ANALYSIS OF FRACTIONAL ORDER SCHR$\ddot{O}$DINGER EQUATION WITH SINGULAR AND NON-SINGULAR KERNEL DERIVATIVES VIA NOVEL HYBRID SCHEME

Shelly Arora; S. S. Dhaliwal; Wen Xiu Ma; Atul Pasrija

doi:10.11948/20240246

2025 Volume 15 Issue 2

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Shelly Arora, S. S. Dhaliwal, Wen Xiu Ma, Atul Pasrija. ANALYSIS OF FRACTIONAL ORDER SCHR$\ddot{O}$DINGER EQUATION WITH SINGULAR AND NON-SINGULAR KERNEL DERIVATIVES VIA NOVEL HYBRID SCHEME[J]. Journal of Applied Analysis & Computation, 2025, 15(2): 1039-1067. doi: 10.11948/20240246

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Shelly Arora, S. S. Dhaliwal, Wen Xiu Ma, Atul Pasrija. ANALYSIS OF FRACTIONAL ORDER SCHR$\ddot{O}$DINGER EQUATION WITH SINGULAR AND NON-SINGULAR KERNEL DERIVATIVES VIA NOVEL HYBRID SCHEME[J]. Journal of Applied Analysis & Computation, 2025, 15(2): 1039-1067. doi: 10.11948/20240246

ANALYSIS OF FRACTIONAL ORDER SCHR$\ddot{O}$DINGER EQUATION WITH SINGULAR AND NON-SINGULAR KERNEL DERIVATIVES VIA NOVEL HYBRID SCHEME

1.
Department of Mathematics, Punjabi University, Patiala 147002, India
2.
Department of Mathematics, SLIET, Longowal, Sangrur 148106, India
3.
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, China
4.
Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia
5.
Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, USA
6.
Material Science Innovation and Modelling, Department of Mathematical Sciences, Northwest University, Mafikeng Campus, Mmabatho 2735, South Africa
7.
Department of Mathematics, Punjabi University, Patiala 147002, India

Author Bio: Email: aroshelly@pbi.ac.in(S. Arora); Email: sukhjit_d@sliet.ac.in(S. S. Dhaliwal)
Corresponding authors: Email: mawx@cas.usf.edu(W. X. Ma); Email: atulpasrija_rs@pbi.ac.in(A. Pasrija)

Abstract

Abstract

In the present study, a novel semi-analytic scheme is proposed to obtain exact and approximate series solutions for the time fractional linear and non-linear Schr$ \ddot{o} $dinger equation. This hybrid scheme employs the general bivariate transform followed by the homotopy perturbation method to formulate the recurrence relation. The recurrence relation leads to a system of linear differential equations that associates with the desired components of the series solution. To characterize the considered model with memory effects, the fractional temporal order is considered in the Caputo, Caputo-Fabrizio, and Atangana-Baleanu in Caputo senses. The adapted scheme appears efficient and competent in identifying a diverse collection of trigonometric, wave, and soliton solutions with the availability of initial data. Configurational variations in the governing phenomena with alterations in the fractional order are addressed through graphical illustrations. The potential of the developed regime is affirmed through the uniqueness and convergence analysis of the acquired results. Numerical results are found to be in accordance with existing results in terms of absolute error norms. The main highlight of the proposed scheme is its efficacy and simplicity in constructing a series solution that rapidly converges to the exact solution.
- Fractional derivative /
- partial differential equations /
- integral transform /
- homotopy perturbation method
MSC: 26A33, 35A22, 35C05, 35C20

