Kısmi türevli kesirli mertebeden lineer schrödinger denklemlerinin sayısal çözümleri

Abstract

Bu tezde ksmi türevli zaman-kesirli mertebeden lineer Scrödinger denklemi ile ifade edilen problem ele alnmıştr. Problem, Caputo kesirli türev tanımının uygulanmasyla tamsayılı mertebeden lineer Scrödinger denklemi haline getirildikten sonra Komkakt Sonlu Farklar(KSF) ve Ortalama Vektör Alan (OVA) metodları ile çözülmüştür. Tezde ayrca uzay-kesirli mertebeden difüzyon denklemi de Caputo kesirli türev tanımının ardından KSF ve OVA metodları ile çözülmüştür. Ayrca kısmi türevli zaman-kesirli mertebeden lineer Scrödinger denklemine uygulanan her iki metod için de dağılım analizi yaplmıştr.

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 denition, the problem was solved via Compact Finite Differences (CFD) and Average Vector Field (AVF) methods. Additionally, the problem in the form of space-fractional diffusion equation was solved via CFD and AVF methods after application of Caputo derivative definition, so it was indicated that CFD and AVF methods were applicable for space-fractional dierential equations. Dispersion analysis forboth methods were also carried out.