FBE- Matematik Mühendisliği Lisansüstü Programı - Yüksek Lisans

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  • Öge
    Stability analysis of a mathematical model of Crimean Congo haemorrhagic fever disease
    (Graduate School Of Science Engineering And Technology, 2020) Alın, Miray ; Özer, Saadet Seher ; 637195 ; Department of Mathematical Engineering
    Today, ticks are harmful parasitic creatures feared by humans. Ticks do not always carry dangerous diseases. However, we should not ignore the pathogens and viruses that may be carried because these creatures can carry various viruses and seriously threaten human health. If it is not diagnosed early, it can result in fatal consequences. Ticks can get viruses from their hosts at various stages of their lives. Ticks can transmit these viruses to humans in the adult tick stage. Here we can say that the animals that ticks use as hosts are only vectors. Cattle, bovine or chickens do not show symptoms of diseases which are caused by ticks. In this thesis, the spread of Crimean-Congo haemorrhagic fever disease is investigated by considering the problem as an epidemic model. Before stating the problem, in first chapter, some information about dynamic systems is given. The definition of systems of differential equations and their stability analysis are mentioned. Besides, the autonomous systems of equations are briefly explained. And how their stability can be analysed is mentioned. Then, to guide our own problem, information about the well-known SI, SIR, SIS epidemic models and Prey-Predator model and their stability is given in the second chapter. And finally in the third chapter the original problem of the thesis is examined. The system of equation of these models is non-linear. After writing system of equation we found the equilibrium points first. Then, we do linearisation by substituting the equilibrium point in to the Jacobian matrix. We investigated sign of the eigenvalues of these Jacobian matrices which are evaluated by equilibrium points of epidemic models. If all eigenvalues are negative the equilibrium point is stable. If at least one eigenvalue is positive, then the equilibrium point is called unstable. It is not always possible to determine the sign of eigenvalues. In such a case, we could talk about basic reproduction number. Basic reproduction number is represented by R0. If R0 < 1, all eigenvalues are negative and the equilibrium point is a stable equilibrium point. The disease disappear over time. Otherwise, if R0 > 1, at least one of the eigenvalues is positive. Also, the endemic equilibrium point exist when R0 > 1. In addition to, when R0 > 1 disease free equilibrium point is unstable and endemic equilibrium point is stable. And the disease becomes endemic. The problem is expressed as the combination of the variation of population dynamics of human, tick and birds(chicken). In all dynamics of human and tick we considered the in and outs to the compartments, outs as both in the meaning of transfers between compartments and removals such as death. The inputs to the system are either taken constants or logistic growth effects. In this thesis, we investigate the problem in three different ways. • The model which takes logistic growth both in tick and chicken populations, • The model which takes logistic growth only in chicken population, • The model which takes logistic growth only in tick population. We use a system of five ODEs to represent the interaction between chicken population, susceptible and infected populations of humans and ticks. It can be said that there is SI model between infected tick and susceptible tick, SIS model between infected human and susceptible human, and Prey-Predator model between tick and chicken. We have determined the equilibrium points for each model and investigate the stability of the equilibrium points. During the studies the reproduction numbers were found and the stability is investigated with respect to the reproduction numbers. The bifurcation analysis has also been done for tick logistic - chicken logistic model and tick logistic -chicken constant model. According to the results of the first and second models, it was observed that there was a decrease in the number of ticks when the chicken population in the environment was increased. In addition, if the frequency of unleashing of chickens into the environment is increased, then ticks can be more likely to increase among chickens is. Therefore, the number of ticks in the environment may decrease. Due to this decrease, it has mathematically shown that the Crimean Congo Haemorrhagic Fever disease decreases over time.
  • Öge
    Group classification for a higher-order boussinesq equation
    (Fen Bilimleri Enstitüsü, 2020) Hasanoğlu, Yasin ; Özdemir, Cihangir ; 637222 ; Matematik Mühendisliği
    Lie symmetry analysis of partial differential equations (PDE) is a connection for many mathematical fields, including Lie algebras, Lie groups,differential geometry, ordinary differential equations, partial differential equations and mathematical physics. This list can be extended according to the research topic, type of the PDE and so on. Finding analytical solution of a PDE is not easy in general. A powerful tool which is used by both mathematicians and physicists to find analytical solution of a PDE is transformation groups. Transformation groups, simply, can be defined as groups of which action leave the solution space of an equation invariant. One can reduce the number of independent variables of a PDE by using Lie groups and Lie algebras. The Lie algorithm to find symmetry generators can be summarized as follows: First, one generates the determining equations for the symmetries of the system. Second, these equations are solved manually or with a computer package to determine the explicit forms of the vector fields of which flows generate the transformation groups. By using Lie series and commutation relations, one can compute adjoint representations, determine the structure of the Lie algebra of the equation. From the Lie algebras, symmetry groups are obtained and actions of these symmetry groups leave the solution space of the PDE invariant.One can use Lie theory to classify differential equations. The procedure for the classification of symmetry algebras can be summarized as follows: First, find equivalence transformation of the equation. Second, find non-equivalent forms of the symmetry generator. Last, determine the invariance algebra of the equation from two and higher dimensional Lie algebras (the well-known structural results on the classification of low dimensional Lie algebras make this procedure possible). The result of this procedure is a list of representative equations with canonical invariance algebras, classified up to equivalence transformations.Symmetry classification of PDEs are studied by both mathematicians and physicists. Some mathematicians focus on Lie symmetry classification itself since it can be useful for finding integrable systems of PDEs. This thesis can be seen as an application of Lie symmetry analysis which is described above. In this thesis, a family of higher-order Boussinesq (HBq) equations of the form u_{tt}=η1 u_{xxtt}-η2 u_{xxxxtt}+(f(u))_{xx} where f(u) is an arbitary function, is considered to be classified according to the Lie symmetry algebras the equation admits depending on the formulation of the nonlinearity f(u). In Chapter 1, the literature about HBq is reviewed and main results of the thesis are given. In Chapter 2, some fundamental definitions, theorems and notations regarding Lie group analysis of differential equations is introduced. In Chapter 3, the main result of the thesis is proved, and three possible canonical forms of f(u) is obtained so that the equation admits finite-dimensional Lie algebras. In Chapter 4, some exact solutions to HBq is found by focusing on traveling wave solutions which is widely concerned in literature. Through this thesis, we believe that we contribute to the current literature on symmetry algebras of Boussinesq-type equations and also on the solutions of this PDE.
