Browsing by Author "Yalçinbas, S"

Now showing 1 - 17 of 17
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    A collocation method using Hermite polynomials for approximate solution of pantograph equations
    Yalçinbas, S; Aynigül, M; Sezer, M
    In this paper, a numerical method based on polynomial approximation, using Hermite polynomial basis, to obtain the approximate solution of generalized pantograph equations with variable coefficients is presented. The technique we have used is an improved collocation method. Some numerical examples, which consist of initial conditions, are given to illustrate the reality and efficiency of the method. In addition, some numerical examples are presented to show the properties of the given method; the present method has been compared with other methods and the results are discussed. (C) 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
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    Numeric solutions for the pantograph type delay differential equation using First Boubaker polynomials
    Akkaya, T; Yalçinbas, S; Sezer, M
    A numerical method is applied to solve the pantograph equation with proportional delay under the mixed conditions. The method is based on the truncated First Boubaker series. The solution is obtained in terms of First Boubaker polynomials. Also, illustrative examples are included to demonstrate the validity and applicability of the technique. The results obtained are compared by the known results. (C) 2013 Elsevier Inc. All rights reserved.
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    A numerical approach for solving linear integro-differential-difference equations with Boubaker polynomial bases
    Yalçinbas, S; Akkaya, T
    In this paper, a new collocation method, which is based on Boubaker polynomials, is introduced for the approximate solutions of mixed linear integro-differential-difference equations under the mixed conditions. The aim of this article is to present the applicability and validity of the technique and the comparisons are made with the existing results. The results demonstrate the accuracy and efficiency of the present work. (C) 2012 Ain Shams University. Production and hosting by Elsevier B.V. All rights reserved.
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    The approximate solution of high-order linear Volterra-Fredholm integro-differential equations in terms of Taylor polynomials
    Yalçinbas, S; Sezer, M
    In the present paper, a Taylor method is developed to find the approximate solution of high-order linear Volterra-Fredholm integro-differential equations under the mixed conditions in terms of Taylor polynomials about any point, In addition, examples that illustrate the pertinent features of the method are presented, and the results of study are discussed. (C) 2000 Elsevier Science Inc, All rights reserved.
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    APPROXIMATE SOLUTION OF HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS BY MEANS OF A NEW RATIONAL CHEBYSHEV COLLOCATION METHOD
    Yalçinbas, S; Özsoy, N; Sezer, M
    In this paper, a new approximate method for solving higher-order linear ordinary differential equations with variable coefficients under the mixed conditions is presented. The method is based on the rational Chebyshev (RC) Tau, Chebyshev and Taylor collocation methods. The solution is obtained in terms of rational Chebyshev (RC) functions. Also, illustrative examples are given to demonstrate the validity and applicability of the method.
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    Comparison of Legendre polynomial approximation and variational iteration method for the solutions of general linear Fredholm integro-differential equations
    Bildik, N; Konuralp, A; Yalçinbas, S
    In this study it is shown that the numerical solutions of linear Fredholm integro-differential equations obtained by using Legendre polynomials can also be found by using the variational iteration method. Furthermore the numerical solutions of the given problems which are solved by the variational iteration method obviously converge rapidly to exact solutions better than the Legendre polynomial technique. Additionally, although the powerful effect of the applied processes in Legendre polynomial approach arises in the situations where the initial approximation value is unknown, it is shown by the examples that the variational iteration method produces more certain solutions where the first initial function approximation value is estimated. In this paper, the Legendre polynomial approximation (LPA) and the variational iteration method (VIM) are implemented to obtain the solutions of the linear Fredholm integro-differential equations and the numerical solutions with respect to these methods are compared. (C) 2009 Elsevier Ltd. All rights reserved.
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    A Bernoulli polynomial approach with residual correction for solving mixed linear Fredholm integro-differential-difference equations
    Erdem, K; Yalçinbas, S; Sezer, M
    In this study, an approximate method based on Bernoulli polynomials and collocation points has been presented to obtain the solution of higher order linear Fredholm integro-differential-difference equations with the mixed conditions. The method we have used consists of reducing the problem to a matrix equation which corresponds to a system of linear algebraic equations. The obtained matrix equation is based on the matrix forms of Bernoulli polynomials and their derivatives by means of collocations. The solutions are obtained as the truncated Bernoulli series which are defined in the interval [a,b]. To illustrate the method, it is applied to the initial and boundary values. Also error analysis and numerical examples are included to demonstrate the validity and applicability of the technique.
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    Numerical Solution of Telegraph Equation Using Bernoulli Collocation Method
    Biçer, KE; Yalçinbas, S
    Using Bernoulli collocation method, an approximate solution of the telegraph equation has been proposed. The numerical results and comparisons with the exact solution demonstrate the validity and applicability of the technique.
