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

Hojjat Habibzadeh; Mohammad Shahriari; Ali Shokri; Mohammad Mehdizadeh Khalsaraeı

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

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.

Keywords

References

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Details

Primary Language

English

Subjects

Numerical Analysis, Ordinary Differential Equations, Difference Equations and Dynamical Systems, Partial Differential Equations

Journal Section

Research Article

Authors

Hojjat Habibzadeh This is me
0009-0004-1444-2157
Iran

Mohammad Shahriari
0000-0002-8982-2451
Iran

Ali Shokri ^*
0000-0003-2699-1490
Iran

Mohammad Mehdizadeh Khalsaraeı
0000-0002-0130-0449
Iran

Publication Date

September 7, 2026

Submission Date

March 15, 2025

Acceptance Date

May 12, 2026

Published in Issue

Year 2026 Volume: 16 Number: 9

IZ

https://izlik.org/JA26XJ27YM

Cite

RIS / Bibtex

APA

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, 16(9), 1135-1171. https://izlik.org/JA26XJ27YM

AMA

1.Habibzadeh H, Shahriari M, Shokri A, Khalsaraeı MM. A HIGH-ORDER CAPUTO-BASED NUMERICAL SCHEME FOR APPROXIMATING THE ONE-DIMENSIONAL TIME-INDEPENDENT FRACTIONAL SCHRÖDINGER EQUATION. JAEM. 2026;16(9):1135-1171. https://izlik.org/JA26XJ27YM

Chicago

Habibzadeh, Hojjat, Mohammad Shahriari, Ali Shokri, and Mohammad Mehdizadeh Khalsaraeı. 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 16 (9): 1135-71. https://izlik.org/JA26XJ27YM.

EndNote

Habibzadeh H, Shahriari M, Shokri A, Khalsaraeı MM (September 1, 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 16 9 1135–1171.

IEEE

[1]H. Habibzadeh, M. Shahriari, A. Shokri, and M. M. Khalsaraeı, “A HIGH-ORDER CAPUTO-BASED NUMERICAL SCHEME FOR APPROXIMATING THE ONE-DIMENSIONAL TIME-INDEPENDENT FRACTIONAL SCHRÖDINGER EQUATION”, JAEM, vol. 16, no. 9, pp. 1135–1171, Sept. 2026, [Online]. Available: https://izlik.org/JA26XJ27YM

ISNAD

Habibzadeh, Hojjat - Shahriari, Mohammad - Shokri, Ali - Khalsaraeı, Mohammad Mehdizadeh. “A HIGH-ORDER CAPUTO-BASED NUMERICAL SCHEME FOR APPROXIMATING THE ONE-DIMENSIONAL TIME-INDEPENDENT FRACTIONAL SCHRÖDINGER EQUATION”. TWMS Journal of Applied and Engineering Mathematics 16/9 (September 1, 2026): 1135-1171. https://izlik.org/JA26XJ27YM.

JAMA

1.Habibzadeh H, Shahriari M, Shokri A, Khalsaraeı MM. A HIGH-ORDER CAPUTO-BASED NUMERICAL SCHEME FOR APPROXIMATING THE ONE-DIMENSIONAL TIME-INDEPENDENT FRACTIONAL SCHRÖDINGER EQUATION. JAEM. 2026;16:1135–1171.

MLA

Habibzadeh, Hojjat, et al. “A HIGH-ORDER CAPUTO-BASED NUMERICAL SCHEME FOR APPROXIMATING THE ONE-DIMENSIONAL TIME-INDEPENDENT FRACTIONAL SCHRÖDINGER EQUATION”. TWMS Journal of Applied and Engineering Mathematics, vol. 16, no. 9, Sept. 2026, pp. 1135-71, https://izlik.org/JA26XJ27YM.

Vancouver

1.Hojjat Habibzadeh, Mohammad Shahriari, Ali Shokri, Mohammad Mehdizadeh Khalsaraeı. A HIGH-ORDER CAPUTO-BASED NUMERICAL SCHEME FOR APPROXIMATING THE ONE-DIMENSIONAL TIME-INDEPENDENT FRACTIONAL SCHRÖDINGER EQUATION. JAEM [Internet]. 2026 Sep. 1;16(9):1135-71. Available from: https://izlik.org/JA26XJ27YM