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-splinebased differential quadrature method (FDM-DQM) for getting the numericalsolutions for the nonlinear Schrödinger (NLS) equation. To solve complex NLSequation firstly we have separated NLS equation into the two real value partialdifferential equations. After that they are discretized in time using special typeof classical finite difference method namely, Crank-Nicolson scheme. Then, forspace 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 L 2and L ∞ and two lowest invariants I 1 and I 2 and relative changes of invariantshave been calculated. As a last step, the numerical result already obtained havebeen compared with earlier studies by using same parameters. The comparisonhas clearly indicated that the presently used method, namely FDM-DQM, isan appropriate and accurate numerical scheme and allowed us to present forsolving a wide class of partial differential equations.