Browsing by Subject "Polynomial approximation"

Now showing 1 - 7 of 7
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    Estimating roots of polynomials using perturbation theory
    (2007) Pakdemirli M.; Yurtsever H.A.
    Perturbation theory and the order of magnitude of terms are employed to develop two theorems. The theorems may be useful to estimate the order of magnitude of the roots of a polynomial a priori before solving the equation. The theorems are developed for two special types of polynomials of arbitrary order with their coefficients satisfying certain conditions. Numerical applications of the theorems are presented as examples. © 2006 Elsevier Inc. All rights reserved.
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    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.
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    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.
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    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.
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    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.
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    On the numerical solution of fractional differential equations with cubic nonlinearity via matching polynomial of complete graph
    (Springer, 2019) Kürkçü Ö.K.; Aslan E.İ.; Sezer M.
    This study deals with a generalized form of fractional differential equations with cubic nonlinearity, employing a matrix-collocation method dependent on the matching polynomial of complete graph. The method presents a simple and efficient algorithmic infrastructure, which contains a unified matrix expansion of fractional-order derivatives and a general matrix relation for cubic nonlinearity. The method also performs a sustainable approximation for high value of computation limit, thanks to the inclusion of the matching polynomial in matrix system. Using the residual function, the convergence and error estimation are investigated via the second mean value theorem having a weight function. In comparison with the existing results, highly accurate results are obtained. Moreover, the oscillatory solutions of some model problems arising in several applied sciences are simulated. It is verified that the proposed method is reliable, efficient and productive. © 2019, Indian Academy of Sciences.
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    Computing the weighted neighbor isolated tenacity of interval graphs in polynomial time
    (John Wiley and Sons Inc, 2021) Aslan E.; Tosun M.A.
    Weighted graphs in graph theory are created by weighing different values depending on the importance of connections or centers in a graph model. Networks can be modeled with graphs such that the devices and centers correspond to the vertices and connections correspond to the edges. In these networks, weight can be assigned to the vertices for the workload and importance of the devices and centers, so that planning such as security and cost can be made in advance in the design of the network. Network reliability and security is an important issue in the computing area. There are several parameters for vulnerability measurement values of these networks modeled with graphs. We recommend the weighted conversion of the neighbor isolated tenacity parameter for this topic. It is known that tenacity, which is the basis of this parameter, is NP-hard. But polynomial solutions can be created in interval graphs, which is a special graph from the perfect graph class. In this article, polynomial time algorithm is given to calculate weighted neighbor isolated tenacity of the interval graphs. © 2020 Wiley Periodicals LLC