New solutions of time-space fractional coupled Schrödinger systems

Mubashir Qayyum; Efaza Ahmad; Hijaz Ahmad; Bandar Almohsen; Mubashir Qayyum; Efaza Ahmad; Hijaz Ahmad; Bandar Almohsen

doi:10.3934/math.20231383

AIMS Mathematics

2023, Volume 8, Issue 11: 27033-27051. doi: 10.3934/math.20231383

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New solutions of time-space fractional coupled Schrödinger systems

1.
Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Lahore, Pakistan
2.
Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Roma, Italy
3.
Near East University, Operational Research Center in Healthcare, Nicosia, PC: 99138, TRNC Mersin 10, Turkey
4.
Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon
5.
Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi Arabia

Received: 25 July 2023 Revised: 26 August 2023 Accepted: 03 September 2023 Published: 22 September 2023
MSC : 35G50, 35C10

The current manuscript focuses on the solution and analysis of space and time fractional coupled Schrödinger system that belongs to a class of evolution equations. These systems encounter in different fields related to plasma waves, optics, and quantum physics. The fractional He-Laplace approach is proposed for the series form solutions of fractional systems. This approach contains hybrid of Laplace transform and homotopy perturbation along with Caputo fractional derivative. The current study provide new results on time and space fractional coupled Schrödinger systems which are not captured in existing literature. Reliability of proposed algorithm in both time and space fractional scenarios is observed through residual error concept throughout fractional domains. The effect of fractional parameters on wave profiles are analyzed numerically and graphically as 2D and 3D illustrations. Analysis reveals that proposed algorithm is suitable for non-linear time-space fractional systems encountering in different fields of sciences.
- time-space fractional,
- Schrödinger system,
- Laplace transform,
- homotopy perturbation
Citation: Mubashir Qayyum, Efaza Ahmad, Hijaz Ahmad, Bandar Almohsen. New solutions of time-space fractional coupled Schrödinger systems[J]. AIMS Mathematics, 2023, 8(11): 27033-27051. doi: 10.3934/math.20231383

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Abstract

The current manuscript focuses on the solution and analysis of space and time fractional coupled Schrödinger system that belongs to a class of evolution equations. These systems encounter in different fields related to plasma waves, optics, and quantum physics. The fractional He-Laplace approach is proposed for the series form solutions of fractional systems. This approach contains hybrid of Laplace transform and homotopy perturbation along with Caputo fractional derivative. The current study provide new results on time and space fractional coupled Schrödinger systems which are not captured in existing literature. Reliability of proposed algorithm in both time and space fractional scenarios is observed through residual error concept throughout fractional domains. The effect of fractional parameters on wave profiles are analyzed numerically and graphically as 2D and 3D illustrations. Analysis reveals that proposed algorithm is suitable for non-linear time-space fractional systems encountering in different fields of sciences.

