Benjamin-bona-mahony denklemleri

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: BBM denklemi, varlık ve teklik, BBMB denklemi, sürekli bağımlılık Bu tez 6 bölümden oluşmaktadır. Tezin birinci bölümünde, Benjamin-Bona-Mahony denklemleri ile ilgili yapılan geçmiş çalışmalar hakkında bilgi verilmiştir. İkinci bölümde, bu tezde kullanılan temel tanım ve kavramlara yer verilmiştir. Üçüncü bölümde, Bo Lu, Guanxiu Yuan ve Jinku Yang tarafından yazılan "A Class of Exact Solutions of the BBM Equations" isimli makale incelenmiştir. Dördüncü bölümde, L. A. Medeiros ve G. Perla Menzala tarafından yazılan "Existence and Uniqueness for Periodic Solutions of the Benjamin-Bona-Mahony Equation'' isimli makale incelenmiştir. Beşinci bölümde ise daha önce çalışılmamış olan Benjamin-Bona-Mahony-Burger denkleminin çözümlerinin katsayılara sürekli bağımlılığı incelenmiştir. Altıncı bölümde ise tez çalışmasından elde edilen sonuçlar belirtilmiştir. Çalışma literatürde bilinen sonuçlar araştırılarak oluşturulmuştur.Keywords: BBM Equation, Existence and Uniqueness Theorem, BBMB Equation, Continuous Dependence This thesis consists of six chapters. In the first chapter, information about past studies on Benjamin-Bona-Mahony equations is given. In the second chapter, the basic definitions and concepts used in this thesis are given. In the third chapter, the article entitled "A Class of Exact Solutions of the BBM Equations" written by Bo Lu, Guanxiu Yuan and Jinku Yang is investigated. In the fourth chapter, the article entitled "Existence and Uniqueness for Periodic Solutions of the Benjamin-Bona-Mahony Equation'' written by L. A. Medeiros and G. Perla Menzala is examined. In the fifth chapter, continuous dependence on the coefficients of the Benjamin-Bona-Mahony-Burger equation solutions which has not been studied before is examined. Finally in the sixth of chapter, the results obtained from the thesis are stated. This work is performed by investigating the results known in literature

Finite volume methods for unidirectional dispersive wave models

We extend the framework of the finite volume method to dispersive unidirectional water wave propagation in one space dimension. In particular we consider a KdV-BBM type equation. Explicit and IMEX Runge-Kutta type methods are used for time discretizations. The fully discrete schemes are validated by direct comparisons to analytic solutions. Invariants conservation properties are also studied. Main applications include important nonlinear phenomena such as dispersive shock wave formation, solitary waves and their various interactions.Comment: 25 pages, 12 figures, 51 references. Other authors papers can be downloaded at http://www.lama.univ-savoie.fr/~dutykh

Finite volume methods for unidirectional dispersive wave model

We extend the framework of the finite volume method to dispersive unidirectional water wave propagation in one space dimension. In particular, we consider a KdV–BBM-type equation. Explicit and implicit–explicit Runge–Kutta-type methods are used for time discretizations. The fully discrete schemes are validated by direct comparisons to analytic solutions. Invariants’ conservation properties are also studied. Main applications include important nonlinear phenomena such as dispersive shock wave formation, solitary waves, and their various interaction

Unique continuation property and control for the Benjamin-Bona-Mahony equation on the torus

We consider the Benjamin-Bona-Mahony (BBM) equation on the one dimensional torus T = R/(2{\pi}Z). We prove a Unique Continuation Property (UCP) for small data in H^1(T) with nonnegative zero means. Next we extend the UCP to certain BBM-like equations, including the equal width wave equation and the KdV-BBM equation. Applications to the stabilization of the above equations are given. In particular, we show that when an internal control acting on a moving interval is applied in BBM equation, then a semiglobal exponential stabilization can be derived in H^s(T) for any s \geq 1. Furthermore, we prove that the BBM equation with a moving control is also locally exactly controllable in H^s(T) for any s \geq 0 and globally exactly controllable in H s (T) for any s \geq 1

The Whitham Equation as a Model for Surface Water Waves

The Whitham equation was proposed as an alternate model equation for the simplified description of uni-directional wave motion at the surface of an inviscid fluid. As the Whitham equation incorporates the full linear dispersion relation of the water wave problem, it is thought to provide a more faithful description of shorter waves of small amplitude than traditional long wave models such as the KdV equation. In this work, we identify a scaling regime in which the Whitham equation can be derived from the Hamiltonian theory of surface water waves. The Whitham equation is integrated numerically, and it is shown that the equation gives a close approximation of inviscid free surface dynamics as described by the Euler equations. The performance of the Whitham equation as a model for free surface dynamics is also compared to two standard free surface models: the KdV and the BBM equation. It is found that in a wide parameter range of amplitudes and wavelengths, the Whitham equation performs on par with or better than both the KdV and BBM equations.Comment: 14 pages, 4 figure

Solitary waves, periodic and elliptic solutions to the Benjamin, Bona & Mahony (BBM) equation modified by viscosity

In this paper, we use a traveling wave reduction or a so-called spatial approximation to comprehensively investigate periodic and solitary wave solutions of the modified Benjamin, Bona & Mahony equation (BBM) to include both dissipative and dispersive effects of viscous boundary layers. Under certain circumstances that depend on the traveling wave velocity, classes of periodic and solitary wave like solutions are obtained in terms of Jacobi elliptic functions. An ad-hoc theory based on the dissipative term is presented, in which we have found a set of solutions in terms of an implicit function. Using dynamical systems theory we prove that the solutions of \eqref{BBMv} experience a transcritical bifurcation for a certain velocity of the traveling wave. Finally, we present qualitative numerical results.Comment: 14 pages, 11 figure

Nongauge bright soliton of the nonlinear Schrodinger (NLS) equation and a family of generalized NLS equations

We present an approach to the bright soliton solution of the NLS equation from the standpoint of introducing a constant potential term in the equation. We discuss a `nongauge' bright soliton for which both the envelope and the phase depend only on the traveling variable. We also construct a family of generalized NLS equations with solitonic sech^p solutions in the traveling variable and find an exact equivalence with other nonlinear equations, such as the Korteveg-de Vries and Benjamin-Bona-Mahony equations when p=2Comment: ~4 pages, 3 figures, 16 references, published versio