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; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
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.
An Operator Splitting Approximation Combined With the Supg Method for Transport Equations With Nonlinear Reaction Term
(Tech Science Press, 2012) Baysal, Onur; Tanoğlu, Gamze; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
In this work, an operator splitting method is proposed in order to obtain a stable numerical solution for transport equation with non-linear reaction term. We split the transport equation into a reaction part and an advection diffusion part. The former one which becomes a nonlinear ordinary differential equation can be approximated by the simple higher order integrator or solved exactly. The later one is approximated by the Streamline-Upwind Petrov-Galerkin (SUPG) method combined with the generalized Euler time integration (q-method). Numerical results that illustrate the good performance of this method are reported.
Citation - Scopus: 1
Pseudo-Multi Functions for the Stabilization of Convection-Diffusion Equations on Rectangular Grids
(Begell House Inc., 2013) Neslitürk, Ali İhsan; Baysal, Onur; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
We propose a finite element method of Petrov-Galerkin type for a singularly perturbed convection diffusion problem on a discretization consisting of rectangular elements. The method is based on enriching the finite-element space with a combination of multiscale and residual-free bubble functions. These functions require the solution of the original differential problem, which makes the method quite expensive, especially in two dimensions. Therefore, we instead employ their cheap, yet efficient approximations, using only a few nodes in each element. Several numerical tests confirm the good performance of the corresponding numerical method.
Stabilized finite element methods for time dependent convection-diffusion equations
(Izmir Institute of Technology, 2012) Baysal, Onur; Tanoğlu, Gamze; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
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.