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References

[1]	S. Abbasbandy, Iterated He's homotopy perturbation method for quadratic Riccati differential equation, Applied Mathematics and Computation, 2006, 175(1), 581–589. doi: 10.1016/j.amc.2005.07.035 CrossRef Google Scholar
[2]	R. K. Adivi Sri Venkata, A. Kirubanandam and R. Kondooru, Numerical solutions of time fractional Sawada Kotera Ito equation via natural transform decomposition method with singular and non-singular kernel derivatives, Mathematical Methods in the Applied Sciences, 2021, 44(18), 14025–14040. doi: 10.1002/mma.7672 CrossRef Google Scholar
[3]	L. Akinyemi, K. S. Nisar, C. A. Saleel, H. Rezazadeh, P. Veeresha, M. M. Khater and M. Inc, Novel approach to the analysis of fifth-order weakly non-local fractional Schrödinger equation with Caputo derivative, Results in Physics, 2021, 31, 104958. doi: 10.1016/j.rinp.2021.104958 CrossRef Google Scholar
[4]	G. Akram, M. Sadaf and I. Zainab, Effect of a new local derivative on space-time fractional non-linear Schrödinger equation and its stability analysis, Optical and Quantum Electronics, 2023, 55(9), 834. doi: 10.1007/s11082-023-05009-y CrossRef Google Scholar
[5]	A. Ali, J. Ahmad and S. Javed, Exploring the dynamic nature of soliton solutions to the fractional coupled non-linear Schrödinger model with their sensitivity analysis, Optical and Quantum Electronics, 2023, 55(9), 810. doi: 10.1007/s11082-023-05033-y CrossRef Google Scholar
[6]	A. Ali, A. R. Seadawy and D. Lu, Soliton solutions of the non-linear Schrödinger equation with the dual power law non-linearity and resonant non-linear Schrödinger equation and their modulation instability analysis, Optik, 2017, 145, 79–88. doi: 10.1016/j.ijleo.2017.07.016 CrossRef Google Scholar
[7]	K. K. Ali and M. Maneea, Optical solitons using optimal homotopy analysis method for time-fractional (1+1)-dimensional coupled non-linear Schrödinger equations, Optik, 2023, 283, 170907. doi: 10.1016/j.ijleo.2023.170907 CrossRef Google Scholar
[8]	K. K. Ali, M. Maneea and M. S. Mohamed, Solving nonlinear fractional models in superconductivity using the q-homotopy analysis transform method, Journal of Mathematics, 2023, 2023, 6647375. Google Scholar
[9]	S. A. Alsallami, M. Maneea, E. M. Khalil, S. Abdel-Khalek and K. K. Ali, Insights into time fractional dynamics in the Belousov-Zhabotinsky system through singular and non-singular kernels, Scientific Reports, 2023, 13(1), 22347. doi: 10.1038/s41598-023-49577-1 CrossRef Google Scholar
[10]	I. G. Ameen, R. O. A. Taie and H. M. Ali, Two effective methods for solving non-linear coupled time-fractional Schrödinger equations, Alexandria Engineering Journal, 2023, 70, 331–347. doi: 10.1016/j.aej.2023.02.046 CrossRef Google Scholar
[11]	V. D. Ao, D. V. Tran, K. T. Pham, D. M. Nguyen, H. D. Tran, T. K. Do, V. H. Do and T. V. Phan, A Schrödinger equation for evolutionary dynamics, Quantum Reports, 2023, 5(4), 659–682. doi: 10.3390/quantum5040042 CrossRef Google Scholar
[12]	V. E. Arkhincheev, Anomalous diffusion in inhomogeneous media: Some exact results, Modelling Measurement and Control a General Physics Electronics and Electrical Engineering, 1993, 49, 11. Google Scholar
[13]	S. Arora and A. Pasrija, A novel integral transform operator and its applications, Iranian Journal of Numerical Analysis and Optimization, 2023, 13(3), 553–575. Google Scholar
[14]	S. Arshad, I. Saleem, A. Akgül, J. Huang, Y. Tang and S. M. Eldin, A novel numerical method for solving the Caputo-Fabrizio fractional differential equation, AIMS Math., 2023, 8(4), 9535–9556. doi: 10.3934/math.2023481 CrossRef Google Scholar
[15]	A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel, Thermal Science, 2016, 20(2), 763–769. doi: 10.2298/TSCI160111018A CrossRef Google Scholar
[16]	R. P. Bell, The application of quantum mechanics to chemical kinetics, Proceedings of the Royal Society of London, 1933, 139(838), 466–474. Google Scholar
