Akademik Veri Yönetim Sistemi
Tezin Türü: Doktora
Tezin Yürütüldüğü Kurum: Eskişehir Osmangazi Üniversitesi, FEN BİLİMLERİ ENSTİTÜSÜ, Fen Bil.Enst.Md.Lüğü, Türkiye
Tezin Onay Tarihi: 2020
Tezin Dili: Türkçe
Öğrenci: AYSUN TOK ONARCAN
Asıl Danışman (Eş Danışmanlı Tezler İçin): Nihat Adar
Eş Danışman: İdiris Dağ
Özet:
In this thesis, it is aimed to develop algorithms for numerical solutions of reaction-diffusion equation systems which are defined to mathematically model some problems in many sciences such as physics, chemistry and biology by using trigonometric B-spline base functions of different degrees. The results, obtained by applying the proposed solution method on four different examples of reaction diffusion equation systems in the literature, were compared. Among these systems, besides the linear problem whose analytical solution is known, Brusselator model, Schnakenberg model and Gray-Scott model were chosen as test problems. In the study, collocation method, one of the finite element methods, was applied in order to obtain the approximate solution of reaction-diffusion equation systems and Crank-Nicolson method was used for the time discretization of the equation system. In the first part of the thesis, definition of the study is implemented and the purpose of the subject has been mentioned. Accordingly, the reaction-diffusion equation system and test problems are defined. The studies in the literature on approximate solutions of reaction-diffusion equation systems are given and different solution methods of these problems are mentioned. After than, the basic concepts and methods that are used in the study are discussed. The Crank-Nicolson method which is used for time discretization from finite difference methods and the collocation method, which is used to obtain approximate solution from finite element methods, are explained. Base functions at different degrees of collocation method which are applied in the solution of the model system; trigonometric quadratic, trigonometric cubic, trigonometric quartic and trigonometric quintic B-spline functions are defined. The equation system that models all test problems in general has been created, also the matrix solution algorithms which is used to solve this equation system are given. The approximate solutions obtained by applying the proposed method on the selected test problems are shown with the help of tables and graphs. In order to determine the accuracy of the method, error norms are used for linear problem and relative error is used for nonlinear problems also the stability of the matrix system is analysed. In the last part of the thesis, there are comments and suggestions about the results obtained.