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    LEGENDRE SERIES SOLUTIONS OF FREDHOLM INTEGRAL EQUATIONS
    Yalçinbas, S; Aynigül, M; Akkaya, T
    A matrix method for approximately solving linear Fredholm integral equations of the second kind is presented. The solution involves a truncated Legendre series approximation. The method is based on first taking the truncated Legendre series expansions of the functions in equation and then substituting their matrix forms into the equation. Thereby the equation reduces to a matrix equation, which corresponds to a linear system of algebraic equations with unknown Legendre coefficients. In addition, some equations considered by other authors are solved in terms of Legendre polynomials and the results are compared.
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    Numerical approach of linear delay difference equations with variable coefficients in terms of Bernoulli polynomials
    Erdem, K; Yalçinbas, S
    The aim of this study is to give a Bernoulli polynomial approximation for the solution of linear delay difference equations with variable coefficients under the mixed conditions about any point. For this purpose, Bernoulli matrix method is introduced. This method transforms linear delay difference equations and the given conditions into matrix equation which corresponds to a system of linear algebraic equation. In addition, examples that illustrate the pertinent features of the method are presented, and the results of study are discussed. Also we have discussed the accuracy of the method.
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    Approximate solutions of linear Volterra integral equation systems with variable coefficients
    Sorkun, HH; Yalçinbas, S
    In this paper, a new approximate method has been presented to solve the linear Volterra integral equation systems (VIEs). This method transforms the integral system into the matrix equation with the help of Taylor series. By merging these results, a new system which corresponds to a system of linear algebraic equations is obtained. The solution of this system yields the Taylor coefficients of the solution function. Also, this method gives the analytic solution when the exact solutions are polynomials. So as to show this capability and robustness, some systems of VIEs are solved by the presented method in order to obtain their approximate solutions. Crown Copyright (C) 2010 Published by Elsevier Inc. All rights reserved.
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    Boubaker Polynomial Approach for Solving High-Order Linear Differential-Difference Equations
    Akkaya, T; Yalçinbas, S
    A numerical method is applied to solve the pantograph equation with proportional delay under the mixed conditions. The method is based on first taking the truncated Boubaker series of the functions in the differential difference equations and then substituting their matrix forms into the equation. Hence, the result matrix equation can be solved and the unknown Boubaker coefficients can be found approximately. The solution is obtained in terms of Boubaker polynomials. Also, illustrative examples are included to demonstrate the validity and applicability of the technique. The results obtained are compared by the known results.
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    APPROXIMATE SOLUTIONS OF NONLINEAR VOLTERRA INTEGRAL EQUATION SYSTEMS
    Yalçinbas, S; Erdem, K
    The purpose of this study is to implement a new approximate method for solving system of nonlinear Volterra integral equations. The technique is based on, first, differentiating both sides of integral equations n times and then substituting the Taylor series the unknown functions in the resulting equation and later, transforming to a matrix equation. By merging these results, a new system which corresponds to a system of linear algebraic equations is obtained. The solution of this system yields the Taylor coefficients of the solution function. Some numerical results are also given to illustrate the efficiency of the method.
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    Bernoulli Polynomial Approach to High-Order Linear Differential-Difference Equations
    Erdem, K; Yalçinbas, S
    In this paper, a Bernoulli matrix method is developed to find an approximate solution of high- order linear differential difference equations with variable coeffcients under the mixed conditions. The solution is obtained in terms of Bernoulli polynomials.
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    Legendre polynomial solutions of high-order linear Fredholm integro-differential equations
    Yalçinbas, S; Sezer, M; Sorkun, HH
    In this study, a Legendre collocation matrix method is presented to solve high-order Linear Fredholm integro-differential equations under the mixed conditions in terms of Legendre polynomials. The proposed method converts the equation and conditions to matrix equations, by means of collocation points on the interval [-1,1], which corresponding to systems of linear algebraic equations with Legendre coefficients. Thus, by solving the matrix equation, Legendre coefficients and polynomial approach are obtained. Also examples that illustrate the pertinent features of the method are presented and by using the error analysis, the results are discussed. (c) 2009 Elsevier Inc. All rights reserved.
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    A Taylor polynomial approach for solving generalized pantograph equations with nonhomogenous term
    Sezer, M; Yalçinbas, S; Gülsu, M
    A numerical method for solving the generalized ( retarded or advanced) pantograph equation with constant and variable coefficients under mixed conditions is presented. The method is based on the truncated Taylor polynomials. The solution is obtained in terms of Taylor polynomials. The method is illustrated by studying an initial value problem. IIIustrative examples are included to demonstrate the validity and applicability of the technique. The results obtained are compared to the known results.
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    A Collocation Method to Solve Higher Order Linear Complex Differential Equations in Rectangular Domains
    Sezer, M; Yalçinbas, S
    In this article, a collocation method is developed to find an approximate solution of higher order linear complex differential equations with variable coefficients in rectangular domains. This method is essentially based on the matrix representations of the truncated Taylor series of the expressions in equation and their derivates, which consist of collocation points defined in the given domain. Some numerical examples with initial and boundary conditions are given to show the properties of the method. All results were computed using a program written in scientific Work Place v5.5 and Maple v12. (C) 2009 Wiley Periodicals. Inc. Numer Methods Partial Differential Eq 26: 596-611, 2010