Department of Mathematics
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Browsing Department of Mathematics by Author "Aydın, Ayhan"
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Item AVERAGE VECTOR FIELD METHOD FOR HAMILTONIAN SYSTEMS(2022-01-11) Sabawe, Bahaa Ahmed Khalaf; Aydın, AyhanIn this thesis, we present and analyze four energy preserving methods for the numer ical solution of initial value problems of Hamiltonian type. In particular, the average vector field (AVF) and partitioned AVF (PAVF) methods are used to drive energy preserving methods. In addition to these two energy preserving methods, two en ergy persevering PAVF composition (PAVF-C) and PAVF plus (AVF-P) methods are presented. The thesis accompanied numerical result for Zakharov system that demon strate remarkable properties of the proposed energy persevering methods. In this the sis, this is the first time that energy persevering AVF, PAVF, PAVF-C and PAVF-P methods are proposed for Zakharov system. It is shown that PAVF and PAVF-C meth ods for Zakharov system are linearly implicit methods that have remarkable lower cost than the original AVF method. In addition, we further show that the PAVF meth ods preserve the mass conservation of the Zakharov system while the AVF method cannot.Item EXPONENTIAL FINITE DIFFERENCE METHOD FOR NONLINEAR BLACK-SCHOLES EQUATION(2017-04-04) Omar, Fathia; Aksoy, Ümit; Aydın, AyhanIn this thesis, we investigate exponential finite difference method for nonlinear Black Scholes equation arising in an illiquid market. Chapter 1 is devoted to the literature survey with some basic definitions and terminology of the option pricing problem. In Chapter 2 we review the Black-Scholes model and finite difference methods for Black Scholes equation. In Chapter 3, an explicit finite difference method for a nonlinear Black-Scholes equation is studied with the monotonicity, stability and consistency re sults. In Chapter 4, we apply the exponential finite difference method to linear and nonlinear Black-Scholes equations. Moreover, we investigate consistency and con vergence of the method. Numerical experiments are performed to verify theoretical results. Exponential finite difference method is compared with an explicit finite dif ference method proposed for linear and nonlinear Black-Scholes equation. Numerical results show that exponential finite difference method exhibits better performance then explicit method. Finally, we give a brief conclusion in Chapter 5.