Browsing by Author "Yalçinbaş S."
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Item The approximate solution of high-order linear Volterra-Fredholm integro-differential equations in terms of Taylor polynomials(Elsevier Inc., 2000) Yalçinbaş 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.Item On the stability of delay integro-differential equations(2007) Yeniçerioǧlu A.F.; Yalçinbaş S.Some new stability results are given for a delay integro-differential equation. A basis theorem on the behavior of solutions of delay integro-differential equations is established. As a consequence of this theorem, a stability criterion is obtained. © Association for Scientific Research.Item A Taylor polynomial approach for solving generalized pantograph equations with nonhomogenous term(2008) Sezer M.; Yalçinbaş 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.Item Legendre polynomial solutions of high-order linear Fredholm integro-differential equations(2009) Yalçinbaş S.; Sezer M.; Sorkun H.H.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. © 2009 Elsevier Inc.Item Approximate solutions of linear Volterra integral equation systems with variable coefficients(2010) Sorkun H.H.; Yalçinbaş 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. © 2010.Item A collocation method to solve higher order linear complex differential equations in rectangular domains(2010) Sezer M.; Yalçinbaş 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 WorkPlace v5.5 and Maple v12. © 2009 Wiley Periodicals, Inc.Item Comparison of Legendre polynomial approximation and variational iteration method for the solutions of general linear Fredholm integro-differential equations(2010) Bildik N.; Konuralp A.; Yalçinbaş 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. © 2009 Elsevier Ltd. All rights reserved.Item Hermite series solutions of linear fredholm integral equations(Association for Scientific Research, 2011) Yalçinbaş S.; Aynigül M.A matrix method for approximately solving linear Fredholm integral equations of the second kind is presented. The solution involves a truncated Hermite series approximation. The method is based on first taking the truncated Hermite 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 Hermite coefficients. In addition, some equations considered by other authors are solved in terms of Hermite polynomials and the results are compared. Copyright © Association for Scientific Research.Item Bernoulli polynomial approach to high-order linear differential-difference equations(2012) Erdem K.; Yalçinbaş 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. © 2012 American Institute of Physics.Item Boubaker polynomial approach for solving high-order linear differential-difference equations(2012) Akkaya T.; Yalçinbaş 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. © 2012 American Institute of Physics.Item Numerical approach of linear delay difference equations with variable coefficients in terms of Bernoulli polynomials(2012) Erdem K.; Yalçinbaş 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. © 2012 American Institute of Physics.Item A numerical approach for solving linear integro-differential-difference equations with Boubaker polynomial bases(Ain Shams University, 2012) Yalçinbaş 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. © 2011 Ain Shams University. Production and hosting by Elsevier B.V. All rights reserved.Item Fibonacci collocation method for solving linear differential-difference equations(Association for Scientific Research, 2013) Kurt A.; Yalçinbaş S.; Sezer M.This study presents a new method for the solution of mth-order linear differential-difference equations with variable coefficients under the mixed conditions. We introduce a Fibonacci collocation method based on the Fibonacci polynomials for the approximate solution. Numerical examples are included to demonstrate the applicability of the technique. The obtained results are compared by the known results.Item Taylor collocation method for solving a class of the first order nonlinear differential equations(Association for Scientific Research, 2013) Taştekin D.; Yalçinbaş S.; Sezer M.In this study, we present a reliable numerical approximation of the some first order nonlinear ordinary differential equations with the mixed condition by the using a new Taylor collocation method. The solution is obtained in the form of a truncated Taylor series with easily determined components. Also, the method can be used to solve Riccati equation. The numerical results show the effectuality of the method for this type of equations. Comparing the methodology with some known techniques shows that the existing approximation is relatively easy and highly accurate.Item Legendre collocation method for solving nonlinear differential equations(Association for Scientific Research, 2013) Güner A.; Yalçinbaş S.In this study, a matrix method based on Legendre collocation points on interval [-1,1] is proposed for the approximate solution of the some first order nonlinear ordinary differential equations with the mixed conditions in terms of Legendre polynomials. The method by means of Legendre collocation points, transforms the differential equation to a matrix equation which corresponds to a system of nonlinear algebraic equations with unknown Legendre coefficients. Also, the method can be used for solving Riccati equation. The numerical results show the effectuality of the method for this type of equations. Comparisons are made between the obtained solution and the exact solution.Item A Bernoulli polynomial approach with residual correction for solving mixed linear Fredholm integro-differential-difference equations(2013) Erdem K.; Yalçinbaş 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. © 2013 Copyright Taylor and Francis Group, LLC.Item Approximate solutions of linear Fredholm integral equations system with variable coefficients(Association for Scientific Research, 2013) Yalçinbaş S.In this paper, a new approximate method has been presented to solve the linear Fredholm integral equations system (FIEs). 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. Also, this method gives the analytic solution when the exact solutions are polynomials. So as to Show this capability and robustness, some systems of FIEs are solved by the presented method in order to obtain their approximate solutions.Item Numeric solutions for the pantograph type delay differential equation using First Boubaker polynomials(2013) Akkaya T.; Yalçinbaş 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. © 2013 Elsevier Inc. All rights reserved.Item Bernstein collocation method for solving the first order Nonlinear differential equations with the mixed Non-Linear conditions(Association for Scientific Research, 2015) Yalçinbaş S.; Görler H.In this study, we present the Bernstein matrix method to solve the first order nonlinear ordinary differential equations with the mixed non-linear conditions. By using this method, we obtain the approximate solutions in form of the Bernstein polynomials [1,2,16,17]. The method reduces the problem to a system of the nonlinear algebraic equations by means of the required matrix relations of the solutions form. By solving this system, the approximate solution is obtained. Finally, the method will be illustrated on the examples.Item Fermat collocation method for nonlinear system of first order boundary value problems(Association for Scientific Research, 2015) Taştekin D.; Yalçinbaş S.In this study, a numerical approach is proposed to obtain approximate solutions of nonlinear system of first order boundary value problem. This technique is essentially based on the truncated Fermat series and its matrix representations with collocation points. Using the matrix method, we reduce the problem to a system of nonlinear algebraic equations. Numerical examples are also given to demonstrate the validity and applicability of the presented technique. The method is easy to implement and produces accurate results. © 2015, Association for Scientific Research. All rights reserved.