[17]	M. Bilal, J. Ren, M. Inc and R. K. Alhefthi, Optical soliton and other solutions to the non-linear dynamical system via two efficient analytical mathematical schemes, Optical and Quantum Electronics, 2023, 55(11), 938. doi: 10.1007/s11082-023-05103-1 CrossRef Google Scholar
[18]	M. Born, Quantenmechanik der stoßvorgänge, Zeitschrift für Physik, 1926, 38(11), 803–827. Google Scholar
[19]	M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation & Applications, 2015, 1(2), 73–85. Google Scholar
[20]	A. Chabchoub and R. H. Grimshaw, The hydrodynamic non-linear Schrödinger equation: Space and time, Fluids, 2016, 1(3), 23. doi: 10.3390/fluids1030023 CrossRef Google Scholar
[21]	K. S. Cole, Electric Conductance of Biological Systems, Cold Spring Harbor Laboratory Press, NY, USA, 1933. Google Scholar
[22]	V. D. Djordjevic and T. M. Atanackovic, Similarity solutions to non-linear heat conduction and Burgers/Korteweg–deVries fractional equations, Journal of Computational and Applied Mathematics, 2008, 222(2), 701–714. doi: 10.1016/j.cam.2007.12.013 CrossRef Google Scholar
[23]	S. O. Edeki, G. O. Akinlabi and S. A. Adeosun, Analytic and numerical solutions of time-fractional linear Schrödinger equation, Communications in Mathematics and Applications, 2016, 7(1), 1–10. Google Scholar
[24]	Z. Y. Fan, K. K. Ali, M. Maneea, M. Inc and S. W. Yao, Solution of time fractional Fitzhugh–Nagumo equation using semi analytical techniques, Results in Physics, 2023, 51, 106679. doi: 10.1016/j.rinp.2023.106679 CrossRef Google Scholar
[25]	A. Ghorbani, Beyond Adomian polynomials: He polynomials, Chaos, Solitons & Fractals, 2009, 39(3), 1486–1492. Google Scholar
[26]	W. G. Glöckle and T. F. Nonnenmacher, A fractional calculus approach to self-similar protein dynamics, Biophysical Journal, 1995, 68(1), 46–53. doi: 10.1016/S0006-3495(95)80157-8 CrossRef Google Scholar
[27]	S. H. Hamed, E. A. Yousif and A. I. Arbab, Analytic and approximate solutions of the space-time fractional Schrödinger equations by homotopy perturbation Sumudu transform method, in Abstract and Applied Analysis, 2014, 2014, 863015. Google Scholar
[28]	J. H. He, Homotopy perturbation method: A new non-linear analytical technique, Applied Mathematics and Computation, 2003, 135(1), 73–79. doi: 10.1016/S0096-3003(01)00312-5 CrossRef Google Scholar
[29]	B. Hong, J. Wang and C. Li, Analytical solutions to a class of fractional coupled non-linear Schrödinger equations via Laplace-HPM technique, AIMS Math., 2023, 8(7), 15670–15688. doi: 10.3934/math.2023800 CrossRef Google Scholar
[30]	Z. Z. Kang, T. C. Xia and W. X. Ma, Riemann–Hilbert approach and N-soliton solution for an eighth-order non-linear Schrödinger equation in an optical fiber, Advances in Difference Equations, 2019, 2019, 188. doi: 10.1186/s13662-019-2121-5 CrossRef Google Scholar
[31]	M. Kapoor, Analytical approach for solution of linear and non-linear time-fractional Schrödinger equations by employing Sumudu transform iterative method, International Journal of Applied and Computational Mathematics, 2023, 9(3), 38. doi: 10.1007/s40819-023-01508-4 CrossRef Google Scholar
[32]	A. Khan, A. Ali, S. Ahmad, S. Saifullah, K. Nonlaopon and A. Akgül, Non-linear Schrödinger equation under non-singular fractional operators: A computational study, Results in Physics, 2022, 43, 106062. doi: 10.1016/j.rinp.2022.106062 CrossRef Google Scholar
[33]	A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, the Netherlands, 2006. Google Scholar
[34]	Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, Academic Press, USA, 2003. Google Scholar
[35]	N. M. Kocherginsky, Interpretation of Schrödinger equation based on classical mechanics and spin, Quantum Studies: Mathematics and Foundations, 2021, 8(2), 217–227. doi: 10.1007/s40509-020-00240-8 CrossRef Google Scholar
