Quantum solutions of a nonlinear Schrödinger equation

Document Type : Research Article

Authors

• S. Arfaoui ^{1, 2}
• Ben Mabrouk ^{3, 4, 5}

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

² nstitut Supérieur d’Informatique du Kef, Université de Jendouba, 5 Rue Saleh Ayech - 7100 Kef, Tunisia

³ Laboratory of Algebra, Number Theory and Nonlinear Analysis, Department of Math- ematics, Faculty of Sciences, University of Monastir, Avenue of the Environment, 5019 Monastir, Tunisia.

⁴ Department of Mathematics, Higher Institute of Applied Mathematics and Computer Science, University of Kairouan, Street Assad Ibn Alfourat, Kairouan 3100, Tunisia.

⁵ Department of Mathematics, Faculty of Sciences, University of Tabuk, King Faisal Road, Tabuk 47512, Saudi Arabia.

10.22067/ijnao.2023.84855.1324

Abstract

In the present paper, we precisely conduct a quantum calculus method for the numerical solutions of PDEs. A nonlinear Schrödinger equation is considered. Instead of the known classical discretization methods based on the finite difference scheme, Adomian method, and third modified ver-sions, we consider 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 effi-cient numerical solutions for PDEs. The q-discretization leads effectively to higher orders of convergence provided with faster algorithms. The numer-ical tests are applied to both propagation and interaction of soliton-type solutions.

Keywords

• NLS equation
• Quantum calculus
• Numerical solution
• Error estimates

Main Subjects

• Numerical methods for differential equations and its applications

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