References

[1]	S. Nadeem, W. Fuzhang, F. M. Alharbi, F. Sajid, N. Abbas, A. S. El-Shafay, et al., Numerical computations for Buongiorno nano fluid model on the boundary layer flow of viscoelastic fluid towards a nonlinear stretching sheet, Alex. Eng. J., 61 (2022), 1769–1778. https://doi.org/10.1016/j.aej.2021.11.013 doi: 10.1016/j.aej.2021.11.013
[2]	M. Qayyum, S. Afzal, E. Ahmad, M. B. Riaz, Fractional modeling and analysis of unsteady squeezing flow of Casson nanofluid via extended He-Laplace algorithm in Liouville-Caputo sense, Alex. Eng. J., 73 (2023), 579–591. https://doi.org/10.1016/j.aej.2023.05.010 doi: 10.1016/j.aej.2023.05.010
[3]	A. E. Aboanber, A. A. Nahla, A. M. El-Mhlawy, O. Maher, An efficient exponential representation for solving the two-energy group point telegraph kinetics model, Ann. Nucl. Energy, 166 (2022), 108698. https://doi.org/10.1016/j.anucene.2021.108698 doi: 10.1016/j.anucene.2021.108698
[4]	C. Villa, A. Gerisch, M. A. J. Chaplain, A novel nonlocal partial differential equation model of endothelial progenitor cell cluster formation during the early stages of vasculogenesis, J. Theor. Biol., 534 (2022), 110963. https://doi.org/10.1016/j.jtbi.2021.110963 doi: 10.1016/j.jtbi.2021.110963
[5]	I. Ahmad, H. Ahmad, P. Thounthong, Y. Chu, C. Cesarano, Solution of multi-term time-fractional PDE models arising in mathematical biology and physics by local meshless method, Symmetry, 12 (2020), 1195. https://doi.org/10.1016/j.heliyon.2023.e16522 doi: 10.1016/j.heliyon.2023.e16522
[6]	O. D. Adeyemo, C. M. Khalique, Lie group classification of generalized variable coefficient Korteweg-de Vries equation with dual power-law nonlinearities with linear damping and dispersion in quantum field theory, Symmetry, 14 (2022), 83. https://doi.org/10.3390/sym14010083 doi: 10.3390/sym14010083
[7]	S. Afzal, M. Qayyum, M. B. Riaz, A. Wojciechowski, Modeling and simulation of blood flow under the influence of radioactive materials having slip with MHD and nonlinear mixed convection, Alex. Eng. J., 69 (2023), 9–24. https://doi.org/10.1016/j.aej.2023.01.013 doi: 10.1016/j.aej.2023.01.013
[8]	L. Guo, H. Wu, T. Zhou, Normalizing field flows: Solving forward and inverse stochastic differential equations using physics-informed flow models, J. Comput. Phys., 461 (2022), 111202. https://doi.org/10.1016/j.jcp.2022.111202 doi: 10.1016/j.jcp.2022.111202
[9]	S. P. Joseph, New traveling wave exact solutions to the coupled Klein-Gordon system of equations, PDE Appl. Math., 5 (2022), 100208. https://doi.org/10.1016/j.padiff.2021.100208 doi: 10.1016/j.padiff.2021.100208
[10]	M. Farman, A. Akgül, S. Askar, T. Botmart, A. Ahmad, H. Ahmad, Modeling and analysis of fractional order Zika model, AIMS Math., 7 (2022), 3912–3938. https://doi.org/10.3934/math.2022216 doi: 10.3934/math.2022216
[11]	J. Liouville, Mémoire sur quelques questions de géométrie et de mécanique, et sur un nouveau genre de calcul pour résoudre ces questions, J. Éc. Polytech. Math., 1832.
[12]	K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Hoboken: Wiley, 1993.
[13]	B. Riemann, Versuch einer allgemeinen auffassung der integration und differentiation, Gesammelte Werke, 62 (1876), 1876.
[14]	M. Caputo, Elasticita e Dissipazione, Bologna: Zanichelli, 1969.
[15]	H. Ahmad, N. Alam, M. Omri, New computational results for a prototype of an excitable system, Results Phys., 28 (2021). https://doi.org/10.1016/j.rinp.2021.104666
[16]	M. Qayyum, E. Ahmad, S. Afzal, T. Sajid, W. Jamshed, A. Musa, et al., Fractional analysis of unsteady squeezing flow of casson fluid via homotopy perturbation method, Sci. Rep., 12 (2022), 18406. https://doi.org/10.1038/s41598-022-23239-0 doi: 10.1038/s41598-022-23239-0
