Numerical Approximation for 2-Dimensional Schrödinger Equation Using Modified Gauss Elimination Method

Year 2022, Issue: 35, 653 - 658, 07.05.2022

Tarkan Aydın , Ali Sırma

https://doi.org/10.31590/ejosat.1018127

Abstract

In this study, modifed Gauss elimination method will be used to obtain solution of first order Rothe difference scheme and second order Crank-Nicholson difference scheme for numerical approximation of two-dimensional Schrödinger equation in space variable. One example is given and approximate solution is found by three methods. Modified Gauss elimination method is used with respect to time variable and with respect to space variable. In order to compare the difference schemes are also solved by the classical inverse matrix method.

Keywords

Modified Gauss elimination method, Rothe difference scheme, Self-adjoint operator

Modifiye Gauss Eleme Yöntemi Kullanarak 2-Boyutlu Schrödinger Denklemine Sayısal Yaklaşım

Abstract

Bu çalışmada, uzay değişkeninde iki boyutlu Schrödinger denkleminin sayısal yaklaşımı için birinci mertebeden Rothe fark şemasının ve ikinci mertebeden Crank-Nicholson fark şemasının çözümünü elde etmek için modifiye Gauss eliminasyon yöntemi kullanılmıştır. Bir örnek verilmiş ve üç yöntemle yaklaşık çözüm bulunmuştur. Modifiye Gauss eliminasyon yöntemi, zaman değişkenine ve uzay değişkenine göre kullanılmıştır. Karşılaştırma yapmak için fark şemaları, klasik ters matris yöntemi ile de çözülmüştür.

Keywords

Modifiye Gauss eleme metodu, Rothe fark şeması, Öz-eşlenik operator

References

• Ashyralyev A. & Sırma A., (2008). Nonlocal boundary value problems for the Schrödinger equation. Computers & Mathematics with Applications, 55, 392-407.
• Ashyralyev A. & Sırma A., (2009). Modified Crank-Nicolson difference schemes for nonlocal boundary value problems for the Schrödinger equation. Discrete Dynamics in Nature and Society, 2009, 1-15.
• Ashyralyev A. & Hiçdurmaz B., (2011). A note on fractional Schrödinger differential equations. Kybernetes, 40, 736-750.
• Ashyralyev A. & Özdemir Y., (2005). Stability of difference schemes for hyperbolic-parabolic equations. Computers & Mathematics with Applications, 50, 1443-1476.
• Ashyralyyev, C., (2017). Numerical solution to Bitsadze-Samarskii type elliptic overdetermined multipoint NBVP. Boundary Value Problems, 74, 1-22.
• Ashyralyyev, C. & Akyuz G., (2018). Finite difference method for Bitsadze-Samarskii type overdetermined elliptic problem with Dirichlet conditions. Filomat, 32, 859-872.
• Ashyralyyev, C. & Cay A., (2020). Numerical solution to elliptic inverse problem with Neumann-type integral condition and overdetermination. Karaganda University-Mathematics, 99, 5-17.
• Ozdemir, Y, (2007). Nonlocal boundary value problem for hyperbolic-parabolic differential and difference equations, PhD Thesis, Gebze High Technology University, Graduate School of Science and Technology, Izmit, 152.
• Sırma A., (2021). A single step second order of accuracy difference scheme for integral type nonlocal boundary value Schrödinger problem. Tbilisi Mathematical Journal, Special Issue 8-2021, 13-20.