FBE- Matematik Mühendisliği Lisansüstü Programı
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Matematik Mühendisliği Ana Bilim Dalı altında bir lisansüstü programı olup, yüksek lisans ve doktora düzeyinde eğitim vermektedir.
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Sustainable Development Goal "none" ile FBE- Matematik Mühendisliği Lisansüstü Programı'a göz atma
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ÖgeGardner Tipi Denklemler İçin Whitham Modülasyon Teorisi(Lisansüstü Eğitim Enstitüsü, 2021) Aslanova, Günay ; Ahmetolan, Semra ; 692598 ; Matematik MühendisliğiDispersif şok dalgaları, teorideki kısaltmayla DSW'lar, okyanuslardan atmosfere, optik fiberlere kadar birçok uygulamada görülür. Bu dalgaları incelemek için dalga teorisindeki en önemli araştırmalardan biri olarak görülen Whitham Modülasyon Teorisi (WMT) geliştirilmiştir. Bu teori ile dalga trenlerinin yavaş modülasyonunun analizi yapılabilmektedir. Modülasyon teorisi, genlik, frekans ve dalga sayısı kavramlarının zamandaki yavaş değişimini içermektedir. Bu teori ile yavaş değişkenler için kısmi diferansiyel denklemler elde edilir. "Whitham modülasyon denklemleri" veya "Whitham sistemi" olarak adlandırılan bu denklemler, oldukça zengin bir matematiksel yapıya sahiptir ve aynı zamanda dispersif şok dalgalarının tanımlanması için güçlü bir analitik araçtır. Bu tez çalışmasında ilk olarak, (2+1) boyutlu Gardner-KP denklemi için bir benzerlik dönüşümü uygulanmasıyla (1+1) boyutlu silindirik Gardner (cG) denklemi elde edilmiştir. Elde edilen bu denklem için dispersif şok dalgası çözümünü betimleyen Whitham sistemi, tanımlanan uygun Riemann değişkenleri cinsinden türetilmiştir. cG denkleminin Whitham sisteminin sayısal çözümleriyle elde edilen DSW çözümü (asimptotik çözüm) ve cG denkleminin doğrudan sayısal çözümü karşılaştırılarak aralarında uyumlu sonuçlar oluştuğu gözlemlenmiştir. Bu çalışmada incelenen ikinci problemde, (3+1) boyutlu Gardner-KP denklemi için benzer analiz yapılarak uygun başlangıç koşulu ile bu denklem (1+1) boyutlu küresel Gardner (sG) denklemine indirgenmiştir. İndirgenen denklem için WMT uygulanarak modülasyon denklemleri elde edilmiştir. Whitham sisteminin sayısal çözümlerinin bulunması sonucunda asimptotik çözümle sG denkleminin sayısal simülasyonları karşılaştırılarak aralarında tutarlılık olduğu gözlemlenmiştir. Böylece, uygun koşullar altında yüksek boyutlu denklemler için de DSW analizinin başarılı sonuçlar verdiği görülmüştür.
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ÖgeGroup classification for a higher-order boussinesq equation(Fen Bilimleri Enstitüsü, 2020) Hasanoğlu, Yasin ; Özdemir, Cihangir ; 637222 ; Matematik MühendisliğiLie 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.
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ÖgeStability analysis and HOPF bifurcation in a delay-dynamical system( 2020) Çalış, Yasemin ; Özdemir, Cihangir ; Demirci, Ali ; 637228 ; Matematik MühendisliğiNonlinear 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.