Quantum solutions of a nonlinear Schr\"odinger equation

Iranian Journal of Numerical Analysis and Optimization

مقاله 1، دوره 14، Issue 2, IN PROGRESS - شماره پیاپی 29، شهریور 2024، صفحه 330-346   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.2023.84855.1324

نویسندگان

Sabrine Arfaoui^{1، 2}؛ Benmabrouk Anouar^{* 1، 3، 4}

¹Laboratory of Algebra, Number Theory and Nonlinear Analysis, Department of Mathematics, Faculty of Sciences, University of Monastir, Avenue of the Environment, 5019 Monastir, Tunisia.

²Institut Sup´erieur d’Informatique du Kef, Universit´e de Jendouba, 5 Rue Saleh Ayech 7100 Kef, Tunisia.

³Department of Mathematics, Higher Institute of Applied Mathematics and Computer Science, University of Kairouan, Street Assad Ibn Alfourat, Kairouan 3100, Tunisia.

⁴Department of Mathematics, Faculty of Sciences, University of Tabuk, King Faisal Road, Tabuk 47512, Saudi Arabia.

چکیده

In the present paper, we precisely conduct a quantum calculus method for the numerical solutions of PDEs. A nonlinear Schr\"odinger equation is considered. Instead of the known classical discretization methods based on the finite difference scheme, Adomian method, and their modified versions, we consider instead a discretization scheme leading to subdomains according to $q$-calculus and provide an approximate solution due to a specific value of the parameter $q$. Error estimates show that $q$-calculus may produce efficient numerical solutions for PDEs. The $q$-discretization leads effectively to higher orders of convergence provided with faster algorithms. The numerical tests are applied to both propagation and interaction of soliton-type solutions.

کلیدواژه‌ها

NLS equation؛ Quantum calculus؛ Numerical solution؛ Error estimates