Abstract
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.
Highlights
Received: 25 January 2021Accepted: 11 February 2021Published: 21 February 2021Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.Copyright: c 2021 by the authors.Licensee MDPI, Basel, Switzerland.Many real life phenomena can be modeled by systems of ordinary differential equations (ODEs)
We present two new methods, namely generalized Bernstein function (GBF) tau and GBF collocation methods, to numerically solve the systems of ODEs
Let us call u j,m and ubj,m as Bernstein polynomials (BPs) series solution obtained by collocation method (BPSSC) or tau method (BPSST) and GBF series solution obtained by collocation method (GBFSSC) or tau method (GBFSST)
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Isik et al [8] presented an approximate method based on the Bernstein polynomials for solving high order linear differential equations. Yuzbasi [10] and Baleanu et al [11] presented approximate analytical methods constituted of the Bernstein polynomials for solving fractional Riccati type differential equations. Maleknejad et al [15] proposed a numerical method for solving the systems of high order linear Volterra–Fredholm integro-differential equations by using Bernstein operational matrices. An approximate solution method, called multistage Bernstein collocation method, to solve strongly nonlinear damped systems was given in [20]. We present two new methods, namely generalized Bernstein function (GBF) tau and GBF collocation methods, to numerically solve the systems of ODEs. The methods are obtained by a special generalization of m-th degree Bernstein polynomials and collocation or tau methods. We apply the methods for different values of m to show the dependency of m values
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