New numerical analysis of Riemann-Liouville time-fractional Schrödinger with power, exponential decay, and Mittag-Leffler laws

Volume 10, Issue 8, pp 4231--4243 http://dx.doi.org/10.22436/jnsa.010.08.18

Publication Date: August 16, 2017 Submission Date: June 02, 2017

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Authors

Badr Saad T. Alkahtani - Department of mathematics, colle of science, King Saud University, P. O. Box 1142, Riyadh 11989, Saudi Arabia. Ilknur Koca - Department of Mathematics, Faculty of Sciences, Mehmet Akif Ersoy University, 15100, Burdur, Turkey. Abdon Atangana - Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, 9300, Bloemfontein, South Africa.

Abstract

The mathematical equation that describes how the quantum state of a quantum system changes during time was considered within the framework of fractional differentiation with three different derivatives in Riemann-Liouville sense. The fractional derivatives used in this work are constructed based on power, exponential decay, and Mittag-Leffler law. A new numerical scheme for fractional derivative in Riemann-Liouville sense is presented and used to solve numerically the Schrödinger equation. The stability analysis of each model is presented in detail.

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Keywords

• Power law
• exponential decay law
• Mittag-Leffler law
• numerical scheme
• Schrödinger equation.

MSC

•  26A33
•  33E12
•  35J10

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