Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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Article Citation - WoS: 7Citation - Scopus: 7Complex Ginzburg–landau Equations With Dynamic Boundary Conditions(Elsevier Ltd., 2018) Corrêa, Wellington José; Özsarı, TürkerThe initial-dynamic boundary value problem (idbvp) for the complex Ginzburg–Landau equation (CGLE) on bounded domains of RN is studied by converting the given mathematical model into a Wentzell initial–boundary value problem (ibvp). First, the corresponding linear homogeneous idbvp is considered. Secondly, the forced linear idbvp with both interior and boundary forcings is studied. Then, the nonlinear idbvp with Lipschitz nonlinearity in the interior and monotone nonlinearity on the boundary is analyzed. The local well-posedness of the idbvp for the CGLE with power type nonlinearities is obtained via a contraction mapping argument. Global well-posedness for strong solutions is shown. Global existence and uniqueness of weak solutions are proven. Smoothing effect of the corresponding evolution operator is proved. This helps to get better well-posedness results than the known results on idbvp for nonlinear Schrödinger equations (NLS). An interesting result of this paper is proving that solutions of NLS subject to dynamic boundary conditions can be obtained as inviscid limits of the solutions of the CGLE subject to same type of boundary conditions. Finally, long time behavior of solutions is characterized and exponential decay rates are obtained at the energy level by using control theoretic tools.Article Citation - WoS: 9Citation - Scopus: 9Multi-Wave and Rational Solutions for Nonlinear Evolution Equations(Walter de Gruyter GmbH, 2010) Aslan, İsmailNonlinear evolution equations always admit multi-soliton and rational solutions. The Burgers equation is used as an example, and the exp-function method is used to eluciadte the solution procedure.Article Citation - WoS: 6Citation - Scopus: 6Construction of Exact Solutions for Fractional-Type Difference-Differential Equations Via Symbolic Computation(Elsevier Ltd., 2013) Aslan, İsmailThis paper deals with fractional-type difference-differential equations by means of the extended simplest equation method. First, an equation related to the discrete KdV equation is considered. Second, a system related to the well-known self-dual network equations through a real discrete Miura transformation is analyzed. As a consequence, three types of exact solutions (with the aid of symbolic computation) emerged; hyperbolic, trigonometric and rational which have not been reported before. Our results could be used as a starting point for numerical procedures as well.Article Citation - WoS: 3Citation - Scopus: 2Application of the Exp-Function Method To the (2+1)-Dimensional Boiti-Leon Equation Using Symbolic Computation(Taylor and Francis Ltd., 2011) Aslan, İsmailLocate full-text(opens in a new window)|Full Text(opens in a new window)|View at Publisher| Export | Download | Add to List | More... International Journal of Computer Mathematics Volume 88, Issue 4, March 2011, Pages 747-761 Application of the Exp-function method to the (2+1)-dimensional Boiti-Leon-Pempinelli equation using symbolic computation (Article) Aslan, I. Department of Mathematics, Izmir Institute of Technology, Urla, Izmir 35430, Turkey View references (47) Abstract This paper deals with the so-called Exp-function method for studying a particular nonlinear partial differential equation (PDE): the (2+1)-dimensional Boiti-Leon-Pempinelli equation. The method is constructive and can be carried out in a computer with the aid of a computer algebra system. The obtained generalized solitary wave solutions contain more arbitrary parameters compared with the earlier works, and thus, they are wider. This means that our method is effective and powerful for constructing exact and explicit analytic solutions to nonlinear PDEs.Article Citation - WoS: 23Citation - Scopus: 26Travelling Wave Solutions To Nonlinear Physical Models by Means of the First Integral Method(Springer Verlag, 2011) Aslan, İsmailThis paper presents the first integral method to carry out the integration of nonlinear partial differential equations in terms of travelling wave solutions. For illustration, three important equations of mathematical physics are analytically investigated. Through the established first integrals, exact solutions are successfully constructed for the equations considered. © Indian Academy of Sciences.Article Second Order Diffraction of Water Waves by a Bottom Mounted Vertical Circular Cylinder and Some Related Numerical Problems(The American Society of Mechanical Engineers(ASME), 2007) Yılmaz, OğuzA Hankel transformation is used to obtain the second order diffraction solution of vertical cylinder of circular cross section. The improper integral over the free surface is tackled carefully. The singularity at the free surface is overcome effectively using a third order nonlinear transformation. Numerical results for free surface elevations compare well with the published data.