  • Öge
    Stability analysis and HOPF bifurcation in a delay-dynamical system
    ( 2020) Çalış, Yasemin ; Özdemir, Cihangir ; Demirci, Ali ; 637228 ; Matematik Mühendisliği
    Nonlinear dynamical systems have had an important place in the financial science for the last decades. These developments have helped the community understand the internal complexity of financial and economical models especially through stability, bifurcation and chaos theory. In literature, there is a great deal of studies and dynamical systems on this field. In this thesis work, the following dynamical system is considered x'=z(t)+[y(t)-a]x(t)+u(t) (1a), y'=1-by(t)-x^{2}(t)+K[y(t)-y(t-\tau)] (1b), z'=-x(t)-cz(t) (1c), u'=-dx(t)y(t)-ku(t) (1d) where a,b,c,d,k are nonnegative parameters of the system. Here K is the feedback strength and τ is time delay term, K,τϵR and K,τ≥0. State variables of the systems represent the interest rate x, the investment demand y, the price index z and average profit margin u. The main purpose of this study is to investigate the dynamic response of the system with average profit margin variable and time delay. The topics covered in the thesis study are as follows: In Section 1, we introduce the model we are considering and we present information on the properties of this system. We give a brief overview on the other financial dynamical systems available in the literature. In Section 2, we review some basic information about nonlinear stability analysis of dynamical systems, in non-delay and delay case. Section 3 includes the main work that was carried out in this thesis study. A financial model with the delayed feedback term is considered and the fixed points of this system are obtained. The distributions of the roots of the transcendental type characteristic equation is analyzed at the fixed points. After stability analysis, we determine a critical value for the time delay τ, which we name as τ _{0}. We show that the system undergoes a Hopf bifurcation at τ _{0} theoretically, switching its dynamics from stability to instability under some conditions on the parameters. Furthermore, the information obtained theoretically is represented by numerical simulations. We exhibit the stability condition of the system at the different τ values by graphs. In Section 4, we summarize our results and we conclude by some future recommendations.
  • Öge
    A dynamical systems approach to the interplay between tobacco smokers, electronic-cigarette smokers and smoking quitters
    ( 2020-07) Yıldız, Esmanur ; Özer, Saadet Seher ; Şengül, Mustafa Taylan ; 641335 ; Department of Mathematical Engineering
    In this thesis, the effect of e-cigarettes on smoking cessation is studied using the tools of dynamical systems theory. The purpose here is to examine this efficacy by representing and analysing a non-linear ODE system modelling potential smokers, tobacco smokers, e-cigarette smokers and quitters. Fundamental theories required for the interpretation of the behaviour of dynamical systems are given and some epidemiological models are analyzed. The natural behaviour of some linear physical systems is quite predictable. Contrary to that, many natural phenomena are unpredictable. So, we employ non-linear systems which are more complex and are not exactly suitable for the solution to the problem at hand as opposed to linear systems. Non-linear systems are ubiquitous throughout the natural world. As presented in this work, biological systems can be represented by non-linear systems. For instance, several disease models are generally investigated by using non-linear mathematical models. From a wider perspective, mathematical modelling is significant in describing the smoking cessation models. These models have been examined using ODE systems in view of the fact that we can analyse the spread and control of smoking with these systems. It is well known that smoking is a common social phenomenon in today's world. Since smoking is an addiction, some individuals see the use of electronic cigarette as a way of quitting tobacco smoking. We also know that the prevalence of smoking extremely affects the social behaviour of people in a population.
  • Öge
    Pseudo conharmonically symmetric manifolds
    (Institute of Science And Technology, 2014-01-21) Taşçı Yavuz, Ayşe ; Zengin, Füsun ; 509111071 ; Mathematics Engineering ; Matematik Mühendisliği
    The main concern of this thesis is to investigate an n-dimensional pseudo conharmonically symmetric Riemannian manifold (M,g) (non-conharmonically flat) whose the conharmonic curvature tensor H satisfies the condition of pseudo symmetric. Such a manifold is called a pseudo conharmonically symmetric manifold and denoted by (PCHS)n.