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Öğe An effective approach to numerical soliton solutions for the Schrodinger equation via modified cubic B-spline differential quadrature method(Pergamon-Elsevier Science Ltd, 2017) Bashan, Ali; Yagmurlu, Nuri Murat; Ucar, Yusuf; Esen, AlaattinIn this study, an effective differential quadrature method (DQM) which is based on modified cubic B-spline (MCB) has been implemented to obtain the numerical solutions for the nonlinear Schrodinger (NLS) equation. After separating the Schrodinger equation into coupled real value differential equations,we have discretized using DQM and then obtained ordinary differential equation systems. For time integration, low storage strong stability-preserving Runge-Kutta method has been used. Numerical solutions of five different test problems have been obtained. The efficiency and accuracy of the method have been measured by calculating error norms L2 and Linfinity and two lowest invariants I1 and I2. Also relative changes of invariants are given. The newly obtained numerical results have been compared with the published numerical results and a comparison has shown that the MCB-DQM is an effective numerical scheme to solve the nonlinear Schrodinger equation. (C) 2017 Elsevier Ltd. All rights reserved.Öğe 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, AlaattinObtaining 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.Öğe A Fresh Look To Exact Solutions of Some Coupled Equations(E D P Sciences, 2018) Karaagac, Berat; Yagmurlu, Nuri Murat; Esen, Alaattin; Kutluay, SelcukThis 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.Öğe A Lumped Galerkin finite element method for the generalized Hirota-Satsuma coupled KdV and coupled MKdV equations(Tbilisi Centre Math Sci, 2019) Yagmurlu, Nuri Murat; Karaagac, Berat; Esen, AlaattinIn the present study, a Lumped Galerkin finite element method using quadratic B-splines has been applied to the generalized Hirota-Satsuma coupled Korteweg de Vries (KdV) and coupled modified Korteweg-de Vries (mKdV) equations. The numerical solutions of discretized equations using Lumped Galerkin finite element method have been obtained using the fourth order Runge-Kutta method which is widely used for the solution of ordinary differential equation system. The numerical solutions obtained for various space and time values have been compared with exact ones using the error norms L-2 and L-infinity. Lumped Galerkin finite element method is an effective one which can be applied to a wide range of nonlinear evolution equations.Öğe 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 MuratIn 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.Öğe A new numerical approach to Gardner Kawahara equation in magneto-acoustic waves in plasma physics(Wiley, 2023) Ucar, Yusuf; Yagmurlu, Nuri Murat; Esen, Alaattin; Karaagac, BeratThe basic idea of this article is to investigate the numerical solutions of Gardner Kawahara equation, a particular case of extended Korteweg-de Vries equation, by means of finite element method. For this purpose, a collocation finite element method based on trigono-metric quintic B-spline basis functions is presented. The standard finite difference method is used to discretize time derivative and Crank-Nicolson approach is used to obtain more accurate numerical results. Then, von Neumann stability analysis is performed for the numerical scheme obtained using collocation finite element method. Several numerical examples are presented and discussed to exhibit the feasibility and capability of the finite element method and trigonometric B-spline basis functions. More specifically, the error norms L-2 and L-infinity are reported for numerous time and space discretization values in tables. Graphical representations of the solutions describing motion of wave are presented.Öğe A new outlook for analysis of Noyes-Field model for the nonlinear Belousov-Zhabotinsky reaction using operator splitting method(Pergamon-Elsevier Science Ltd, 2023) Karaagac, Berat; Esen, Alaattin; Ucar, Yusuf; Yagmurlu, Nuri MuratThe main idea of this paper is to investigate numerical solutions of Noyes Field model for Belousov-Zhabotinsky reaction by implementing the combination of two well-known numerical techniques. The proposed methods are collocation method based on finite elements, which is a useful and very flexible approach for solving partial differential equations (PDE), and operator splitting method which is a widely used procedure in the numerical solution of initial and boundary value problems for PDEs. Especially, for this paper, the application of collocation methods are based on trigonometric cubic B-splines. With the help of two techniques discrete schemes are investigated. Next, we presented stability of discrete schemes with Von- Neumann stability analysis. Also, we present the result of applying methods to Noyes Field model and the error norms L-2 and L-infinity to show how accurate numerical solutions to exact ones and graphical representations associated numerical results are shown.Öğe A new perspective for the numerical solution of the Modified Equal Width wave equation(Wiley, 2021) Bashan, Ali; Yagmurlu, Nuri Murat; Ucar, Yusuf; Esen, AlaattinFinding the approximate solutions to natural systems in the branch of mathematical modelling has become increasingly important and for this end various methods have been proposed. The purpose of the present paper is to obtain and analyze the numerical solutions of Modified Equal Width equation (MEW). This equation is one of those equations used to model nonlinear phenomena which has a significant role in several branches of science such as plasma physics, fluid mechanics, optics and kinetics. Firstly, for the discretization of spatial derivatives, a fifth-order quantic B-spline based scheme is directly implemented. Secondly, a forward finite difference formula is applied for the temporal discretization of derivatives with respect to time. Simulation results establish the validity and applicability of the suggested technique for a wide range of nonlinear equations. Then, the newly obtained theoretical consequences are numerically justified by the simulations and test problems. These illustrative test problems are presented verifying the superiority of the newly presented scheme compared to other existing schemes and techniques. The suggested method with symbolic computational software such as, Matlab, is proven as an effective way to obtain the soliton solutions of several nonlinear partial differential equations (PDEs). Finally, the newly obtained results are presented graphically to justify the approximate findings.