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 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 Exponential Jacobi spectral method for hyperbolic partial differential equations

2019 ◽  
Vol 13 (4) ◽  
pp. 347-354 ◽  
Author(s):  
Y. H. Youssri ◽  
R. M. Hafez
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Collocation Method ◽  
Initial Conditions ◽  
Operational Matrix ◽  
Hyperbolic Partial Differential Equations ◽  
Partial Differential ◽  
Linear System Of Equations ◽  
Operational Matrix Of Differentiation ◽  
Jacobi Functions

Abstract Herein, we have proposed a scheme for numerically solving hyperbolic partial differential equations (HPDEs) with given initial conditions. The operational matrix of differentiation for exponential Jacobi functions was derived, and then a collocation method was used to transform the given HPDE into a linear system of equations. The preferences of using the exponential Jacobi spectral collocation method over other techniques were discussed. The convergence and error analyses were discussed in detail. The validity and accuracy of the proposed method are investigated and checked through numerical experiments.

scholarly journals A Collocation Method Based on the Bernoulli Operational Matrix for Solving Nonlinear BVPs Which Arise from the Problems in Calculus of Variation

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Emran Tohidi ◽  
Adem Kılıçman
Keyword(s):  
Collocation Method ◽  
Necessary Conditions ◽  
Operational Matrix ◽  
Bernoulli Polynomials ◽  
Variational Problems ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Lagrange Equations ◽  
Calculus Of Variation ◽  
Operational Matrix Of Differentiation

A new collocation method is developed for solving BVPs which arise from the problems in calculus of variation. These BVPs result from the Euler-Lagrange equations, which are the necessary conditions of the extremums of problems in calculus of variation. The proposed method is based upon the Bernoulli polynomials approximation together with their operational matrix of differentiation. After imposing the collocation nodes to the main BVPs, we reduce the variational problems to the solution of algebraic equations. It should be noted that the robustness of operational matrices of differentiation with respect to the integration ones is shown through illustrative examples. Complete comparisons with other methods and superior results confirm the validity and applicability of the presented 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.

scholarly journals A numerical technique for a general form of nonlinear fractional-order differential equations with the linear functional argument

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Khalid K. Ali ◽  
Mohamed A. Abd El salam ◽  
Emad M. H. Mohamed
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Fractional Order ◽  
Linear Functional ◽  
Numerical Technique ◽  
Operational Matrix ◽  
Fractional Order Differential Equations ◽  
The Given ◽  
Functional Argument ◽  
Special Case

AbstractIn this paper, a numerical technique for a general form of nonlinear fractional-order differential equations with a linear functional argument using Chebyshev series is presented. The proposed equation with its linear functional argument represents a general form of delay and advanced nonlinear fractional-order differential equations. The spectral collocation method is extended to study this problem as a discretization scheme, where the fractional derivatives are defined in the Caputo sense. The collocation method transforms the given equation and conditions to algebraic nonlinear systems of equations with unknown Chebyshev coefficients. Additionally, we present a general form of the operational matrix for derivatives. A general form of the operational matrix to derivatives includes the fractional-order derivatives and the operational matrix of an ordinary derivative as a special case. To the best of our knowledge, there is no other work discussed this point. Numerical examples are given, and the obtained results show that the proposed method is very effective and convenient.

scholarly journals Two matrix methods for solution of nonlinear and linear Lane-Emden type equations with mixed condition by operational matrix

2015 ◽  
Vol 4 (3) ◽  
pp. 420 ◽  
Author(s):  
Behrooz Basirat ◽  
Mohammad Amin Shahdadi
Keyword(s):  
Bernstein Polynomials ◽  
Operational Matrix ◽  
Numerical Procedure ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Mixed Condition ◽  
Matrix Methods ◽  
Numerical Examples ◽  
Linear Algebraic Equations ◽  
The Given

<p>The aim of this article is to present an efficient numerical procedure for solving Lane-Emden type equations. We present two practical matrix method for solving Lane-Emden type equations with mixed conditions by Bernstein polynomials operational matrices (BPOMs) on interval [<em>a; b</em>]. This methods transforms Lane-Emden type equations and the given conditions into matrix equation which corresponds to a system of linear algebraic equations. We also give some numerical examples to demonstrate the efficiency and validity of the operational matrices for solving Lane-Emden type equations (LEEs).</p>

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