A mixed method approach to Schrödinger equation: Finite difference method and quartic B-spline based differential quadrature method

Abstract

The present manuscript includes finite difference method and quartic B-spline based differential quadrature method (FDM-DQM) for getting the numerical solutions for the nonlinear Schrödinger (NLS) equation. To solve complex NLS equation firstly we have separated NLS equation into the two real value partial differential equations. After that they are discretized in time using special type of classical finite difference method namely, Crank-Nicolson scheme. Then, for space integration differential quadrature method has been implemented. So, partial differential equation turn into simple a system of algebraic equations. To display the accuracy of the present hybrid method, the error norms L2 and L∞ and two lowest invariants I1 and I2 and relative changes of invariants have been calculated. As a last step, the numerical result already obtained have been compared with earlier studies by using same parameters. The comparison has clearly indicated that the presently used method, namely FDM-DQM, is an appropriate and accurate numerical scheme and allowed us to present for solving a wide class of partial differential equations.