A Hybrid Approach to Approximate and Exact Solutions for Linear and Nonlinear Fractional-Order Schrödinger Equations with Conformable Fractional Derivatives
Authors
-
Muhammad Imran Liaqat
Abdus Salam School of Mathematical Sciences, Government College University, Lahore 54600, Pakistan; National College of Business Administration & Economics, 54000 Lahore, Pakistan
https://orcid.org/0000-0002-5732-9689
https://doi.org/10.61383/ejam.20242371
Keywords:
approximate solutions; Elzaki transform; exact solutions; Adomian decomposition method; Schr\Abstract
Fractional-order Schrödinger differential equations extend the classical Schrödinger equation by incorporating fractional calculus to describe more complex physical phenomena. The Schrödinger equations are solved using fractional derivatives expressed through the Caputo derivative. However, there is limited research on exact and approximate solutions involving conformable fractional derivatives. This study aims to address this gap by employing a hybrid approach that combines the Elzaki transform with the decomposition technique to solve the Schrödinger equation with conformable fractional derivatives, considering both zero and nonzero trapping potentials. The efficiency of this approach is evaluated through the analysis of relative and absolute errors, confirming its accuracy. Our method serves as a viable alternative to Caputo-based approaches for solving time-fractional Schrödinger equations. Moreover, we conclude that the conformable derivative is a suitable alternative to the Caputo derivative in modeling such systems.
References
Chen, W., Sun, H., & Li, X. (2022). Fractional derivative modeling in mechanics and engineering. Springer Nature.
Dumitru, B., & Agarwal, R. P. (2021). Fractional calculus in the sky. Adv. differ. equ., 2021(1).
Giusti, A., Colombaro, I., Garra, R., Garrappa, R., Polito, F., Popolizio, M., & Mainardi, F. (2020). A practical guide to Prabhakar fractional calculus. Fract. Calc. Appl. Anal., 23(1), 9-54.
Valentim, C. A., Rabi, J. A., & David, S. A. (2021). Fractional mathematical oncology: On the potential of non-integer order calculus applied to interdisciplinary models. Biosystems, 204, 104377.
Liaqat, M. I., Khan, A., Alqudah, M. A., & Abdeljawad, T. (2023). Adapted homotopy perturbation method with Shehu transform for solving conformable fractional nonlinear partial differential equations. Fractals, 31(02), 2340027.
MohammedDjaouti, A., Khan, Z. A., Imran Liaqat, M., & Al-Quran, A. (2024). A novel technique for solving the nonlinear fractional-order smoking model. Fractal Fract., 8(5), 286.
Huang, G., Qin, H. Y., Chen, Q., Shi, Z., Jiang, S., & Huang, C. (2024). Research on Application of Fractional Calculus Operator in Image Underlying Processing. Fractal Fract., 8(1), 37.
Arora, S., Mathur, T., Agarwal, S., Tiwari, K., & Gupta, P. (2022). Applications of fractional calculus in computer vision: a survey. Neurocomputing, 489, 407-428.
Alinei-Poiana, T., Dulf, E. H., & Kovacs, L. (2023). Fractional calculus in mathematical oncology. Sci. Rep., 13(1), 10083.
Zhang, T., Qu, H., & Zhou, J. (2023). Asymptotically almost periodic synchronization in fuzzy competitive neural networks with Caputo-Fabrizio operator. Fuzzy Sets Syst., 471, 108676.
Luo, H., & Zhang, T. (2022). Equilibrium point, exponential stability and synchronization of numerical fractional-order shunting inhibitory cellular neural networks with piecewise feature. Proceedings of the Institution of Mechanical Engineers, Part I: Syst. Sci. Control Eng., 236(10), 1908-1921.
Liaqat, M. I., Akgül, A., De la Sen, M., & Bayram, M. (2023). Approximate and exact solutions in the sense of conformable derivatives of quantum mechanics models using a novel algorithm. Symmetry, 15(3), 744.
Zhang,T., &Li, Y.(2022). S-asymptotically periodic fractional functional differential equations with off-diagonal matrix Mittag-Leffler function kernels. Math. Comput. Simul., 193, 331-347.
Abdelouahab, M. S., & Hamri, N. E. (2016). The Grünwald-Letnikov fractional-order derivative with fixed memory length. Mediterr. J. Math., 13(2), 557-572.
Ledesma, C. E. T., & Bonilla, M. C. M. (2021). Fractional Sobolev space with Riemann-Liouville fractional derivative and application to a fractional concave-convex problem. Adv. Oper. Theory, 6(4), 65.
Zhang, T., & Li, Y. (2022). Global exponential stability of discrete-time almost automorphic Caputo-Fabrizio BAMfuzzyneural networks via exponential Euler technique. Knowl. Based Syst., 246, 108675.
Zhang, T., & Li, Y. (2022). Exponential Euler scheme of multi-delay Caputo-Fabrizio fractional-order differential equations. Appl. Math. Lett., 124, 107709.
Liaqat, M. I., & Akgül, A. (2022). A novel approach for solving linear and nonlinear time-fractional Schrödinger equations. Chaos Solit. Fract., 162, 112487.
Zhang, T., Li, Y., & Zhou, J. (2023). Almost automorphic strong oscillation in time-fractional parabolic equations. Fractal Fract., 7(1), 88.
Zhang, T., & Xiong, L. (2020). Periodic motion for impulsive fractional functional differential equations with piecewise Caputo derivative. Appl. Math. Lett., 101, 106072.
Syam,M.I., &Al-Refai, M.(2019).Fractional differential equations with Atangana-Baleanu fractional derivative: analysis and applications. Chaos Solit. Fract.: X, 2, 100013.
