Browsing by Author "Sezer M."
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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 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 Approximate solution of multi-pantograph equation with variable coefficients(2008) Sezer M.; yalçinbaş S.; Şahin N.This paper deals with the approximate solution of multi-pantograph equation with nonhomogenous term in terms of Taylor polynomials. The technique we have used is based on a Taylor matrix method. In addition, some numerical examples are presented to show the properties of the given method and the results are discussed. © 2007 Elsevier B.V. All rights reserved.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 solution of higher order linear differential equations by means of a new rational Chebyshev collocation method(Association for Scientific Research, 2010) 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. © Association for Scientific Research.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 A collocation method using Hermite polynomials for approximate solution of pantograph equations(2011) Yalçinbaç 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. © 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.Item An exponential approximation for solutions of generalized pantograph-delay differential equations(2013) Yüzbaşi Ş.; Sezer M.In this paper, a new matrix method based on exponential polynomials and collocation points is proposed for solutions of pantograph equations with linear functional arguments under the mixed conditions. Also, an error analysis technique based on residual function is developed for the suggested method. Some examples are given to demonstrate the validity and applicability of the method and the comparisons are made with existing results. © 2013 Elsevier Inc.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 A Chebyshev series approximation for linear second- order partial differential equations with complicated conditions(Gazi Universitesi, 2013) Yuksel G.; Sezer M.The purpose of this study is to present a new collocation method for the solution of second-order, linear partial differential equations (PDEs) under the most general conditions. The method has improved from Chebyshev matrix method, which has been given for solving of ordinary differential, integral and integro-differential equations. The method is based on the approximation by the truncated bivariate Chebyshev series. PDEs and conditions are transformed into the matrix equations, which corresponds to a system of linear algebraic equations with the unknown Chebyshev coefficients, via Chebyshev collocation points. Combining these matrix equations and then solving the system yields the Chebyshev coefficients of the solution function. Finally, the effectiveness of the method is illustrated in several numerical experiments and error analysis is performed.Item Müntz-Legendre polynomial solutions of linear delay Fredholm integro-differential equations and residual correction(Association for Scientific Research, 2013) Yüzbaşi S.; Gök E.; Sezer M.In this paper, we consider the Müntz-Legendre polynomial solutions of the linear delay Fredholm integro-differential equations and residual correction. Firstly, the linear delay Fredholm integro-differential equations are transformed into a system of linear algebraic equations by using by the matrix operations of the Müntz-Legendre polynomials and the collocation points. When this system is solved, the Müntz-Legendre polynomial solution is obtained. Then, an error estimation is presented by means of the residual function and the Müntz-Legendre polynomial solutions are improved by the residual correction method. The technique is illustrated by studying the problem for an example. The obtained results show that error estimation and the residual correction method is very effective.Item Exponential collocation method for solutions of singularly perturbed delay differential equations(2013) Yüzbaşi Ş.; Sezer M.This paper deals with the singularly perturbed delay differential equations under boundary conditions. A numerical approximation based on the exponential functions is proposed to solve the singularly perturbed delay differential equations. By aid of the collocation points and the matrix operations, the suggested scheme converts singularly perturbed problem into a matrix equation, and this matrix equation corresponds to a system of linear algebraic equations. Also, an error analysis technique based on the residual function is introduced for the method. Four examples are considered to demonstrate the performance of the proposed scheme, and the results are discussed. © 2013 Şuayip Yüzbaşi and Mehmet Sezer.Item Numerical solution of duffing equation by using an improved taylor matrix method(2013) Bülbül B.; Sezer M.We have suggested a numerical approach, which is based on an improved Taylor matrix method, for solving Duffing differential equations. The method is based on the approximation by the truncated Taylor series about center zero. Duffing equation and conditions are transformed into the matrix equations, which corresponds to a system of nonlinear algebraic equations with the unknown coefficients, via collocation points. Combining these matrix equations and then solving the system yield the unknown coefficients of the solution function. Numerical examples are included to demonstrate the validity and the applicability of the technique. The results show the efficiency and the accuracy of the present work. Also, the method can be easily applied to engineering and science problems. © 2013 Berna Bülbül and Mehmet Sezer.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 A collocation method for solving fractional riccati differential equation(2013) Öztürk Y.; Anapali A.; Gülsu M.; Sezer M.We have introduced a Taylor collocation method, which is based on collocation method for solving fractional Riccati differential equation with delay term. This method is based on first taking the truncated Taylor expansions of the solution function in the fractional Riccati differential equation and then substituting their matrix forms into the equation. Using collocation points, we have the system of nonlinear algebraic equation. Then, we solve the system of nonlinear algebraic equation using Maple 13, and we have the coefficients of the truncated Taylor sum. In addition, illustrative examples are presented to demonstrate the effectiveness of the proposed method. Comparing the methodology with some known techniques shows that the present approach is relatively easy and highly accurate. © 2013 Yalçin Öztürk et al.Item A new approach to numerical solution of nonlinear Klein-Gordon equation(2013) Bülbül B.; Sezer M.A numerical method based on collocation points is developed to solve the nonlinear Klein-Gordon equations by using the Taylor matrix method. The method is applied to some test examples and the numerical results are compared with the exact solutions. The results reveal that the method is very effective, simple, and convenient. In addition, an error estimation of proposed method is presented. © 2013 Berna Bülbül and Mehmet Sezer.Item Fibonacci collocation method for solving high-order linear fredholm integro-differential-difference equations(2013) Kurt A.; Yalçnbaş S.; Sezer M.A new collocation method based on the Fibonacci polynomials is introduced for the approximate solution of high order-linear Fredholm integro-differential- difference equations with the mixed conditions. The proposed method is analyzed to show the convergence of the method. Some further numerical experiments are carried out to demonstrate the method. © 2013 Ayşe Kurt et al.Item A collocation method to find solutions of linear complex differential equations in circular domains(2013) Yüzbaşi Ş.; Sezer M.In this study, we introduce a collocation approach for solving high-order linear complex differential equations in circular domain. By using collocation points defined in a circular domain and Bessel functions of the first kind, this method transforms the linear complex differential equations into a matrix equation. The matrix equation corresponds to a system of linear equations with the unknown Bessel coefficients. Proposed method gives the analytic solution when the exact solutions are polynomials. Numerical examples are given to demonstrate the validity and applicability of the technique and the comparisons are made with existing results. The results obtained from the examples demonstrate the efficiency and accuracy of the present work. All of the numerical computations have been computed on computer using a code written in Matlab. © 2013 Elsevier Inc. All rights reserved.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.