s conformable derivative, Waves in Random and Complex Media, 29, 2019, 678-693.
[51] Anderson, D.R., Ulness, D.J., Newly defined conformable derivatives, Advances in Dynamical Systems and Applications, 10, 2015, 109-137.
[52] Zhang, S., A generalized exp-function method for fractional Riccati differential equations, Communications in Theoretical Physics, 39, 2010, 405.
[53] Topsakal, M., Guner, O., Bekir, A., Unsal, O., Exact solutions of some fractional differential equations by various expansion methods, Journal of Physics: Conference Series, 766, 2016, 012035.