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Dergiler / TWMS Journal of Applied and Engineering Mathematics / 2026 / Volume: 16 Issue: 9

A HIGH-ORDER CAPUTO-BASED NUMERICAL SCHEME FOR APPROXIMATING THE ONE-DIMENSIONAL TIME-INDEPENDENT FRACTIONAL SCHRÖDINGER EQUATION

Hojjat Habibzadeh(University of Maragheh)0009-0004-1444-2157
Mohammad Shahriari(University of Maragheh)0000-0002-8982-2451
Ali Shokri(Sahand University of Technology)0000-0003-2699-1490
Mohammad Mehdizadeh Khalsaraeı(University of Maragheh)0000-0002-0130-0449
Sayfa
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Yayın Tarihi
2026-09-07
DOI
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Yapay Zeka Hızlı Bakış & Çıkarımlar

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⚡ TL;DR(1 Cümlede Hızlı Çıkarım)

The findings indicate that the developed high-order Caputo-based scheme is a powerful and reliable tool for solving fractional Schrödinger equations, offering significant improvements in accuracy for a wide range of fractional orders.

📌 3 Maddede Bu Makale

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Amaç & Hipotez:The time-independent fractional Schrödinger equation is pivotal for modeling quantum systems exhibiting nonlocal interactions and anomalous dispersion, for which analytical solutions are often intractable.
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Temel Bulgular & Veri:This paper presents a high-order numerical scheme to approximate the solution of the one-dimensional time-independent Schrödinger equation incorporating Caputo fractional derivatives of order $\alpha$, comprehensively covering the ranges $0 < \alpha \le 1$ and $1 < \alpha \le 2$.
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Sonuç & Katkı:The findings indicate that the developed high-order Caputo-based scheme is a powerful and reliable tool for solving fractional Schrödinger equations, offering significant improvements in accuracy for a wide range of fractional orders.
🔬Tür:Modelleme & Sayısal Simülasyon
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:Fractional Schrödinger equation · Caputo derivative · Finite difference method

Abstract

The time-independent fractional Schrödinger equation is pivotal for modeling quantum systems exhibiting nonlocal interactions and anomalous dispersion, for which analytical solutions are often intractable. This paper presents a high-order numerical scheme to approximate the solution of the one-dimensional time-independent Schrödinger equation incorporating Caputo fractional derivatives of order $\alpha$, comprehensively covering the ranges $0 < \alpha \le 1$ and $1 < \alpha \le 2$. The core of our approach lies in the development of novel finite difference discretizations. For $0 < \alpha \le 1$, we employ a second-order weighted-shifted Grünwald approximation to achieve higher accuracy than standard schemes. For $1 < \alpha \le 2$, a distinct high-order approximation is derived, which is further extended to a fourth-order scheme for enhanced precision. A rigorous theoretical analysis is provided, including a detailed error bound proof that establishes the scheme's accuracy and stability. We also investigate the convergence rate and the impact of numerical errors on long-term simulations, demonstrating the method's robustness. The validity and efficiency of the proposed method are confirmed through several numerical examples. The results show excellent agreement with exact solutions in limiting cases (e.g., $\alpha \to 2$) and confirm that the observed convergence orders align with our theoretical predictions. The findings indicate that the developed high-order Caputo-based scheme is a powerful and reliable tool for solving fractional Schrödinger equations, offering significant improvements in accuracy for a wide range of fractional orders.

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Bu Makaleye Atıf Yap

Habibzadeh, H., Shahriari, M., Shokri, A., Khalsaraeı, M. M. (2026). A HIGH-ORDER CAPUTO-BASED NUMERICAL SCHEME FOR APPROXIMATING THE ONE-DIMENSIONAL TIME-INDEPENDENT FRACTIONAL SCHRÖDINGER EQUATION. TWMS Journal of Applied and Engineering Mathematics, Volume: 16 Issue: 9. https://ajindex.com/articles/1604198

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Kaynakça (48)

