scholarly journals Residual Correction Procedure with Bernstein Polynomials for Solving Important Systems of Ordinary Differential Equations

2020 ◽  
Vol 64 (1) ◽  
pp. 63-80
Author(s):  
M.H.T. Alshbool ◽  
W. Shatanawi ◽  
I. Hashim ◽  
M. Sarr
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Bernstein Polynomials ◽  
Correction Procedure ◽  
Residual Correction

scholarly journals An Efficient Scheme for Time-Dependent Emden-Fowler Type Equations Based on Two-Dimensional Bernstein Polynomials

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1473
Author(s):  
Ahmad Sami Bataineh ◽  
Osman Rasit Isik ◽  
Abedel-Karrem Alomari ◽  
Mohammad Shatnawi ◽  
Ishak Hashim
Keyword(s):  
Numerical Solutions ◽  
Bernstein Polynomials ◽  
Adomian Decomposition Method ◽  
Approximate Solutions ◽  
Time Dependent ◽  
Operational Matrices ◽  
Correction Procedure ◽  
Adomian Decomposition ◽  
Special Cases ◽  
Residual Correction

In this study, we introduce an efficient computational method to obtain an approximate solution of the time-dependent Emden-Fowler type equations. The method is based on the 2D-Bernstein polynomials (2D-BPs) and their operational matrices. In the cases of time-dependent Lane–Emden type problems and wave-type equations which are the special cases of the problem, the method converts the problem to a linear system of algebraic equations. If the problem has a nonlinear part, the final system is nonlinear. We analyzed the error and give a theorem for the convergence. To estimate the error for the numerical solutions and then obtain more accurate approximate solutions, we give the residual correction procedure for the method. To show the effectiveness of the method, we apply the method to some test examples. The method gives more accurate results whenever increasing n,m for linear problems. For the nonlinear problems, the method also works well. For linear and nonlinear cases, the residual correction procedure estimates the error and yields the corrected approximations that give good approximation results. We compare the results with the results of the methods, the homotopy analysis method, homotopy perturbation method, Adomian decomposition method, and variational iteration method, on the nodes. Numerical results reveal that the method using 2D-BPs is more effective and simple for obtaining approximate solutions of the time-dependent Emden-Fowler type equations and the method presents a good accuracy.

scholarly journals An Enhanced Adaptive Bernstein Collocation Method for Solving Systems of ODEs

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 425 ◽  
Author(s):  
Ahmad Sami Bataineh ◽  
Osman Rasit Isik ◽  
Moa’ath Oqielat ◽  
Ishak Hashim
Keyword(s):  
Linear System ◽  
Collocation Method ◽  
Bernstein Polynomials ◽  
Operational Matrix ◽  
Correction Procedure ◽  
Stiff Systems ◽  
Bernstein Functions ◽  
Operational Matrix Of Differentiation ◽  
Systems Of Odes ◽  
Residual Correction

In this paper, we introduce two new methods to solve systems of ordinary differential equations. The first method is constituted of the generalized Bernstein functions, which are obtained by Bernstein polynomials, and operational matrix of differentiation with collocation method. The second method depends on tau method, the generalized Bernstein functions and operational matrix of differentiation. These methods produce a series which is obtained by non-polynomial functions set. We give the standard Bernstein polynomials to explain the generalizations for both methods. By applying the residual correction procedure to the methods, one can estimate the absolute errors for both methods and may obtain more accurate results. We apply the methods to some test examples including linear system, non-homogeneous linear system, nonlinear stiff systems, non-homogeneous nonlinear system and chaotic Genesio system. The numerical shows that the methods are efficient and work well. Increasing m yields a decrease on the errors for all methods. One can estimate the errors by using the residual correction procedure.

scholarly journals Numerical solution of the first order nonlinear differen-tial equations with the mixed nonlinear conditions by using PLSM(comparison with Bernstein polynomials method)

2019 ◽  
Vol 29 ◽  
pp. 01014
Author(s):  
Marioara Lăpădat ◽  
Mohsen Razzaghi ◽  
Mădălina Sofia Paşca
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Least Squares ◽  
Ordinary Differential Equations ◽  
Least Squares Method ◽  
Bernstein Polynomials ◽  
Approximate Solutions ◽  
Nonlinear Ordinary Differential Equations ◽  
First Order ◽  
Approximate Polynomial

We use the Polynomial Least Squares Method (PLSM), which allows us to compute analytical approximate polynomial solutions for nonlinear ordinary differential equations with the mixed nonlinear conditions. The accuracy of the method is illustrated by a comparison with approximate solutions previously computed using Bernstein polynomials method.

scholarly journals Bernstein Series Solution of a Class of Lane-Emden Type Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Osman Rasit Isik ◽  
Mehmet Sezer
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Numerical Experiments ◽  
Series Solution ◽  
Bernstein Polynomials ◽  
Absolute Error ◽  
Correction Procedure ◽  
Collocation Points ◽  
Residual Correction ◽  
Theoretical Results

The purpose of this study is to present an approximate solution that depends on collocation points and Bernstein polynomials for a class of Lane-Emden type equations with mixed conditions. The method is given with some priori error estimate. Even the exact solution is unknown, an upper bound based on the regularity of the exact solution will be obtained. By using the residual correction procedure, the absolute error can be estimated. Also, one can specify the optimal truncation limitnwhich gives a better result in any norm. Finally, the effectiveness of the method is illustrated by some numerical experiments. Numerical results are consistent with the theoretical results.

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