Yazar "Bashan, Ali" seçeneğine göre listele

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    Approximation of the KdVB equation by the quintic B-spline differential quadrature method
    (Academic Publication Council, 2015) Bashan, Ali; Karakoc, Seydi Battal Gazi; Geyikli, Turabi
    In this paper, the Korteweg-de Vries-Burgers' (KdVB) equation is solved numerically by a new differential quadrature method based on quintic B-spline functions. The weighting coefficients are obtained by semi-explicit algorithm including an algebraic system with five-band coefficient matrix. The L-2 and L-infinity error norms and lowest three invariants I-1, I-2 and I-3 have computed to compare with some earlier studies. Stability analysis of the method is also given. The obtained numerical results show that the present method performs better than the most of the methods available in the literature.
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    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, Alaattin
    In 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.
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    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, Alaattin
    The 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.
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    A mixed method approach to the solitary wave, undular bore and boundary-forced solutions of the Regularized Long Wave equation
    (Springer Heidelberg, 2022) Bashan, Ali; Yagmurlu, N. Murat
    The aim of the present work is to obtain numerical solutions of the solitary wave, undular bore and boundary-forced problems for Regularized Long Wave (RLW) equation. For this purpose, low-order modified cubic B-spline is chosen as base functions. Crank-Nicolson formulae combined with efficient space discretization method have been applied. With the aid of Rubin and Graves type linearization technique, nonlinear terms are linearized and a solvable linear equation system has been obtained. Three significant test problems in the literature are solved successfully. The present algorithm has obtained high accurate numerical solutions of the RLW equation. Numerical results are compared with those of some earlier ones and given. The rates of the convergence are also investigated.
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    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, Alaattin
    In 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.
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    A new perspective for the numerical solution of the Modified Equal Width wave equation
    (Wiley, 2021) Bashan, Ali; Yagmurlu, Nuri Murat; Ucar, Yusuf; Esen, Alaattin
    Finding 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.
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    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, Alaattin
    In 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.
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    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, Alaattin
    The 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.
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    Numerical solution of the complex modified Korteweg-de Vries equation by DQM
    (Iop Publishing Ltd, 2016) Bashan, Ali; Ucar, Yusuf; Yagmurlu, N. Murat; Esen, Alaattin
    In 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.
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    A NUMERICAL SOLUTION OF THE MODIFIED REGULARIZED LONG WAVE (MRLW) EQUATION USING QUARTIC B-SPLINES
    (Turkic World Mathematical Soc, 2013) Karakoc, S. Battal Gazi; Geyikli, Turabi; Bashan, Ali
    In this paper, a numerical solution of the modified regularized long wave (MRLW) equation is obtained by subdomain finite element method using quartic B-spline functions. Solitary wave motion, interaction of two and three solitary waves and the development of the Maxwellian initial condition into solitary waves are studied using the proposed method. Accuracy and efficiency of the proposed method are tested by calculating the numerical conserved laws and error norms L-2 and L-infinity. The obtained results show that the method is an effective numerical scheme to solve the MRLW equation. In addition, a linear stability analysis of the scheme is found to be unconditionally stable.
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    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, Ali
    The 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.
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    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, Alaattin
    In 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.
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    Single soliton and double soliton solutions of the quadratic-nonlinear Korteweg-de Vries equation for small and long-times
    (Wiley, 2021) Bashan, Ali; Esen, Alaattin
    In this article, numerical solutions of the seven different forms of the single soliton and double soliton solutions of the Korteweg-de Vries equation are investigated. Since numerical solution of the six test problems for small-times do not exist in the literature, the present numerical results firstly are reported with exact solutions. Besides small-time solutions, long-time solutions of all test problems are obtained and compared with some of the earlier works. Present algorithm which is based on combination of finite difference method and differential quadrature method have obtained superior results than those in the given literature. Numerical and exact solutions for small-time of all test problems are plotted together for all test problems. Since the numerical results are too close to exact solutions the graphs are indistinguishable. Numerical simulations for long-time solutions are plotted and the error graphs are plotted for the end of the simulations of all test problems.
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    Two Different Methods for Numerical Solution of the Modified Burgers' Equation
    (Hindawi Ltd, 2014) Karakoc, Seydi Battal Gazi; Bashan, Ali; Geyikli, Turabi
    A numerical solution of the modified Burgers' equation (MBE) is obtained by using quartic B-spline subdomain finite element method (SFEM) over which the nonlinear term is locally linearized and using quartic B-spline differential quadrature (QBDQM) method. The accuracy and efficiency of the methods are discussed by computing L-2 and L-infinity error norms. Comparisons are made with those of some earlier papers. The obtained numerical results show that the methods are effective numerical schemes to solve the MBE. A linear stability analysis, based on the von Neumann scheme, shows the SFEM is unconditionally stable. A rate of convergence analysis is also given for the DQM.