A new approach in handling one-dimensional time-fractional Schrödinger equations

Ahmad El-Ajou; Rania Saadeh; Moawaih Akhu Dunia; Ahmad Qazza; Zeyad Al-Zhour; Ahmad El-Ajou; Rania Saadeh; Moawaih Akhu Dunia; Ahmad Qazza; Zeyad Al-Zhour

doi:10.3934/math.2024515

AIMS Mathematics

2024, Volume 9, Issue 5: 10536-10560. doi: 10.3934/math.2024515

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A new approach in handling one-dimensional time-fractional Schrödinger equations

1.
Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Salt 19117, Jordan
2.
Department of Mathematics, Faculty of Science, Zarqa University, Zarqa 13110, Jordan
3.
Department of Basic Engineering Sciences, College of Engineering, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam 31441, Saudi Arabia

Received: 22 November 2023 Revised: 01 February 2024 Accepted: 07 February 2024 Published: 18 March 2024
MSC : 32A05, 35R11, 41A58

Our aim of this paper was to present the accurate analytical approximate series solutions to the time-fractional Schrödinger equations via the Caputo fractional operator using the Laplace residual power series technique. Furthermore, three important and interesting applications were given, tested, and compared with four well-known methods (Adomian decomposition, homotopy perturbation, homotopy analysis, and variational iteration methods) to show that the proposed technique was simple, accurate, efficient, and applicable. When there was a pattern between the terms of the series, we could obtain the exact solutions; otherwise, we provided the approximate series solutions. Finally, graphical results were presented and analyzed. Mathematica software was used to calculate numerical and symbolic quantities.
- fractional operators,
- fractional Schrödinger equation,
- multiple fractional power series,
- Laplace residual power series method
Citation: Ahmad El-Ajou, Rania Saadeh, Moawaih Akhu Dunia, Ahmad Qazza, Zeyad Al-Zhour. A new approach in handling one-dimensional time-fractional Schrödinger equations[J]. AIMS Mathematics, 2024, 9(5): 10536-10560. doi: 10.3934/math.2024515

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Abstract

Our aim of this paper was to present the accurate analytical approximate series solutions to the time-fractional Schrödinger equations via the Caputo fractional operator using the Laplace residual power series technique. Furthermore, three important and interesting applications were given, tested, and compared with four well-known methods (Adomian decomposition, homotopy perturbation, homotopy analysis, and variational iteration methods) to show that the proposed technique was simple, accurate, efficient, and applicable. When there was a pattern between the terms of the series, we could obtain the exact solutions; otherwise, we provided the approximate series solutions. Finally, graphical results were presented and analyzed. Mathematica software was used to calculate numerical and symbolic quantities.

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