Approximate Solutions of Singular Differential Equations With Estimation Error by Using Bernstein Polynomials

Alshbool, Mohammed; Bataineh, A. Sami; Hashim, Ishak; Işık, Osman Raşit

doi:10.12732/ijpam.v100i1.10

Cited by 12 publications

(3 citation statements)

References 15 publications

(7 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In earlier studies [33,34], the fractional Riccati differential equations have been solved by ADM. There are some methods in previous research [35][36][37][38][39][40][41][42] that are based on Bernstein polynomials.…”

Section: Introductionmentioning

confidence: 99%

The general Bernstein function: Application to χ‐fractional differential equations

Sadek,

Sami Bataineh

2024

Math Methods in App Sciences

View full text Add to dashboard Cite

In this paper, we present the general Bernstein functions for the first time. The properties of generalized Bernstein basis functions are given and demonstrated. The classical Bernstein polynomial bases are merely a subset of the general Bernstein functions. Based on the new Bernstein base functions and the collocation method, we present a numerical method for solving linear and nonlinear ‐fractional differential equations ( ‐FDEs) with variable coefficients. The fractional derivative used in this work is the ‐Caputo fractional derivative sense ( ‐CFD). Combining the Bernstein functions basis and the collocation methods yields the approximation solution of nonlinear differential equations. These base functions can be used to solve many problems in applied mathematics, including calculus of variations, differential equations, optimal control, and integral equations. Furthermore, the convergence of the method is rigorously justified and supported by numerical experiments.

show abstract

Section: Introductionmentioning

confidence: 99%

The general Bernstein function: Application to χ‐fractional differential equations

Sadek,

Sami Bataineh

2024

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…Pandey and Kumar [7] used Bernstein operational matrices to solve Emden type equations. Alshbool et al [8] proposed Bernstein polynomial approximation solutions for singular differential equations. Chen et al [9] used Bernstein polynomials to present a numerical solution to the variable order linear cable equation.…”

Section: Introductionmentioning

confidence: 99%

A Reliable Numerical Treatment of Differential Equations via Hybrid Bernstein and Improved Block-Pulse Functions

Ramadan

Osheba

2022

Preprint

View full text Add to dashboard Cite

Recently, the authors attempted to solve integral equations using hybrid improved block pulse and Bernstein functions. However, this is the first article to present a technical coupling between improved block-pulse functions and Bernstein functions for solving differential equations. The current method converts differential equations into an algebraic system that can be solved using traditional methods. To demonstrate the validity of this new approach, some numerical tests are provided. The findings demonstrated that the method is both promising and highly accurate.

show abstract

“…(2011, 2014); Yuksel et al. (2015); Isik and Turkoglu (2015); Alshbool et al. (2015); Bataineh et al.…”

Section: Introductionmentioning

confidence: 99%

Multistage Bernstein collocation method for solving strongly nonlinear damped systems

Bataineh

Al-Omari

Işık

et al. 2018

Journal of Vibration and Control

View full text Add to dashboard Cite

In this paper, we propose an approximate solution method, called multistage Bernstein collocation method (MBCM), to solve strongly nonlinear damped systems. The method is given with an error analysis. The systems investigated include step function external excitation and periodical function external excitation. We found that the MBCM gives more accurate approximate solutions than the standard collocation method. Since MBCM splits the domain of the problem [Formula: see text] into n parts, one can overcome oscillations arising from the standard collocation method by increasing n. The results obtained from MBCM compare favorably with that of the fourth-order Runge–Kutta method (RK4).

show abstract

Approximate Solutions of Singular Differential Equations With Estimation Error by Using Bernstein Polynomials

Cited by 12 publications

References 15 publications

The general Bernstein function: Application to χ‐fractional differential equations

The general Bernstein function: Application to χ‐fractional differential equations

A Reliable Numerical Treatment of Differential Equations via Hybrid Bernstein and Improved Block-Pulse Functions

Multistage Bernstein collocation method for solving strongly nonlinear damped systems

Contact Info

Product

Resources

About