Öğe A novel perspective for simulations of the MEW equation by trigonometric cubic B-spline collocation method based on Rubin-Graves type linearization(Univ Tabriz, 2022) Yagmurlu, Nuri Murat; Karakas, Ali SercanIn the present study, the Modified Equal Width (MEW) wave equation is going to be solved numerically by presenting a new technique based on the collocation finite element method in which trigonometric cubic B-splines are used as approximate functions. In order to support the present study, three test problems; namely, the motion of a single solitary wave, the interaction of two solitary waves, and the birth of solitons are studied. The newly obtained results are compared with some of the other published numerical solutions available in the literature. The accuracy of the proposed method is discussed by computing the numerical conserved laws as well as the error norms L2 and Loo.Öğe 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 SercanIn 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.Öğe Numerical approximation to a solution of the modified regularized long wave equation using quintic B-splines(Springeropen, 2013) Karakoc, Seydi Battal Gazi; Yagmurlu, Nuri Murat; Ucar, YusufIn this work, a numerical solution of the modified regularized long wave (MRLW) equation is obtained by the method based on collocation of quintic B-splines over the finite elements. A linear stability analysis shows that the numerical scheme based on Von Neumann approximation theory is unconditionally stable. Test problems including the solitary wave motion, the interaction of two and three solitary waves and the Maxwellian initial condition are solved to validate the proposed method by calculating error norms and that are found to be marginally accurate and efficient. The three invariants of the motion have been calculated to determine the conservation properties of the scheme. The obtained results are compared with other earlier results. MSC: 97N40, 65N30, 65D07, 76B25, 74S05.Öğe Numerical solution of the coupled Burgers equation by trigonometric B-spline collocation method(Wiley, 2023) Ucar, Yusuf; Yagmurlu, Nuri Murat; Yigit, Mehmet KeremIn the present study, the coupled Burgers equation is going to be solved numerically by presenting a new technique based on collocation finite element method in which cubic trigonometric and quintic B-splines are used as approximate functions. In order to support the present study, three test problems given with appropriate initial and boundary conditions are going to be investigated. The newly obtained results are compared with some of the other published numerical solutions available in the literature. The accuracy of the proposed method is discussed by computing the error norms L2$$ {L}_2 $$ and L infinity$$ {L}_{\infty } $$. A linear stability analysis of the approximation obtained by the scheme shows that the method is unconditionally stable.Öğe Numerical solutions of nonhomogeneous Rosenau type equations by quintic B-spline collocation method(Wiley, 2022) Ozer, Sibel; Yagmurlu, Nuri MuratIn this study, a numerical scheme based on a collocation finite element method using quintic B-spline functions for getting approximate solutions of nonhomogeneous Rosenau type equations prescribed by initial and boundary conditions is proposed. The numerical scheme is tested on four model problems with known exact solutions. To show how accurate results the proposed scheme produces, the error norms defined by L-2 and L-infinity are calculated. Additionally, the stability analysis of the scheme is done by means of the von Neuman method.Öğe Numerical solutions of the equal width equation by trigonometric cubic B-spline collocation method based on Rubin-Graves type linearization(Wiley, 2020) Yagmurlu, Nuri Murat; Karakas, Ali SercanIn this article, the equal width (EW) equation is going to be solved numerically. In order to show the accuracy of the presented method, six test problems namely single solitary wave, interaction of two solitary waves, interaction of three solitary waves, Maxwellian initial condition, undular bore, and soliton collision are going to be solved. For the first test problem, since it has exact solution, the error norms L-2 and L-infinity are going to be calculated and compared with some of the earlier studies existing in the literature. Moreover, the three invariants I-1, I-2, and I-3 of the given problems during the simulations are calculated and tabulated. Besides those comparisons, the relative changes of the invariants are given. Finally, a comparison of those error norms and invariants has clearly shown that the present approach obtained compatible and better results than most of the earlier works by using the same parameters.Öğe OPERATOR SPLITTING FOR NUMERICAL SOLUTIONS OF THE RLW EQUATION(Wilmington Scientific Publisher, Llc, 2018) Yagmurlu, Nuri Murat; Ucar, Yusuf; Celikkaya, IhsanIn this study, the numerical behavior of the one-dimensional Regularized Long Wave (RLW) equation has been sought by the Strang splitting technique with respect to time. For this purpose, cubic B-spline functions are used with the finite element collocation method. Then, single solitary wave motion, the interaction of two solitary waves and undular bore problems have been studied and the effectiveness of the method has been investigated. The new results have been compared with those of some of the previous studies available in the literature. The stability analysis has also been taken into account by the von Neumann method.