Yao, Z., Yang, Z., & Gao, J. (2023). Unconditional stability analysis of Grünwald Letnikov method for fractional order delay differential equations. Chaos Solit. Fract., 177, 114193.
Sevinik Adigüzel, R., Aksoy, Ü., Karapinar, E., & Erhan, ˙ I. M. (2024). On the solution of a boundary value problem associated with a fractional differential equation. Math. Method Appl. Sci., 47(13), 10928-10939.
Dzherbashian, M. M., & Nersesian, A. B. (2020). Fractional derivatives and Cauchy problem for differential equations of fractional order. Fract. Calc. Appl. Anal., 23(6), 1810-1836.
Anakira, N., Chebana, Z., Oussaeif, T. E., Batiha, I. M., & Ouannas, A. (2022). A study of a weak solution of a diffusion problem for a temporal fractional differential equation. Nonlinear Funct. Anal. Appl., 679-689.
Nane, E., Nwaeze, E. R., & Omaba, M. E. (2020). Asymptotic behaviour of solution and non-existence of global solution to a class of conformable time-fractional stochastic equation. Stat. Probab. Lett., 163, 108792.
Khan, A., Liaqat, M. I., Alqudah, M. A., & Abdeljawad, T. (2023). Analysis of the conformable temporal fractional swift-hohenberg equation using a novel computational technique. Fractals, 31(04), 2340050.
Wu, W. Z., Zeng, L., Liu, C., Xie, W., & Goh, M. (2022). A time power-based grey model with conformable fractional derivative and its applications. Chaos Solit. Fract., 155, 111657.
Yavari, M., & Nazemi, A. (2020). On fractional infinite-horizon optimal control problems with a combination of conformable and Caputo-Fabrizio fractional derivatives. ISA Trans., 101, 78-90.
Korpinar, Z., Tchier, F., Inc, M., Bousbahi, F., Tawfiq, F. M., & Akinlar, M. A. (2020). Applicability of time conformable derivative to Wick-fractional-stochastic PDEs. Alex. Eng. J., 59(3), 1485-1493.
Albosaily, S., Elsayed, E. M., Albalwi, M. D., Alesemi, M., & Mohammed, W.W.(2023).TheAnalyticalStochastic Solutions for the Stochastic Potential Yu-Toda-Sasa-Fukuyama Equation with Conformable Derivative Using Different Methods. Fractal Fract., 7(11), 787.
Ashyralyev, A., & Hicdurmaz, B. (2011). A note on the fractional Schrödinger differential equations. Kybernetes, 40(5/6), 736-750.
Djaouti, A. M., Khan, Z. A., Liaqat, M. I., & Al-Quran, A. (2024). Existence uniqueness and averaging principle of fractional neutral stochastic differential equations in the Lp Space with the framework of the Ψ-Caputo derivative. Mathematics, 12(7), 1-21.
Mohammed Djaouti, A., & Imran Liaqat, M. (2024). Qualitative Analysis for the Solutions of Fractional Stochastic Differential Equations. Axioms, 13(7), 438.
Shiri, B., & Baleanu, D. (2019). System of fractional differential algebraic equations with applications. Chaos Solit. Fract., 120, 203-212.
Wang, J. (2021). Symplectic-preserving Fourier spectral scheme for space fractional KleinGordon-Schrödinger equations. Numer. Methods Partial Differ. Equ. , 37(2), 1030-1056.
Zhang, Y., Kumar, A., Kumar, S., Baleanu, D., & Yang, X. J. (2016). Residual power series method for time fractional Schrödinger equations. J. Nonlinear Sci. Appl., 9(11), 5821-5829.
Hussin, C. H. C., Kilicman, A., & Azmi, A. (2018). Analytical solutions of nonlinear Schrödinger equations using multistep modified reduced differential transform method. Compusoft, 7(11), 2939-2944.
Yildirim, A. H. M. E. T. (2009). An algorithm for solving the fractional nonlinear Schrödinger equation by means of the homotopy perturbation method. Int. J. Nonlinear Sci. Numer. Simul., 10(4), 445-450.
Aruna, K., & Ravi Kanth, A. S. V. (2013). Approximate solutions of non-linear fractional Schrödinger equation via differential transform method and modified differential transform method. Natl. Acad. Sci. Lett., 36(2),
-213.
Khan, N. A., Jamil, M., & Ara, A. (2012). Approximate solutions to time-fractional Schrödinger equation via homotopy analysis method. Int. sch. Res. notices, 2012(1), 197068.
Sadighi, A., & Ganji, D. D. (2008). Analytic treatment of linear and nonlinear Schrödinger equations: a study with homotopy-perturbation and Adomian decomposition methods. Phys. Lett., A, 372(4), 465-469.
Ali, L., Zou, G., Li, N., Mehmood, K., Fang, P., & Khan, A. (2024). Analytical treatments of time-fractional seventh-order nonlinear equations via Elzaki transform. J. Eng. Math., 145(1), 1.
Ahmed, S. A. (2024). An Efficient New Technique for Solving Nonlinear Problems Involving the Conformable Fractional Derivatives. J. Appl. Math., 2024(1), 5958560.
Nadeem, M., & Iambor, L. F. (2024). Prospective Analysis of Time-Fractional Emden-Fowler Model Using Elzaki Transform Homotopy Perturbation Method. Fractal Fract., 8(6), 363.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2024 Muhammad Imran Liaqat
This work is licensed under a Creative Commons Attribution 4.0 International License.
Accepted 2024 Aug 25
Published 2024 Sep 03
