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Öğe A Collocation Finite Element Solution for Stefan Problems with Periodic Boundary Conditions(Univ Nis, Fac Sci Math, 2016) Karabenli, Hatice; Ucar, Yusuf; Aksan, E. NesligulIn this study, we are going to obtain some numerical solutions of Stefan problems given together with time-dependent periodic boundary conditions. After using variable space grid method, we have presented a numerical finite element scheme based on collocation finite element method formed with cubic B-splines. The newly obtained numerical results are presented for temperature distribution, the position and the velocity of moving boundary. It is shown that the size of domain, oscillation amplitude and oscillation frequency which are situated at the boundary condition, strongly influence the temperature distribution and position of moving boundary. The numerical results are compared with other numerical solutions obtained by using finite difference method and they are found to be in good agreement with each other.Öğe Dynamics of modified improved Boussinesq equation via Galerkin Finite Element Method(Wiley, 2020) Karaagac, Berat; Ucar, Yusuf; Esen, AlaattinThe aim of this paper is to investigate numerical solutions of modified improved Boussinesq (MIBq) equationutt=uxx+alpha mml:mfenced close=) open=( separators=u3xx+uxxtt, which is a modified type of Boussinesq equations born as an art of modelling water-wave problems in weakly dispersive medium such as surface waves in shallow waters or ion acoustic waves. For this purpose, Lumped Galerkin finite element (LGFE) method, an effective, accurate, and cost-effective method, is applied to model equation by the aid of quadratic B-spline basis. The efficiency and accuracy of the method are tested with two problems, namely, propagation solitary wave and interaction of two solitary waves. The error normsL(2)andL(infinity)have been computed in order to measure how accurate the numerical solutions. Also, the stability analysis has been investigated.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 An efficient Strang splitting technique combined with the multiquadric-radial basis function for the Burgers' equation(Springer Heidelberg, 2021) Seydaoglu, Muaz; Ucar, Yusuf; Kutluay, SelcukIn 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.Öğe Finite difference method combined with differential quadrature method for numerical computation of the modified equal width wave equation(Wiley, 2021) Bashan, Ali; Yagmurlu, N. Murat; Ucar, Yusuf; Esen, AlaattinThe aim of this study is to improve the numerical solution of the modified equal width wave equation. For this purpose, finite difference method combined with differential quadrature method with Rubin and Graves linearizing technique has been used. Modified cubic B-spline base functions are used as base function. By the combination of two numerical methods and effective linearizing technique high accurate numerical algorithm is obtained. Three main test problems are solved for various values of the coefficients. To observe the performance of the present method, the error norms of the single soliton problem which has analytical solution are calculated. Besides these error norms, three invariants are reported. Comparison of the results displays that our algorithm produces superior results than those given in the literature.Öğe A Galerkin Finite Element Method to Solve Fractional Diffusion and Fractional Diffusion-Wave Equations(Vilnius Gediminas Tech Univ, 2013) Esen, Alaattin; Ucar, Yusuf; Yagmurlu, Nuri; Tasbozan, OrkunIn the present study, numerical solutions of the fractional diffusion and fractional diffusion-wave equations where fractional derivatives are considered in the Caputo sense have been obtained by a Galerkin finite element method using quadratic B-spline base functions. For the fractional diffusion equation, the L1 discretizaton formula is applied, whereas the L2 discretizaton formula is applied for the fractional diffusion-wave equation. The error norms L2 and L1 are computed to test the accuracy of the proposed method. It is shown that the present scheme is unconditionally stable by applying a stability analysis to the approximation obtained by the proposed scheme.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 quintic B-spline based Crank-Nicolson-differential quadrature method algorithm for numerical solutions of the nonlinear Schrodinger equation(Springer Heidelberg, 2018) Bashan, Ali; Ucar, Yusuf; Yagmurlu, N. Murat; Esen, AlaattinIn the present paper, a Crank-Nicolson-differential quadrature method (CN-DQM) based on utilizing quintic B-splines as a tool has been carried out to obtain the numerical solutions for the nonlinear Schrodinger (NLS) equation. For this purpose, first of all, the Schrodinger equation has been converted into coupled real value differential equations and then they have been discretized using both the forward difference formula and the Crank-Nicolson method. After that, Rubin and Graves linearization techniques have been utilized and the differential quadrature method has been applied to obtain an algebraic equation system. Next, in order to be able to test the efficiency of the newly applied method, the error norms, L-2 and L-infinity, as well as the two lowest invariants, I-1 and I-2, have been computed. Besides those, the relative changes in those invariants have been presented. Finally, the newly obtained numerical results have been compared with some of those available in the literature for similar parameters. This comparison clearly indicates that the currently utilized method, namely CN-DQM, is an effective and efficient numerical scheme and allows us to propose to solve a wide range of nonlinear equations.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 new perspective for the numerical solutions of the cmKdV equation via modified cubic B-spline differential quadrature method(World Scientific Publ Co Pte Ltd, 2018) Bashan, Ali; Yagmurlu, N. Murat; Ucar, Yusuf; Esen, AlaattinIn the present paper, a novel perspective fundamentally focused on the differential quadrature method using modifi ed cubic B-spline basis functions are going to be applied for obtaining the numerical solutions of the complex modified Korteweg-de Vries (cmKdV) equation. In order to test the effectiveness and effciency of the present approach, three test problems, that is single solitary wave, interaction of two solitary waves and interaction of three solitary waves will be handled. Furthermore, the maximum error norm L-infinity will be calculated for single solitary wave solutions to measure the effciency and the accuracy of the present approach. Meanwhile, the three lowest conservation quantities will be calculated and also used to test the effciency of the method. In addition to these test tools, relative changes of the invariants will be calculated and presented. In the end of these processes, those newly obtained numerical results will be compared with those of some of the published papers. As a conclusion, it can be said that the present approach is an effective and effcient one for solving the cmKdV equation and can also be used for numerical solutions of other problems.