[36]	B. T. Krishna, Studies on fractional order differentiators and integrators: A survey, Signal Processing, 2011, 91(3), 386–426. doi: 10.1016/j.sigpro.2010.06.022 CrossRef Google Scholar
[37]	N. Laskin, Fractional market dynamics, Physica A: Statistical Mechanics and its Applications, 2000, 287(3–4), 482–492. Google Scholar
[38]	N. Laskin, Fractional quantum mechanics and Lévy path integrals, Physics Letters A, 2000, 268(4–6), 298–305. Google Scholar
[39]	N. Laskin, Fractional quantum mechanics, Physical Review E, 2000, 62(3), 3135. doi: 10.1103/PhysRevE.62.3135 CrossRef Google Scholar
[40]	N. Laskin, Fractional Schrödinger equation, Physical Review E, 2002, 66(5), 056108. doi: 10.1103/PhysRevE.66.056108 CrossRef Google Scholar
[41]	M. Lechelon, Y. Meriguet, M. Gori, S. Ruffenach, I. Nardecchia, E. Floriani, D. Coquillat, S. Teppe, S. Mailfert, D. Marguet and P. Ferrier, Experimental evidence for long-distance electrodynamic intermolecular forces, Science Advances, 2022, 8(7), eabl5855. doi: 10.1126/sciadv.abl5855 CrossRef Google Scholar
[42]	E. K. Lenzi, E. C. Gabrick, E. Sayari, A. S. de Castro, J. Trobia and A. M. Batista, Anomalous relaxation and three-level system: A fractional Schrödinger equation approach, Quantum Reports, 2023, 5(2), 442–458. doi: 10.3390/quantum5020029 CrossRef Google Scholar
[43]	W. X. Ma and M. Chen, Direct search for exact solutions to the non-linear Schrödinger equation, Applied Mathematics and Computation, 2009, 215(8), 2835–2842. doi: 10.1016/j.amc.2009.09.024 CrossRef Google Scholar
[44]	K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, 1993. Google Scholar
[45]	I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications, Academic Press, USA, 1998. Google Scholar
[46]	A. Qazza, A. Burqan and R. Saadeh, A new attractive method in solving families of fractional differential equations by a new transform, Mathematics, 2021, 9(23), 3039. doi: 10.3390/math9233039 CrossRef Google Scholar
[47]	S. Z. Rida, H. M. El-Sherbiny and A. A. M. Arafa, On the solution of the fractional non-linear Schrödinger equation, Physics Letters A, 2008, 372(5), 553–558. doi: 10.1016/j.physleta.2007.06.071 CrossRef Google Scholar
[48]	R. Saadeh, A. Qazza and A. Burqan, A new integral transform: ARA transform and its properties and applications, Symmetry, 2020, 12(6), 925. doi: 10.3390/sym12060925 CrossRef Google Scholar
[49]	R. Z. Saadeh and B. F. A. Ghazal, A new approach on transforms: Formable integral transform and its applications, Axioms, 2021, 10(4), 332. doi: 10.3390/axioms10040332 CrossRef Google Scholar
[50]	L. J. Shen, Fractional derivative models for viscoelastic materials at finite deformations, International Journal of Solids and Structures, 2020, 190, 226–237. Google Scholar
[51]	H. Tariq, M. Sadaf, G. Akram, H. Rezazadeh, J. Baili, Y. P. Lv and H. Ahmad, Computational study for the conformable non-linear Schrödinger equation with cubic–quintic–septic non-linearities, Results in Physics, 2021, 30, 104839. Google Scholar
[52]	A. M. Wazwaz and G. Q. Xu, Bright, dark and Gaussons optical solutions for fourth-order Schrödinger equations with cubic–quintic and logarithmic non-linearities, Optik, 2020, 202, 163564. Google Scholar
[53]	Y. Xie, Z. Yang and L. Li, New exact solutions to the high dispersive cubic–quintic non-linear Schrödinger equation, Physics Letters A, 2018, 382(36), 2506–2514. Google Scholar
[54]	T. Xu, B. Tian, L. L. Li, X. Lü and C. Zhang, Dynamics of Alfvén solitons in inhomogeneous plasmas, Physics of Plasmas, 2008, 15(10), 102307. Google Scholar
[55]	N. Zettili, Quantum Mechanics: Concepts and Applications, 2nd Ed., John Wiley & Sons, Chichester, UK, 2009. Google Scholar
[56]	M. X. Zhou, A. R. Kanth, K. Aruna, K. Raghavendar, H. Rezazadeh, M. Inc and A. A. Aly, Numerical solutions of time fractional Zakharov-Kuznetsov equation via natural transform decomposition method with non-singular kernel derivatives, Journal of Function Spaces, 2021, 2021, 9884027. Google Scholar