  1. 1] Caputo, M., Fabrizio, M., (2015), A new definition of Fractional Derivative without Singular Kernel, Progress in fractional differentiation & applications, 1(2), pp. 73–85.
  2. Atangana, A., Baleanu, D., (2016), New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20(2), pp. 763–769.
  3. Nonnenmacher, T. F., Metzler, R., (1998), Applications of fractional calculus ideas to biology.
  4. Wang, H., (2019), Research on application of fractional calculus in signal real-time analysis and processing in stock financial market, Chaos, Solitons and Fractals, 128, pp. 92–97.
  5. Ibrahim, R. W., Darus, M., (2011), Differential operator generalized by fractional derivatives, Miskolc mathematical notes/Mathematical notes, 12(2), pp. 167–167.
  6. Tarasov, V. E., (2017), Interpretation of Fractional Derivatives as Reconstruction from Sequence of Integer Derivatives, Fundamenta Informaticae, 151(1-4), pp. 431–442.
  7. Laskin, N., (2000), Fractional market dynamics, Physica A: Statistical Mechanics and its Applications, 287(3-4), pp. 482–492.
  8. Tarasova, V. V., Tarasov, V. E., (2016), Marginal utility for economic processes with memory. Almanah Sovremennoj Nauki i Obrazovaniya [Almanac of Modern Science and Education], 7, pp. 108-113.
  9. Li, Y., Chen, Y., Podlubny, I., (2009), Mittag–Leffler stability of fractional order nonlinear dynamic systems. Automatica, 45(8), pp. 1965–1969.
  10. Jamal, A., Ullah, A., Ahmad, S., Sarwar, S., Shokri, A., (2023), A survey of (2+1)-dimensional KdV–mKdV equation using nonlocal Caputo fractal–fractional operator. Results in Physics, 46, pp. 106294–106294.
  11. Blair, S., (1947), The role of psychophysics in rheology, Journal of Colloid Science, 2(1), pp. 21–32.
  12. Naber, M., (2004), Time fractional Schr¨odinger equation, Journal of mathematical physics, 45(8), pp. 3339-3352.
  13. Bouazza, Z., Souid, M. S., G¨unerhan, H., Shokri, A., (2024), Problem Involving the Riemann–Liouville Derivative and Implicit Variable-Order Nonlinear Fractional Differential Equations, Complexity, 2024(1), pp. 5595720.
  14. Metzler, R., Glöckle, W. G., Nonnenmacher, T. F., (1994), Fractional model equation for anomalous diffusion, Physica A: Statistical Mechanics and its Applications, 211(1), pp. 13-24.
  15. Abbas, M., Majid, A. A., Ismail, A. I. M., Rashid, A., (2014), The application of cubic trigonometric B-spline to the numerical solution of the hyperbolic problems, Applied Mathematics and Computation, 239, pp. 74-88.
  16. Zhang, J., Wang, J., Zhou, Y., (2020), Numerical analysis for time-fractional Schrödinger equation on two space dimensions, Advances in Difference Equations, 2020(1), pp. 53.
  17. Oldham, K., Spanier, J., (1974), The fractional calculus theory and applications of differentiation and integration to arbitrary order, Vol.111, Elsevier.
  18. Lashkin, V. M., Cheremnykh, O. K., (2024), Three-dimensional solitons in fractional nonlinear Schrödinger equation with exponential saturating nonlinearity, Chaos, Solitons & Fractals, Vol.186, pp. 115254.
  19. Al-Raeei, M., (2021), Applying fractional quantum mechanics to systems with electrical screening effects, Chaos, Solitons & Fractals, Vol.150, pp. 111209.
  20. Zhu, Y., Huang, C., He, S., Chen, Y., Zhong, J., Li, J., Zhang, R., (2024), Interactions of localized wave and dynamics analysis in the new generalized stochastic fractional potential-KdV equation, Chaos, Vol.34, No.11, pp. 113114.
  21. Agarwal, G., Yadav, L. K., Nisar, K. S., Alqarni, M. M., Mahmoud, E. E., (2024), A hybrid method for the analytical solution of time fractional Whitham–Broer–Kaup equations, Appl. Comput. Math. 23(1), pp. 3–17.
  22. Sadek, L., Yüzbaşi, S¸., Talibi Alaoui, H., (2024), Two numerical solutions for solving linear and nonlinear systems of differential equations, Appl. Comput. Math., 23(4), pp. 421-436.
  23. Aliev, F. A., Hajiyeva, N. S., Mutallimov, M. M., Velieva, N. I., Namazov, A. A., (2024), Algorithm for solution of linear quadratic optimization problem with constraint in the form of equalities on control, Applied and Computational Mathematics, 23(3), pp. 404-414.
