A new Fibonacci type collocation procedure for boundary value problems

Koc, Ayse Betul; Çakmak, Musa; Kurnaz, Aydın; Uslu, Kemal

doi:10.1186/1687-1847-2013-262

Cited by 18 publications

(5 citation statements)

References 28 publications

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“…Increasing d yields a small error in the norm for Example . The rational approximation method is given by a comparison with some other numerical methods, including the Taylor matrix method , the exponential approximation , the Fibonacci‐type collocation method , and the Legendre matrix method .…”

Section: Discussionmentioning

confidence: 99%

“…Example Let us consider the following second‐order linear delay difference equation:

y^{^{′′}} (x) + x y^{'} (x) - 2 y (x) = x \cos (x) - 3 \sin (x)

with the conditions y (0) = 0 . The exact solution is y ( x )= sin( x ).…”

Section: Numerical Examplesmentioning

confidence: 99%

“…Moreover, as in Figure , the absolute error is represented exactly by

e_{20, 18}^{*}

. ‐type collocation method and Legendre matrix method on [−1,1]. The results are given in Table .…”

Section: Numerical Examplesmentioning

confidence: 99%

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A rational approximate solution for generalized pantograph‐delay differential equations

Işık

Turkoglu²

2015

Math Methods in App Sciences

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In this paper, a new rational approximation based on a rational interpolation and collocation method is proposed for the solutions of generalized pantograph equations. A comprehensive error analysis is provided. The first part of the error analysis gives an upper bound for the absolute error. The second part is based on residual error procedure that estimates the absolute error. Some numerical examples are given to illustrate the method. The theoretical results support the numerical results.

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Section: Discussionmentioning

confidence: 99%

“…Example Let us consider the following second‐order linear delay difference equation:

y^{^{′′}} (x) + x y^{'} (x) - 2 y (x) = x \cos (x) - 3 \sin (x)

with the conditions y (0) = 0 . The exact solution is y ( x )= sin( x ).…”

Section: Numerical Examplesmentioning

confidence: 99%

See 1 more Smart Citation

A rational approximate solution for generalized pantograph‐delay differential equations

Işık

Turkoglu²

2015

Math Methods in App Sciences

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“…It is seen with the Maple 14 program that arbitrarily selected this point does not affect the result). Hence, at the collocation points in (15), derivatives of the unknown function  can be written in the matrix form as…”

Section: Transactions For the Solution Of The Equationmentioning

confidence: 99%

Fibonacci Operational Matrix Algorithm For Solving Differential Equations Of Lane-Emden Type

Çakmak

2019

Sakarya University Journal of Science

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The aim of this study is to provide an effective and accurate technique for solving differential equations of Lane-Emden type as initial value problems. In this study, a numerical method called Fibonacci polynomial approximation method (FPAM) establish for approximate solution of Lane-Emden type differential equations by using Fibonacci polynomials. A matrix equation can be solved depending on the reduced form of the Lane-Emden type differential equations, which is characterized by an algebraic equation system, with the matrix relations of Fibonacci polynomials and their derivatives and their unknown Fibonacci coefficients. In addition, numerical results are given by comparisons to confirm the reliability of the proposed method for Lane-Emden type differential equations.

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“…Over the last three decades, the scientists have paid much attention to spectral methods due to their high accuracy (see, for instance, [1][2][3][4][5][6][7], and the references therein). On the other hand, spectral methods typically give rise to full matrices, partially negating the gain in efficiency due to the fewer number of grid points.…”

Section: Introductionmentioning

confidence: 99%

A highly accurate Jacobi collocation algorithm for systems of high‐order linear differential–difference equations with mixed initial conditions

Bhrawy

Doha

Băleanu

et al. 2015

Math Methods in App Sciences

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In this paper, a shifted Jacobi–Gauss collocation spectral algorithm is developed for solving numerically systems of high‐order linear retarded and advanced differential–difference equations with variable coefficients subject to mixed initial conditions. The spatial collocation approximation is based upon the use of shifted Jacobi–Gauss interpolation nodes as collocation nodes. The system of differential–difference equations is reduced to a system of algebraic equations in the unknown expansion coefficients of the sought‐for spectral approximations. The convergence is discussed graphically. The proposed method has an exponential convergence rate. The validity and effectiveness of the method are demonstrated by solving several numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier. Copyright © 2015 John Wiley & Sons, Ltd.

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A new Fibonacci type collocation procedure for boundary value problems

Cited by 18 publications

References 28 publications

A rational approximate solution for generalized pantograph‐delay differential equations

A rational approximate solution for generalized pantograph‐delay differential equations

Fibonacci Operational Matrix Algorithm For Solving Differential Equations Of Lane-Emden Type

A highly accurate Jacobi collocation algorithm for systems of high‐order linear differential–difference equations with mixed initial conditions

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