Numerical solution of linear and nonlinear Schrödinger equations via the shifted Chebyshev collocation method

The present study focuses on numerical solutions of linear and nonlinear Schrödinger equations subject to initial and boundary conditions employing the shifted Chebyshev spectral collocation method (SCSCM). In the solution procedure, unknown function and its space derivatives have been approximated employing shifted Chebyshev polynomials and their derivatives, respectively, together with Chebyshev-Gauss-Lobatto points. The present collocation method transforms the Schrödinger equation into a system of ordinary differential equations (ODEs). Thereafter, the obtained system has been solved employing the fourth-order Runge-Kutta scheme. To demonstrate the accuracy and efficiency of the present method, a comparison of the present numerical solutions of different examples of the Schrödinger equation with exact and approximate solutions available in the literature has been discussed. The SCSCM can be implemented to solve second and higher-order linear and nonlinear partial differential equations (PDEs) arising in physics, mechanics, and biophysics.

Keywords:

Shifted Chebyshev polynomials , Spectral collocation method , Schrödinger equation , Runge-Kutta method