Browsing by Author "Baysal, Onur"

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Lower-top and upper-bottom points for any formula in temporal logic
(Izmir Institute of Technology, 2006) Baysal, Onur; Alizade, Rafail
In temporal logic, which is a branch of modal logic, models are constructed on some kind of frames. Common properties of all these frames include totally ordered relations and these frames are bi-directional. These common properties provide the temporal logic time interpretation. By means of this interpretation temporal language has lots of application areas. The main aim of this study is to propose new technic which gets easier proof of some kind of valid formulas in the most popular temporal frame T and to produce new valid formulas with the medium of this new technic. To be able to realize this main aim, first of all the frame T . (N;6;>;R±;R for temporal language has been composed step by step in accordance with principles of modal logic. Then the new terms " lower-top and upper-bottom points for any temporal formula " has been defined in the model M . (T; V ) which is built over the frame T and some propositions of this term have been obtained. At the end of the study it has been presented that proofs of some theorems have been done easier and it has been given possibility to produce the new theorems.Moreover a general investigation about the frame T has been done and presented, furthermore it has been shown that the mirror image of the valid formulas do not have to be valid and it is also possible that the mirror image of non valid formulas can be valid.
Stabilized finite element methods for time dependent convection-diffusion equations
(Izmir Institute of Technology, 2012) Baysal, Onur; Tanoğlu, Gamze
In this thesis, enriched finite element methods are presented for both steady and unsteady convection diffusion equations. For the unsteady case, we follow the method of lines approach that consists of first discretizing in space and then use some time integrator to solve the resulting system of ordinary differential equation. Discretization in time is performed by the generalized Euler finite difference scheme, while for the space discretization the streamline upwind Petrov-Galerkin (SUPG), the Residual free bubble (RFB), the more recent multiscale (MS) and specific combination of RFB with MS (MIX) methods are considered. To apply the RFB and the MS methods, the steady local problem, which is as complicated as the original steady equation, should be solved in each element. That requirement makes these methods quite expensive especially for two dimensional problems. In order to overcome that drawback the pseudo approximation techniques, which employ only a few nodes in each element, are used. Next, for the unsteady problem a proper adaptation recipe, including these approximations combined with the generalized Euler time discretization, is described. For piecewise linear finite element discretization on triangular grid, the SUPG method is used. Then we derive an efficient stability parameter by examining the relation of the RFB and the SUPG methods. Stability and convergence analysis of the SUPG method applied to the unsteady problem is obtained by extending the Burman’s analysis techniques for the pure convection problem. We also suggest a novel operator splitting strategy for the transport equations with nonlinear reaction term. As a result two subproblems are obtained. One of which we may apply using the SUPG stabilization while the other equation can be solved analytically. Lastly, numerical experiments are presented to illustrate the good performance of the method.