Browsing by Author "Gürbüz, B"

Now showing 1 - 11 of 11
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    Laguerre Collocation Method for Solving Fredholm Integro-Differential Equations with Functional Arguments
    Gürbüz, B; Sezer, M; Güler, C
    Laguerre collocation method is applied for solving a class of the Fredholm integro-differential equations with functional arguments. This method transforms the considered problem to a matrix equation which corresponds to a system of linear algebraic equations. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments. Also, the approximate solutions are corrected by using the residual correction method.
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    A numerical technique for solving functional integro-differential equations having variable bounds
    Gökmen, E; Gürbüz, B; Sezer, M
    In this paper, a collocation method based on Taylor polynomials is presented to solve the functional delay integro-differential equations with variable bounds. Using this method, we transform the functional equations to a system of linear algebraic equations. Thus, the unknown coefficients of the approximate solution are determined by solving this system. An error analysis technique based on residual function is developed to improve the numerical solution. Some numerical examples are given to illustrate the accuracy and applicability of the method. Finally, the data are examined according to the residual error estimation. All numerical computations have been performed on the computer programs.
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    Laguerre Polynomial Solutions of a Class of Initial and Boundary Value Problems Arising in Science and Engineering Fields
    Gürbüz, B; Sezer, M
    In this study, we consider high-order nonlinear ordinary differential equations with the initial and boundary conditions. These kinds of differential equations are essential tools for modelling problems in physics, biology, neurology, engineering, ecology, economy, astrophysics, physiology and so forth. Each of the mentioned problems are described by one of the following equations with the specific physical conditions: Riccati, Duffing, Emden-Fowler, Lane Emden type equations. We seek the approximate solution of these special differential equations by means of a operational matrix technique, called the Laguerre collocation method. The proposed method is based on the Laguerre series expansion and the collocation points. By using the method, the mentioned special differential equations together with conditions are transformed into a matrix form which corresponds to a system of nonlinear algebraic equations with unknown Laguerre coefficients, and thereby the problem is approximately solved in terms of Laguerre polynomials. In addition, some numerical examples are presented to demonstrate the efficiency of the proposed method and the obtained results are compared with the existing results in literature.
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    Laguerre Polynomial Solutions of a Class of Delay Partial Functional Differential Equations
    Gürbüz, B; Sezer, M
    In this study, we develop a novel matrix collocation method based on the Laguerre polynomials to find the approximate solutions of some parabolic delay differential equations with integral terms subject to appropriate initial and boundary conditions. The method reduces the solution of the mentioned equations to the solution of a matrix equation which corresponds to system of algebraic equations with unknown Laguerre coefficients. Besides, the error analysis together with numerical results are performed to illustrate the efficiency of our method computationally.
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    Modified Laguerre collocation method for solving 1-dimensional parabolic convection-diffusion problems
    Gürbüz, B; Sezer, M
    In this study, we propose a modified Laguerre collocation method based on operational matrix technique to solve 1-dimensional parabolic convection-diffusion problems arising in applied sciences. The method transforms the equation and mixed conditions of problem into a matrix equation with unknown Laguerre coefficients by means of collocation points and operational matrices. The solution of this matrix equation yields the Laguerre coefficients of the solution function. Thereby, the approximate solution is obtained in the truncated Laguerre series form. Also, to illustrate the usefulness and applicability of the method, we apply it to a test problem together with residual error estimation and compare the results with existing ones. Besides, the algorithm of the present method is given to represent the calculation of approximate solution.
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    Laguerre Polynomial Approach for Solving Nonlinear Klein-Gordon Equations
    Gürbüz, B; Sezer, M
    In this work, we develop a matrix method based on collocation points and Laguerre polynomials to obtain the numerical approximations of the onedimensional nonlinear Klein-Gordon equations. The method is applied on some numerical examples. Also, the numerical results and their exact solutions are compared with other applications. The accuracy and the effectiveness of the method are demonstrated by the results.
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    A New Computational Method Based on Laguerre Polynomials for Solving Certain Nonlinear Partial Integro Differential Equations
    Gürbüz, B; Sezer, M
    In this study, we consider some nonlinear partial integro-differential equations. Most of these equations are used as mathematical models in many problems of physics, biology, chemistry, engineering, and in other areas. Our main purpose is to propose a new numerical method based on the Laguerre and Taylor polynomials, called matrix collocation method, for the numerical solution of the mentioned nonlinear equations under the initial or boundary conditions. To show the effectiveness of this approach, some examples along with error estimations are illustrated by tables and figures.
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    Laguerre polynomial approach for solving Lane-Emden type functional differential equations
    Gürbüz, B; Sezer, M
    In this paper, a numerical method, which is called the Laguerre collocation method, for the approximate solution of Lane-Emden type functional differential equations in terms of Laguerre polynomials are derived. The method is based on the matrix relations of Laguerre polynomials and their derivatives, and reduces the solution of the Lane-Emden type functional differential equation to the solution of a matrix equation corresponding to system of algebraic equations with the unknown Laguerre coefficients. Also, some illustrative examples are included to demonstrate the validity and applicability of the proposed method. (C) 2014 Elsevier Inc. All rights reserved.
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    An Hybrid Numerical Algorithm With Error Estimation For A Class Of Functional Integro-Differential Equations
    Gürbüz, B; Sezer, M
    In this paper, a numerical algorithm based on Laguerre and Taylor polynomials is applied for solving a class of functional integro-differential equations. The considered problem is transfered to a matrix equation which corresponds to a system of linear algebraic equations by Hybrid collocation method under the mixed conditions. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments. Also, the approximate solutions are corrected by using the residual correction.
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    Laguerre Matrix-Collocation Method to Solve Systems of Pantograph Type Delay Differential Equations
    Gürbüz, B; Sezer, M
    In this study, an improved matrix method based on collocation points is developed to obtain the approximate solutions of systems of high-order pantograph type delay differential equations with variable coefficients. These kinds of systems described by the existence of linear functional argument play a critical role in defining many different phenomena and particularly, arise in industrial applications and in studies based on biology, economy, electrodynamics, physics and chemistry. The technique we have used reduces the mentioned delay system solution with the initial conditions to the solution of a matrix equation with the unknown Laguerre coefficients. Thereby, the approximate solution is obtained in terms of Laguerre polynomials. In addition, several examples along with error analysis are given to illustrate the efficiency of the method; the obtained results are scrutinized and interpreted.
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    Modified operational matrix method for second-order nonlinear ordinary differential equations with quadratic and cubic terms
    Gürbüz, B; Sezer, M
    In this study, by means of the matrix relations between the Laguerre polynomials, and their derivatives, a novel matrix method based on collocation points is modified and developed for solving a class of second-order nonlinear ordinary differential equations having quadratic and cubic terms, via mixed conditions. The method reduces the solution of the nonlinear equation to the solution of a matrix equation corresponding to system of nonlinear algebraic equations with the unknown Laguerre coefficients. Also, some illustrative examples along with an error analysis based on residual function are included to demonstrate the validity and applicability of the proposed method.