Quantum solutions of a nonlinear Schr\"odinger equation

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.

Keywords:

NLS equation , Quantum calculus , Numerical Solution , Error estimates