Yazar "Kutluay, Selcuk" seçeneğine göre listele

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    Application of the Exp-function method to the two dimensional sine-Gordon equation
    (Walter De Gruyter & Co, 2009) Esen, Alaattin; Kutluay, Selcuk
    In this paper, the Exp-function method is used to obtain some new generalized solitary wave solutions of the two dimensional sine-Gordon equation. In solving some other nonlinear evolution equations arising in mathematical physics, Exp-function method provides a straightforward and powerful mathematical tool.
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    An efficient Strang splitting technique combined with the multiquadric-radial basis function for the Burgers' equation
    (Springer Heidelberg, 2021) Seydaoglu, Muaz; Ucar, Yusuf; Kutluay, Selcuk
    In the present paper, two effective numerical schemes depending on a second-order Strang splitting technique are presented to obtain approximate solutions of the one-dimensional Burgers' equation utilizing the collocation technique and approximating directly the solution by multiquadric-radial basis function (MQ-RBF) method. To show the performance of both schemes, we have considered two examples of Burgers' equation. The obtained numerical results are compared with the available exact values and also those of other publishedmethods. Moreover, the computed L-2 and L-infinity error norms have been given. It is found that the presented schemes produce better results as compared to those obtained almost all the schemes present in the literature.
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    Exact solutions of nonlinear evolution equations using the extended modified Exp(-?(?)) function method
    (Tbilisi Centre Math Sci, 2019) Karaagac, Berat; Kutluay, Selcuk; Yagmurlu, Nuri Murat; Esen, Alaattin
    Obtaining exact solutions of the evolution equation is one of the very important subjects in mathematics, science and technology. For this purpose, many different methods have been constructed and developed. In this article, a new technique which is called extended modified Exp(-Omega(xi)) function method is going to be studied for seeking new exact solutions of Burger-Fisher equation and Phi Four equation. The method is capable of deriving many number of solutions. With the aid of the method, various exact solutions including trigonometric, hyperbolic and rational solutions have been obtained and using a software the graphical representation of the solutions have been presented. In conclusion, we can say that the present method can also be used for the solutions of a wide range of problems.
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    Exp-function Method for Solving the General Improved KdV Equation
    (Freund Publishing House Ltd, 2009) Kutluay, Selcuk; Esen, Alaattin
    This paper. applies He's Exp-function method to the one-dimensional general improved KdV (GIKdV) equation with nth power nonlinear term to obtain some new generalized solitary solutions and periodic solutions. It is shown that the Exp-function method, with the help of any symbolic computation packages, provides a straightforward and powerful, mathematical tool for solving many generalized nonlinear evolution equations arising in mathematical physics.
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    A Fresh Look To Exact Solutions of Some Coupled Equations
    (E D P Sciences, 2018) Karaagac, Berat; Yagmurlu, Nuri Murat; Esen, Alaattin; Kutluay, Selcuk
    This manuscript is going to seek travelling wave solutions of some coupled partial differential equations with an expansion method known as Sine Gordon-expansion method. Primarily, we are going to employ a wave transformation to partial differential equation to reduce the equations into ordinary differential equations. Then, the solution form of the handled equations is going to be constructed as polynomial of hyperbolic trig or trig functions. Finally, with the aid of symbolic computation, new exact solutions of the partial differentials equations will have been found.
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    A New Highly Accurate Numerical Scheme for Benjamin-Bona-Mahony-Burgers Equation Describing Small Amplitude Long Wave Propagation
    (Springer Basel Ag, 2023) Kutluay, Selcuk; Ozer, Sibel; Yagmurlu, Nuri Murat
    In this article, a new highly accurate numerical scheme is proposed and used for solving the initial-boundary value problem of the Benjamin-Bona-Mahony-Burgers (BBM-Burgers) equation. The BBM-Burgers equation is fully discretized by the Crank-Nicolson type method using the first-order forward finite difference approximation for the derivative in time and the standard second-order central difference approximations for all spatial derivatives. The nonlinear term appearing in the implicit scheme is firstly linearized in terms of a new dependent variable by utilizing the well known Taylor series expansion and then the resulting tri-diagonal linear algebraic equation system is solved by a direct solver method. To test the accuracy and efficiency of the scheme, three experimental test problems are taken into consideration of which the two have analytical solutions and the other one has not an analytical one. The computed results are compared with those of some studies in the literature for the same values of parameters. It is shown that the obtained results from the present method, which is stable and easy-to-use, get closer and closer to the exact solutions when the step sizes refine. This fact is also an other evidence of the accuracy and reliability of the method. Moreover, a low level data storage requirement and easy-to-implement algorithm of the present method can be considered among its notable advantages over other numerical methods. In addition, the unconditionally stability of the present scheme is shown by the von Neumann method.