[17]	K. S. Nisar, K. Logeswari, V. Vijayaraj, H. M. Baskonus, C. Ravichandran, Fractional order modeling the gemini virus in capsicum annuum with optimal control, Fractal Fract., 6 (2022), 61. https://doi.org/10.3390/fractalfract6020061 doi: 10.3390/fractalfract6020061
[18]	A. Yusuf, S. Qureshi, U. T. Mustapha, S. S. Musa, T. A. Sulaiman, Fractional modeling for improving scholastic performance of students with optimal control, Int. J. Appl. Comput. Math., 8 (2022), 37. https://doi.org/10.1007/s40819-021-01177-1 doi: 10.1007/s40819-021-01177-1
[19]	H. Hassani, J. A. Tenreiro Machado, Z. Avazzadeh, E. Naraghirad, S. Mehrabi, Optimal solution of the fractional-order smoking model and its public health implications, Nonlinear Dynam., 108 (2022), 2815–2831. https://doi.org/10.1007/s11071-022-07343-4 doi: 10.1007/s11071-022-07343-4
[20]	M. Qayyum, E. Ahmad, S. T. Saeed, H. Ahmad, S. Askar, Homotopy perturbation method-based soliton solutions of the time-fractional (2+1)-dimensional wu-zhang system describing long dispersive gravity water waves in the ocean, Front. Phys., 11 (2023), 1178154. https://doi.org/10.3389/fphy.2023.1178154 doi: 10.3389/fphy.2023.1178154
[21]	C. Wang, X. Zhou, X. Shi, Y. Jin, Variable fractional order sliding mode control for seismic vibration suppression of uncertain building structure, J. Vib. Eng. Tech., 10 (2021), 299–312. https://doi.org/10.1007/s42417-021-00377-9 doi: 10.1007/s42417-021-00377-9
[22]	I. M. Batiha, S. A. Njadat, R. M. Batyha, A. Zraiqat, A. Dababneh, S. Momani, Design fractional-order PID controllers for single-joint robot arm model, Int. J. Adv. Soft Comput. Appl., 14 (2022), 97–114. https://doi.org/10.15849/ijasca.220720.07 doi: 10.15849/ijasca.220720.07
[23]	M. H. Derakhshan, Existence, uniqueness, Ulam-Hyers stability and numerical simulation of solutions for variable order fractional differential equations in fluid mechanics, J. Appl. Math. Comput., 68 (2021), 403–429. https://doi.org/10.1007/s12190-021-01537-6 doi: 10.1007/s12190-021-01537-6
[24]	A. Cardone, D. Conte, R. D'Ambrosio, B. Paternoster, Multivalue collocation methods for ordinary and fractional differential equations, Mathematics, 10 (2022), 185. https://doi.org/10.3390/math10020185 doi: 10.3390/math10020185
[25]	N. A. Shah, A. Wakif, E. R. El-Zahar, T. Thumma, S. J. Yook, Heat transfers thermodynamic activity of a second-grade ternary nanofluid flow over a vertical plate with Atangana-Baleanu time-fractional integral, Alex. Eng. J., 61 (2022), 10045–10053. https://doi.org/10.1016/j.aej.2022.03.048 doi: 10.1016/j.aej.2022.03.048
[26]	N. P. Dong, H. V. Long, N. L. Giang, The fuzzy fractional SIQR model of computer virus propagation in wireless sensor network using Caputo Atangana–Baleanu derivatives, Fuzzy Sets Syst., 429 (2022), 28–59. https://doi.org/10.1016/j.fss.2021.04.012 doi: 10.1016/j.fss.2021.04.012
[27]	A. Din, F. M. Khan, Z. U. Khan, A. Yusuf, T. Munir, The mathematical study of climate change model under nonlocal fractional derivative, PDE Appl. Math., 5 (2022), 100204. https://doi.org/10.1016/j.padiff.2021.100204 doi: 10.1016/j.padiff.2021.100204
[28]	Y. Gurefe, Y. Pandir, T. Akturk, Analysis of exact solutions of a mathematical model by new function method, Cumhuriyet Sci. J., 43 (2022), 703–707. https://doi.org/10.17776/csj.1083033 doi: 10.17776/csj.1083033
[29]	M. R. Ahamed Fahim, P. R. Kundu, M. E. Islam, M. A. Akbar, M. S. Osman, Wave profile analysis of a couple of (3+1)-dimensional nonlinear evolution equations by sine-Gordon expansion approach, J. Ocean Eng. Sci., 7 (2022), 272–279. https://doi.org/10.1016/j.joes.2021.08.009 doi: 10.1016/j.joes.2021.08.009
