Kısmi türevli denklemlerin çözümlerinin sürekli bağımlılığı

06.03.2018 tarihli ve 30352 sayılı Resmi Gazetede yayımlanan “Yükseköğretim Kanunu İle Bazı Kanun Ve Kanun Hükmünde Kararnamelerde Değişiklik Yapılması Hakkında Kanun” ile 18.06.2018 tarihli “Lisansüstü Tezlerin Elektronik Ortamda Toplanması, Düzenlenmesi ve Erişime Açılmasına İlişkin Yönerge” gereğince tam metin erişime açılmıştır.Anahtar kelimeler: Sürekli Bağımlılık, Suspension Bridge Denklemi Bu tezde, kısmi türevli diferansiyel denklemlerin çözümlerinin katsayılara sürekli bağımlılığı ele alınmıştır. 6 bölümden oluşan bu çalışmanın birinci bölümünde kısmi türevli diferansiyel denklemler ile ilgili yapılan geçmiş çalışmalar hakkında bilgi verilmiştir. İkinci bölümde bu tezde kullanılan temel tanım ve kavramlara yer verilmiştir. Üçüncü bölümde, A.O. Celebi, V.K. Kalantarov, D. Uğurlu tarafından yazılan "On continuous dependence on coefficients of the Brinkman–Forchheimer eqations" adlı makale incelenmiştir. Dördüncü bölümde, G.N. Aliyeva ve V.K. Kalantarov tarafından yazılan "Structural stability for Fitzhugh-Nagumo equations" adlı makale incelenmiştir. Beşinci bölümde ise daha önce çalışılmamış olan asma köprü denkleminin çözümlerinin katsayılarına sürekli bağımlılığı ayrıntılı olarak incelenmiştir.Keywords: Continuous Dependence, Suspension Bridge Equation In this paper, the continuous dependence of the solutions of partial differential equations to the coefficients is discussed. In the first part of this study consisting of 6 chapters, information about previous studies on partial differential equations is given. In the second chapter, the basic definitions and concepts used in this thesis are given. In the third chapter, the article "On continuous dependence on coefficients of the Brinkman Forchheimer equations", written by A.O. Celebi, V.K. Kalantarov, D. Ugurlu, was examined. In the fourth chapter, the article "Structural stability for Fitzhugh-Nagumo equations" written by G.N. Aliyeva and V.K. Kalantarov was examined. In the fifth part, continuous dependence on the coefficients of the solutions of the Suspension bridge equation, which has not been studied before, has been examined in detail

Analysis of the Brinkman-Forchheimer equations with slip boundary conditions

In this work, we study the Brinkman-Forchheimer equations driven under slip boundary conditions of friction type. We prove the existence and uniqueness of weak solutions by means of regularization combined with the Faedo-Galerkin approach. Next we discuss the continuity of the solution with respect to Brinkman's and Forchheimer's coefficients. Finally, we show that the weak solution of the corresponding stationary problem is stable

Analysis of the Brinkman-Forchheimer equations with slip boundary conditions

In this work, we study the Brinkman-Forchheimer equations driven under slip boundary conditions of friction type. We prove the existence and uniqueness of weak solutions by means of regularization combined with the Faedo-Galerkin approach. Next we discuss the continuity of the solution with respect to Brinkman's and Forchheimer's coefficients. Finally, we show that the weak solution of the corresponding stationary problem is stable

Continuous data assimilation for the three-dimensional Brinkman-Forchheimer-extended Darcy model

In this paper we introduce and analyze an algorithm for continuous data assimilation for a three-dimensional Brinkman-Forchheimer-extended Darcy (3D BFeD) model of porous media. This model is believed to be accurate when the flow velocity is too large for Darcy's law to be valid, and additionally the porosity is not too small. The algorithm is inspired by ideas developed for designing finite-parameters feedback control for dissipative systems. It aims to obtaining improved estimates of the state of the physical system by incorporating deterministic or noisy measurements and observations. Specifically, the algorithm involves a feedback control that nudges the large scales of the approximate solution toward those of the reference solution associated with the spatial measurements. In the first part of the paper, we present few results of existence and uniqueness of weak and strong solutions of the 3D BFeD system. The second part is devoted to the setting and convergence analysis of the data assimilation algorithm

Robustness of Regularity for the 33D Convective Brinkman-Forchheimer Equations

We prove a robustness of regularity result for the $3$D convective Brinkman-Forchheimer equations \partial_tu -\mu\Delta u + (u \cdot \nabla)u + \nabla p + \alpha u + \beta\abs{u}^{r - 1}u = f, for the range of the absorption exponent $r \in [1, 3]$ (for $r > 3$ there exist global-in-time regular solutions), i.e. we show that strong solutions of these equations remain strong under small enough changes of the initial condition and forcing function. We provide a smallness condition which is similar to the robustness conditions given for the $3$D incompressible Navier-Stokes equations by Chernyshenko et al. (2007) and Dashti & Robinson (2008).Comment: 22 page