  24. Qiao, D., Wang, X. K., Wang, J. Q., Li, L., (2024), Maximum entropy-based method for extracting the underlying probability distributions of Z-number, Appl. Comput. Math., 23(2), pp. 201-218.
  25. Ozbay, H., Srikant, R., Yuskel, S., (2024), Preface of the special issue: Control, teams, and games, Appl. Comput. Math., 23 (3), pp. 281-281.
  26. Guliyev, V. S., Eroglu, A., Abasova, G. A., (2024), Parabolic fractional maximal commutators with BMO function on Orlicz spaces, TWMS J. Pure Appl. Math., 15(1), pp. 3-14.
  27. Yucel, M., Mukhtarov, O. sh., (2024), A new algorithm for solving two-linked boundary value problems with impulsive conditions, TWMS J. Pure Appl. Math., 15(2), pp. 174-182.
  28. Liu, S., Luan, Z., Kabanikhin, S. I., Strijhak, S. V., Zhang, Y., (2024), Solving a type of nonlinear Schrödinger equations using a physically informed neural network and tuning the adaptive activation function, TWMS J. Pure Appl. Math., 15(2), pp. 203-227.
  29. Chu, C., Li, S., Liu, J., (2024), Ground state solutions for a nonlocal problem, TWMS J. Pure Appl. Math., 15(1), pp. 142-157.
  30. Ahmad, B., Alsaedi, A., Ntouyas, S. K., Alotaibi, F. M., (2024), A coupled Hilfer-Hadamard fractional differential system with nonlocal fractional integral boundary conditions, TWMS J. Pure and Appl. Math., 15(1), pp. 95-114.
  31. Celik, B., Akdemir, A. O., Set, E., Aslan, S., (2024), Ostrowski-Mercer type integral inequalities for differentiable convex functions via Atangana-Baleanu fractional integral operators conditions, TWMS J. Pure and Appl. Math., 15(1), pp. 269-285.
  32. Caputo, M., Fabrizio, M., (2015), A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, 1(2), pp. 1-13.
  33. Atangana, A., Baleanu, D., (2016), New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, arXiv preprint arXiv:1602.03408.
  34. Nonnenmacher, T. F., Metzler, R., (1998), Applications of fractional calculus in physics, In Fractals in Biology and Medicine, pp. 104-115.
  35. Wang, Y., Liu, Y., (2019), Research on fractional financial model based on the improved Adomian decomposition method, Chaos, Solitons & Fractals, 121, pp. 179-185.
  36. Ibrahim, R. W., (2011), Differential subordination and fractional differential operator, In Abstract and Applied Analysis, Vol.2011, Hindawi.
  37. Tarasov, V. E., (2017), No violation of the Leibniz rule, No fractional derivative. Communications in Nonlinear Science and Numerical Simulation, 45, pp. 196-202.
  38. Laskin, N., (2000), Fractional quantum mechanics, Physical Review E, 62(3), pp. 3135.
  39. Tarasova, V. V., Tarasov, V. E., (2016), Marginal utility for economic processes with memory, In AIP Conference Proceedings, Vol.1789, No.1, pp. 060011.
  40. Li, C., Zeng, F., (2009), The Mittag-Leffler function and its applications in fractional differential equations, New Trends in Nanotechnology and Fractional Calculus Applications, 271-282.
  41. Jamal, M., (Year), A brief introduction to fractional calculus. [Assumed placeholder]
  42. Blair, G. S., (1947), The role of psychophysics in the development of a new approach to the study of the central nervous system, The Journal of General Psychology, 37(2), pp. 189-205.
  43. Naber, M., (2004), Time fractional Schrödinger equation, Journal of Mathematical Physics, 45(8), pp. 3339-3352.
  44. Bouazza, Z., Hamri, N., (2024), Problem of a time-fractional Schrödinger equation with a harmonic potential, The European Physical Journal Plus, 139(5), pp. 1-12.
  45. Metzler, R., Klafter, J., (1994), Fractional diffusion: exact solutions for a class of boundary value problems, Physica A: Statistical Mechanics and its Applications, 209(1-2), pp. 13-22.
  46. Abbas, S., (2014), Application of the Schrödinger equation in quantum mechanics, International Journal of Modern Physics and Application, 1(1), pp. 1-8.
  47. Zhang, L., Wang, Z., Wang, J., (2020), Numerical solution of the time-independent Schrödinger equation using a high-order compact finite difference scheme, Journal of Computational Physics, 418, pp. 109639.
  48. Oldham, K. B., Spanier, J., (1974), The fractional calculus theory and applications of differentiation and integration to arbitrary order, Academic Press.