[30]	B. Ghanbari, Employing Hirota's bilinear form to find novel lump waves solutions to an important nonlinear model in fluid mechanics, Results Phys., 29 (2021), 104689. https://doi.org/10.1016/j.rinp.2021.104689 doi: 10.1016/j.rinp.2021.104689
[31]	W. Razzaq, A. Zafar, H. M. Ahmed, W. B. Rabie, Construction solitons for fractional nonlinear Schrodinger equation with $\beta$-time derivative by the new sub-equation method, J. Ocean Eng. Sci., in press, 2022. https://doi.org/10.1016/j.joes.2022.06.013
[32]	G. Akram, M. Sadaf, S. Arshed, F. Sameen, Bright, dark, kink, singular and periodic soliton solutions of Lakshmanan-Porsezian-Daniel model by generalized projective riccati equations method, Optik, 241 (2021), 167051. https://doi.org/10.1016/j.ijleo.2021.167051 doi: 10.1016/j.ijleo.2021.167051
[33]	S. Liao, Homotopy analysis method: A new analytical technique for nonlinear problems, Commun. Nonlinear Sci. Numer. Simul., 2 (1997), 95–100. https://doi.org/10.1016/s1007-5704(97)90047-2 doi: 10.1016/s1007-5704(97)90047-2
[34]	S. Afzal, M. Qayyum, G. Chambashi, Heat and mass transfer with entropy optimization in hybrid nanofluid using heat source and velocity slip: a hamilton–crosser approach, Sci. Rep., 13 (2023), 12392. https://doi.org/10.1038/s41598-023-39176-5 doi: 10.1038/s41598-023-39176-5
[35]	T. Hayat, K. Muhammad, S. Momani, Melting heat and viscous dissipation in flow of hybrid nanomaterial: a numerical study via finite difference method, J. Therm. Anal. Calorime., 147 (2021), 6393–6401. https://doi.org/10.1007/s10973-021-10944-7 doi: 10.1007/s10973-021-10944-7
[36]	H. Ahmad, M. N. Khan, I. Ahmad, M. Omri, M. F. Alotaibi, A meshless method for numerical solutions of linear and nonlinear time-fractional Black-Scholes models, AIMS Math., 8 (2023), 19677–19698. https://doi.org/10.3934/math.20231003 doi: 10.3934/math.20231003
[37]	J. H. He, Homotopy perturbation technique, Comput. Meth. Appl. Mech. Eng., 178 (1999), 257–262. https://doi.org/10.1016/s0045-7825(99)00018-3 doi: 10.1016/s0045-7825(99)00018-3
[38]	R. Amin, K. Shah, H. Ahmad, A. H. Ganie, A. Abdel-Aty, T. Botmart, Haar wavelet method for solution of variable order linear fractional integro-differential equations, AIMS Math., 7 (2022), 5431–5443. https://doi.org/10.3934/math.2022301 doi: 10.3934/math.2022301
[39]	G. Singh, I. Singh, New laplace variational iterative technique to solve twodimensional Schr¨odinger equation, Mater. Today Proc., 62 (2022), 3995–4000. https://doi.org/10.1016/j.matpr.2022.04.585 doi: 10.1016/j.matpr.2022.04.585
[40]	M. Croci, G. R. de Souza, Mixed-precision explicit stabilized Runge-Kutta methods for single- and multi-scale differential equations, J. Comput. Phys., 464 (2022), 111349. https://doi.org/10.1016/j.jcp.2022.111349 doi: 10.1016/j.jcp.2022.111349
[41]	M. Aslam, M. Farman, H. Ahmad, T. N. Gia, A. Ahmad, S. Askar, Fractal fractional derivative on chemistry kinetics hires problem, AIMS Math., 7 (2022), 1155–1184. https://doi.org/10.3934/math.2022068 doi: 10.3934/math.2022068
[42]	J. H. He, M. L. Jiao, K. A. Gepreel, Y. Khan, Homotopy perturbation method for strongly nonlinear oscillators, Math. Comput. Simul., 204 (2023), 243–258. https://doi.org/10.1016/j.matcom.2022.08.005 doi: 10.1016/j.matcom.2022.08.005
[43]	M. Qayyum, E. Ahmad, M. B. Riaz, J. Awrejcewicz, Improved soliton solutions of generalized fifth order time-fractional KdV models: Laplace transform with homotopy perturbation algorithm, Universe, 8 (2022), 563. https://doi.org/10.3390/universe8110563 doi: 10.3390/universe8110563
