Bernstein Operational Matrix with Error Analysis for Solving High Order Delay Differential Equations

2016 ◽  
Vol 3 (3) ◽  
pp. 1749-1762 ◽  
Author(s):  
A. Bataineh ◽  
O. Isik ◽  
N. Aloushoush ◽  
N. Shawagfeh
Keyword(s):  
Differential Equations ◽  
Error Analysis ◽  
Delay Differential Equations ◽  
Operational Matrix ◽  
High Order ◽  
Delay Differential

Numerical Simulation of Fractional Delay Differential Equations Using the Operational Matrix of Fractional Integration for Fractional-Order Taylor Basis

2021 ◽  
Vol 6 (1) ◽  
pp. 10
Author(s):  
İbrahim Avcı 
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Numerical Solutions ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Absolute Error ◽  
Algebraic Equations ◽  
Fractional Delay ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

In this paper, we consider numerical solutions for a general form of fractional delay differential equations (FDDEs) with fractional derivatives defined in the Caputo sense. A fractional integration operational matrix, created using a fractional Taylor basis, is applied to solve these FDDEs. The main characteristic of this approach is, by utilizing the operational matrix of fractional integration, to reduce the given differential equation to a set of algebraic equations with unknown coefficients. This equation system can be solved efficiently using a computer algorithm. A bound on the error for the best approximation and fractional integration are also given. Several examples are given to illustrate the validity and applicability of the technique. The efficiency of the presented method is revealed by comparing results with some existing solutions, the findings of some other approaches from the literature and by plotting absolute error figures.

scholarly journals Legendre wavelet solution of high order nonlinear ordinary delay differential equations

2019 ◽  
Vol 43 (3) ◽  
pp. 1339-1352 ◽  
Author(s):  
Sevin GÜMGÜM ◽  
Demet ERSOY ÖZDEK ◽  
Gökçe ÖZALTUN
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
High Order ◽  
Delay Differential

scholarly journals An Operational Matrix Method Based on Poly-Bernoulli Polynomials for Solving Fractional Delay Differential Equations

Computation ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 82
Author(s):  
Chang Phang ◽  
Yoke Teng Toh ◽  
Farah Suraya Md Nasrudin
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Operational Matrix ◽  
Bernoulli Polynomials ◽  
Variable Coefficients ◽  
Numerical Examples ◽  
Operational Matrix Method ◽  
Fractional Delay ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

In this work, we derive the operational matrix using poly-Bernoulli polynomials. These polynomials generalize the Bernoulli polynomials using a generating function involving a polylogarithm function. We first show some new properties for these poly-Bernoulli polynomials; then we derive new operational matrix based on poly-Bernoulli polynomials for the Atangana–Baleanu derivative. A delay operational matrix based on poly-Bernoulli polynomials is derived. The error bound of this new method is shown. We applied this poly-Bernoulli operational matrix for solving fractional delay differential equations with variable coefficients. The numerical examples show that this method is easy to use and yet able to give accurate results.

scholarly journals A Galerkin-like approach to solve multi-pantograph type delay differential equations

Filomat ◽  
2018 ◽  
Vol 32 (2) ◽  
pp. 409-422 ◽  
Author(s):  
Şuayip Yüzbaşı ◽  
Murat Karaçayır
Keyword(s):  
Approximate Solution ◽  
Linear System ◽  
Differential Equations ◽  
Error Analysis ◽  
Delay Differential Equations ◽  
Inner Product ◽  
Method Error ◽  
Correction Technique ◽  
Residual Correction ◽  
Delay Differential

In this paper, a Galerkin-like approach is presented to numerically solve multi-pantograph type delay differential equations. The method includes taking inner product of a set of monomials with a vector obtained from the equation under consideration. The resulting linear system is then solved, yielding a polynomial as the approximate solution. We also provide an error analysis and discuss the technique of residual correction, which aims to increase the accuracy of the approximate solution. Lastly, the method, error analysis and the residual correction technique are illustrated with several examples. The results are also compared with numerous existing methods from the literature.

scholarly journals Word series high-order averaging of highly oscillatory differential equations with delay

2019 ◽  
Vol 4 (2) ◽  
pp. 445-454 ◽  
Author(s):  
J. M. Sanz-Serna ◽  
Beibei Zhu
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
High Order ◽  
Constant Delay ◽  
Integer Multiple ◽  
Averaged Equations ◽  
Highly Oscillatory ◽  
Highly Oscillatory Differential Equations ◽  
Delay Differential ◽  
Equations With Delay

AbstractWe show that, when the delay is an integer multiple of the forcing period, it is possible to obtain easily high-order averaged versions of periodically forced systems of delay differential equations with constant delay. Our approach is based on the use of word series techniques to obtain high-order averaged equations for differential equations without delay.

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