Kısmi Türevli Kesirli Mertebeden Lineer Schrödinger Denklemlerinin Sayısal Çözümleri

Abstract

Bu tezde kısmi türevli zaman-kesirli mertebeden lineer Scrödinger denklemi ile ifade edilen problem ele al nm ışt r. Problem, Caputo kesirli türev tanı mı nı n uygulanmas yla tamsay ılı mertebeden lineer Scrödinger denklemi haline getirildikten sonra Komkakt Sonlu Farklar (KSF) ve Ortalama Vektör Alan (OVA) metodları ile çözülmüştür. Tezde ayr ca uzay-kesirli mertebeden difüzyon denklemi de Caputo kesirli türev tanı m ın ın ardı ndan KSF ve OVA metodları ile çözülmüştür. Ayr ca k ısmi türevli zaman-kesirli mertebeden lineer Scrödinger denklemine uygulanan her iki metod için de dağılı m analizi yap lm ışt r.

In this thesis, a problem expressed as a time-fractional linear Schrödinger equation was handled to be solved. After transforming the fractional order SE into integer order SE by application of Caputo derivative de nition, the problem was solved via Compact Finite Di fferences (CFD) and Average Vector Field (AVF) methods. Additionally, the problem in the form of space-fractional diff usion equation was solved via CFD and AVF methods after application of Caputo derivative de finition, so it was indicated that CFD and AVF methods were applicable for space-fractional di erential equations. Dispersion analysis for both methods were also carried out.