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    New Solitonary Solutions for the Generalized RLW Equation by He's Exp-function Method
    (Freund Publishing House Ltd, 2009) Esen, Alaattin; Kutluay, Selcuk
    In this paper, He's Exp-function method is applied to the one-dimensional. generalized regularized long wave (RLW) equation with the nth power nonlinear term of dependent variable to obtain its some new generalized solitonary solutions and periodic solutions. It is shown that the Exp-function method, with the help of any symbolic computation packages, provides a straightforward and powerful mathematical tool for solving other generalized nonlinear evolution equations arising in mathematical physics.
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    A novel perspective for simulations of the Modified Equal-Width Wave equation by cubic Hermite B-spline collocation method
    (Elsevier, 2024) Kutluay, Selcuk; Yagmurlu, Nuri Murat; Karakas, Ali Sercan
    In the current study, the Modified Equal -Width (MEW) equation will be handled numerically by a novel technique using collocation finite element method where cubic Hermite B -splines are used as trial functions. To test the accuracy and efficiency of the method, four different experimental problems; namely, the motion of a single solitary wave, interaction of two solitary waves, interaction of three solitary waves and the birth of solitons with the Maxwellian initial condition will be investigated. In order to verify, the validity and reliability of the proposed method, the newly obtained error norms L 2 and L infinity as well as three conservation constants have been compared with some of the other numerical results given in the literature at the same parameters. Furthermore, some wave profiles of the newly obtained numerical results have been given to demonstrate that each test problem exhibits accurate physical simulations. The advantage of the proposed method over other methods is the usage of inner points at Legendre and Chebyshev polynomial roots. This advantage results in better accuracy with less number of elements in spatial direction. The results of the numerical experiments clearly reveal that the presented scheme produces more accurate results even with comparatively coarser grids.
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    Numerical Solutions of the Modified Burgers Equation by a Cubic B-spline Collocation Method
    (Springer, 2016) Kutluay, Selcuk; Ucar, Yusuf; Yagmurlu, N. Murat
    In this paper, a numerical solution of the modified Burgers equation is obtained by a cubic B-spline collocation method. In the solution process, a linearization technique based on quasi-linearization has been applied to deal with the non-linear term appearing in the equation. The computed results are compared with others selected from the available literature. The error norms and are computed and found to be sufficiently small. A Fourier stability analysis of the method is also investigated.
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    Operator time-splitting techniques combined with quintic B-spline collocation method for the generalized Rosenau-KdV equation
    (Wiley, 2019) Kutluay, Selcuk; Karta, Melike; Yagmurlu, Nuri M.
    In this article, the generalized Rosenau-KdV equation is split into two subequations such that one is linear and the other is nonlinear. The resulting subequations with the prescribed initial and boundary conditions are numerically solved by the first order Lie-Trotter and the second-order Strang time-splitting techniques combined with the quintic B-spline collocation by the help of the fourth order Runge-Kutta (RK-4) method. To show the accuracy and reliability of the proposed techniques, two test problems having exact solutions are considered. The computed error norms L-2 and L-infinity with the conservative properties of the discrete mass Q(t) and energy E(t) are compared with those available in the literature. The convergence orders of both techniques have also been calculated. Moreover, the stability analyses of the numerical schemes are investigated.
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    A Quadratic B-Spline Galerkin Approach for Solving a Coupled KdV Equation
    (Vilnius Gediminas Tech Univ, 2013) Kutluay, Selcuk; Ucar, Yusuf
    In this paper, a quadratic B-spline Galerkin finite element approach is applied to one-dimensional coupled KdV equation in order to obtain its numerical solutions. The performance of the method is examined on three test problems. Computed results are compared with the exact results and also other numerical results given in the literature. A Fourier stability analysis of the approach is also done.
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    Strang time-splitting technique for the generalised Rosenau-RLW equation
    (Indian Acad Sciences, 2021) Kutluay, Selcuk; Karta, Melike; Ucar, Yusuf
    In this paper, a numerical scheme based on quintic B-spline collocation method using the Strang splitting technique is presented for solving the generalised Rosenau-regularised long wave (RLW) equation given by appropriate initial-boundary values. For this purpose, firstly the problem is split into two subproblems such that each one includes the derivative in the direction of time. Secondly, each subproblem is reduced to a system of ordinary differential equations (ODEs) using collocation finite-element method with quintic B-splines for spatial integration. Then, the resulting ODEs for time integration are solved using the Strang time-splitting technique with the second order via the usual Runge-Kutta (RK-4) algorithm with the fourth order. To measure the accuracy and efficiency of the present scheme, a model problem with an exact solution is taken into consideration and investigated for various values of the parameter p. The error norms L2 and L infinity together with the invariants of discrete mass Q and discrete energy E have been computed and a comparison is given with other ones found in the literature. The convergence order of the present numerical scheme has also been computed. Furthermore, the stability analysis of the scheme is numerically examined.