Öğe A New Perspective on The Numerical Solution for Fractional Klein Gordon Equation(Gazi Univ, 2019) Karaagac, Berat; Ucar, Yusuf; Yagmurlu, N. Murat; Esen, AlaattinIn the present manuscript, a new numerical scheme is presented for solving the time fractional nonlinear Klein-Gordon equation. The approximate solutions of the fractional equation are based on cubic B-spline collocation finite element method and L2 algorithm. The fractional derivative in the given equation is handled in terms of Caputo sense. Using the methods, fractional differential equation is converted into algebraic equation system that are appropriate for computer coding. Then, two model problems are considered and their error norms are calculated to demonstrate the reliability and efficiency of the proposed method. The newly calculated error norms show that numerical results are in a good agreement with the exact solutions.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 approximation to the MEW equation for the single solitary wave and different types of interactions of the solitary waves(Taylor & Francis Ltd, 2022) Bashan, Ali; Ucar, Yusuf; Yagmurlu, N. Murat; Esen, AlaattinThe main motivation of the current study is to find out better approximate solutions of the modified equal width wave (MEW) equation. In order to achieve this aim, the power of two numerical methods are combined and an extended literature survey has been carried out. Quartic B-spline base functions have been utilized since the first-order and second-order weighting coefficients that are needed for space discretization are obtained directly. As test problems, twelve different applications of single solitary wave and four different applications of the interaction between the two solitary waves are solved successfully. All of the approximate solutions have been compared to nearly fifty various earlier applications existing in the literature. Also, the rate of the convergence is given with error norms. Comparisons show the fact that the current method obtains improved results than most of the common earlier methods.Öğe Numerical solution of the complex modified Korteweg-de Vries equation by DQM(Iop Publishing Ltd, 2016) Bashan, Ali; Ucar, Yusuf; Yagmurlu, N. Murat; Esen, AlaattinIn this paper, a method based on the differential quadrature method with quintic B-spline has been applied to simulate the solitary wave solution of the complex modified Kortewegde Vries equation (CMKdV). Three test problems, namely single solitary wave, interaction of two solitary waves and interaction of three solitary waves have been investigated. The efficiency and accuracy of the method have been measured by calculating maximum error norm L-infinity for single solitary waves having analytical solutions. Also, the three lowest conserved quantities and obtained numerical results have been compared with some of the published numerical results.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 AND STABILITY ANALYSIS OF MODIFIED BURGERS EQUATION VIA MODIFIED CUBIC B-SPLINE DIFFERENTIAL QUADRATURE METHODS(Yildiz Technical Univ, 2019) Ucar, Yusuf; Yagmurlu, N. Murat; Bashan, AliThe purpose of this work is obtain the numerical approximate solutions of the nonlinear modified Burgers' equation (MBE) via the modified cubic B-spline (MCB) differential quadrature methods (DQMs). The accuracy and effectiveness of the methods are measured and reported by finding out error normsL(2) and L-infinity. The present numerical results have been compared with some earlier studies and this comparison clearly indicates that the method is an outstanding numerical scheme for the solution of the MBE. A stability analysis has at the same time been given.Öğe NUMERICAL SOLUTIONS FOR THE FOURTH ORDER EXTENDED FISHER-KOLMOGOROV EQUATION WITH HIGH ACCURACY BY DIFFERENTIAL QUADRATURE METHOD(Yildiz Technical Univ, 2018) Bashan, Ali; Ucar, Yusuf; Yagmurlu, N. Murat; Esen, AlaattinIn this paper, modified cubic B-spline based differential quadrature method (MCB-DQM) has been used to obtain the numerical solutions for the fourth order extended Fisher-Kolmogorov equation (EFK). After using DQM for discretization of the EFK equation, ordinary differential equation systems have been obtained. For time integration, strong stability preserving Runge-Kutta method has been used. Numerical solutions of the three test problems have been investigated. The efficiency and accuracy of the method have been measured by calculating error norms L-2 and L-infinity. The present obtained numerical results have been compared with the published numerical results and the comparison has shown that the method is an effective numerical scheme to solve the EFK equation.Öğe Numerical solutions of Boussinesq equation using Galerkin finite element method(Wiley, 2021) Ucar, Yusuf; Esen, Alaattin; Karaagac, BeratIn this study, Galerkin finite element method has been applied to good Boussinesq (GBq) and bad Boussinesq (BBq) equations which are examples of Boussinesq type equations. To apply the method, cubic B-spline basis functions are taken as element and weight functions. The solutions of the numerical schemes have been obtained using the fourth order Runge-Kutta method. The error norms L-2 and L-infinity have been used to test how compatible they obtained numerical solutions with those of exact ones. As numerical examples, solitary wave motion and interaction of solitary waves have been investigated for both GBq and BBq equation. Also blow-up solutions related to the interaction of two solitary waves are considered for GBq equation.Öğe Numerical Solutions of the Improved Boussinesq Equation by the Galerkin Quadratic B-Spline Finite Element Method(Univ Nis, Fac Sci Math, 2018) Karaagac, Berat; Ucar, Yusuf; Esen, AlaattinIn this paper, we are going to obtain numerical solutions of the improved Boussinesq equation with the aid of Galerkin quadratic B-spline finite element method. To test the accuracy and efficiency of the current method, four test problems have been used. These are solitary wave movement, interaction of two solitary waves, wave break-up and blow-up of solutions. Their results have been compared with those available in the literature for different values of space and time steps. Also, the error norms L-2 and L-infinity have been computed and presented in comparison.
