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

An International Journal of Optimization and Control: Theories & Applications

article

Ali Başhan¹ 

[1] Zonguldak Bulent Ecevit University

关键词: Partial differential equations;    Differential quadrature method;    Quartic B-Splines;    Schr¨odinger equation.;   

DOI  :  10.11121/ijocta.01.2019.00709

学科分类：地球科学(综合)

来源: Balikesir University

【 摘 要 】

The present manuscript include, finite difference method and quartic B-spline based differential quadrature method (FDM-DQM) to obtain the numerical solutions for the nonlinear Schr¨odinger (NLS) equation. For this purpose, firstly Schrödinger equation has been converted into coupled real value differential equations and then they have been discretized using special type of classical finite difference method namely, Crank-Nicolson scheme. After that, Rubin and Graves linearization techniques have been utilized and differential quadrature method has been applied. So, partial differential equation turn into algebraic equation system. Next, in order to be able to test the accuracy of the newly hybrid method, the error norms L2 and L? as well as the two lowest invariants I1 and I2 have been calculated. Besides those, the relative changes in those invariants have been given. Finally, the newly obtained numerical results have been compared with some of those available in the literature for similar parameters. This comparison has clearly indicated that the currently utilized method, namely FDM-DQM, is an effective and efficient numerical schemeand allowed us to propose to solve a wide range of nonlinear equations.

【 授权许可】

CC BY