[44]	Y. Pandir, T. A$\ddot{g}$ir, Genisletilmis deneme denklemi yöntemi ile k$\ddot{u}$bik lineer olmayan Schrödinger denkleminin yeni tam ccöz$\ddot{u}$mleri, Afyon Kocatepe Uni. J. Sci. Eng., 20 (2020), 582–588.
[45]	K. J. Wang, G. D. Wang, Variational theory and new abundant solutions to the (1+2)-dimensional chiral nonlinear Schrödinger equation in optics, Phys. Letters A, 412 (2021), 127588. https://doi.org/10.1016/j.physleta.2021.127588 doi: 10.1016/j.physleta.2021.127588
[46]	M. Al-Smadi, O. A. Arqub, S. Momani, Numerical computations of coupled fractional resonant Schrödinger equations arising in quantum mechanics under conformable fractional derivative sense, Phys. Scripta, 95 (2020), 075218. https://doi.org/10.1088/1402-4896/ab96e0 doi: 10.1088/1402-4896/ab96e0
[47]	S. F. Tian, X. F. Wang, T. T. Zhang, W. H. Qiu, Stability analysis, solitary wave and explicit power series solutions of a (2 + 1)-dimensional nonlinear Schrödinger equation in a multicomponent plasma, Int. J. Numer. Meth. Heat Fluid Flow, 31 (2021), 1732–1748. https://doi.org/10.1108/hff-08-2020-0517 doi: 10.1108/hff-08-2020-0517
[48]	D. F. Li, J. L. Wang, J. W. Zhang, Unconditionally convergent l1-galerkin FEMs for nonlinear time-fractional Schrödinger equations, SIAM J. Sci. Comput., 39 (2017), A3067–A3088. https://doi.org/10.1137/16m1105700 doi: 10.1137/16m1105700
[49]	W. Q. Yuan, C. J. Zhang, D. F. Li, Linearized fast time-stepping schemes for time–space fractional Schrödinger equations, Phys. D Nonlinear Phenomena, 454 (2023), 133865. https://doi.org/10.1016/j.physd.2023.133865 doi: 10.1016/j.physd.2023.133865
[50]	K. Hosseini, E. Hincal, S. Salahshour, M. Mirzazadeh, K. Dehingia, B. J. Nath, On the dynamics of soliton waves in a generalized nonlinear Schrödinger equation, Optik, 272 (2023), 170215. https://doi.org/10.1016/j.ijleo.2022.170215 doi: 10.1016/j.ijleo.2022.170215
[51]	N. A. Kudryashov, Method for finding optical solitons of generalized nonlinear Schrödinger equations, Optik, 261 (2022), 169163. https://doi.org/10.1016/j.ijleo.2022.169163 doi: 10.1016/j.ijleo.2022.169163
[52]	W. Q. Yuan, D. F. Li, C. J. Zhang, Linearized transformed l1 galerkin FEMs with unconditional convergence for nonlinear time fractional Schrödinger equations, Numer. Math. Theory Meth. Appl., 16 (2023), 348–369. https://doi.org/10.4208/nmtma.oa-2022-0087 doi: 10.4208/nmtma.oa-2022-0087
[53]	T. Y. Han, Z. Li, X. Zhang, Bifurcation and new exact traveling wave solutions to time-space coupled fractional nonlinear Schrödinger equation, Phys. Letters A, 395 (2021), 127217. https://doi.org/10.1016/j.physleta.2021.127217 doi: 10.1016/j.physleta.2021.127217
[54]	P. F. Dai, Q. B. Wu, An efficient block Gauss-Seidel iteration method for the space fractional coupled nonlinear Schrödinger equations, Appl. Math. Letters, 117 (2021), 107116. https://doi.org/10.1016/j.aml.2021.107116 doi: 10.1016/j.aml.2021.107116
[55]	K. S. Nisar, S. Ahmad, A. Ullah, K. Shah, H. Alrabaiah, M. Arfan, Mathematical analysis of SIRD model of COVID-19 with Caputo fractional derivative based on real data, Results Phys., 21 (2021), 103772. https://doi.org/10.1016/j.rinp.2020.103772 doi: 10.1016/j.rinp.2020.103772
[56]	T. Bakkyaraj, Lie symmetry analysis of system of nonlinear fractional partial differential equations with Caputo fractional derivative, Eur. Phys. J. Plus, 135 (2020), 126. https://doi.org/10.1140/epjp/s13360-020-00170-9 doi: 10.1140/epjp/s13360-020-00170-9
[57]	N. H. Tuan, H. Mohammadi, S. Rezapour, A mathematical model for COVID-19 transmission by using the Caputo fractional derivative, Chaos Solitons Fract., 140 (2020), 110107. https://doi.org/10.1016/j.chaos.2020.110107 doi: 10.1016/j.chaos.2020.110107

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