Using Nonstandard Finite Difference Methods for Solving Converted Schrodinger Equation to an ODE

In this work, by introducing a transformation, the nonlinear Schrodinger equation is converted to an ordinary differential equation (ODE). Then, two nonstandard finite difference (NSFD) schemes are constructed for studying the reduced equation. It is shown that the methods preserve the positivity and boundedness properties of the original equation and are stable conditionally and consistence. Finally, the results of the methods are compared with each other and also with the results of the standard finite difference scheme at some points. The graphs of the errors of numerical solutions for these schemes are plotted and compared with the exact solutions.