scholarly journals An Optimal Derivative-Free Ostrowski’s Scheme for Multiple Roots of Nonlinear Equations

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1809 ◽  
Author(s):  
Ramandeep Behl ◽  
Samaher Khalaf Alharbi ◽  
Fouad Othman Mallawi ◽  
Mehdi Salimi
Keyword(s):  
Nonlinear Equations ◽  
Real Life ◽  
Higher Order ◽  
Multiple Roots ◽  
Continuous Stirred Tank Reactor ◽  
Computational Mathematics ◽  
Life Problems ◽  
Derivative Free ◽  
Continuous Stirred Tank ◽  
Ostrowski’S Method

Finding higher-order optimal derivative-free methods for multiple roots (m≥2) of nonlinear expressions is one of the most fascinating and difficult problems in the area of numerical analysis and Computational mathematics. In this study, we introduce a new fourth order optimal family of Ostrowski’s method without derivatives for multiple roots of nonlinear equations. Initially the convergence analysis is performed for particular values of multiple roots—afterwards it concludes in general form. Moreover, the applicability and comparison demonstrated on three real life problems (e.g., Continuous stirred tank reactor (CSTR), Plank’s radiation and Van der Waals equation of state) and two standard academic examples that contain the clustering of roots and higher-order multiplicity (m=100) problems, with existing methods. Finally, we observe from the computational results that our methods consume the lowest CPU timing as compared to the existing ones. This illustrates the theoretical outcomes to a great extent of this study.

scholarly journals Derivative-Free King’s Scheme for Multiple Zeros of Nonlinear Functions

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1242
Author(s):  
Ramandeep Behl ◽  
Sonia Bhalla ◽  
Eulalia Martínez ◽  
Majed Aali Alsulami
Keyword(s):  
Iterative Methods ◽  
Fourth Order ◽  
Order Of Convergence ◽  
Real Life ◽  
Multiple Roots ◽  
Computational Results ◽  
Nonlinear Functions ◽  
First Order ◽  
Life Problems ◽  
Derivative Free

There is no doubt that the fourth-order King’s family is one of the important ones among its counterparts. However, it has two major problems: the first one is the calculation of the first-order derivative; secondly, it has a linear order of convergence in the case of multiple roots. In order to improve these complications, we suggested a new King’s family of iterative methods. The main features of our scheme are the optimal convergence order, being free from derivatives, and working for multiple roots (m≥2). In addition, we proposed a main theorem that illustrated the fourth order of convergence. It also satisfied the optimal Kung–Traub conjecture of iterative methods without memory. We compared our scheme with the latest iterative methods of the same order of convergence on several real-life problems. In accordance with the computational results, we concluded that our method showed superior behavior compared to the existing methods.

scholarly journals An Optimal Derivative Free Family of Chebyshev–Halley’s Method for Multiple Zeros

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 546
Author(s):  
Ramandeep Behl ◽  
Sonia Bhalla ◽  
Ángel Alberto Magreñán ◽  
Alejandro Moysi
Keyword(s):  
Nonlinear Equation ◽  
Weight Function ◽  
Convergence Analysis ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Order Of Convergence ◽  
Higher Order ◽  
Multiple Roots ◽  
Iterative Technique ◽  
Derivative Free

In this manuscript, we introduce the higher-order optimal derivative-free family of Chebyshev–Halley’s iterative technique to solve the nonlinear equation having the multiple roots. The designed scheme makes use of the weight function and one parameter α to achieve the fourth-order of convergence. Initially, the convergence analysis is performed for particular values of multiple roots. Afterward, it concludes in general. Moreover, the effectiveness of the presented methods are certified on some applications of nonlinear equations and compared with the earlier derivative and derivative-free schemes. The obtained results depict better performance than the existing methods.

Problem-Based Learning (PBL)

2021 ◽  
pp. 38-56
Author(s):  
Mahmoud Hawamdeh ◽  
Idris Adamu
Keyword(s):  
Problem Solving ◽  
Teaching And Learning ◽  
Real Life ◽  
Problem Based Learning ◽  
Higher Order ◽  
Educational Institutions ◽  
Learning Skills ◽  
Knowledge And Skills ◽  
Life Problems ◽  
Approach To Teaching

This chapter discuss how Problem-Based learning (PBL) helps to achieve this century's approach to teaching and learning for students in higher educational institutions. If adopted, this method of teaching will enable student to attain learning skills (skills, abilities, problem solving, and learning dispositions that have been identified) to acquire a lifelong habit of approaching problems with initiative and diligence and a drive to acquire the knowledge and skills needed for an effective resolution. And they will develop a systematic approach to solving real-life problems using higher-order skills.

scholarly journals A New Higher-Order Iterative Scheme for the Solutions of Nonlinear Systems

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 271 ◽  
Author(s):  
Ramandeep Behl ◽  
Ioannis K. Argyros
Keyword(s):  
Mathematical Modeling ◽  
Banach Space ◽  
Nonlinear Equations ◽  
Numerical Experiments ◽  
Iterative Scheme ◽  
Real Life ◽  
System Of Nonlinear Equations ◽  
Life Problems ◽  
Lipschitz Constants ◽  
Better Than

Many real-life problems can be reduced to scalar and vectorial nonlinear equations by using mathematical modeling. In this paper, we introduce a new iterative family of the sixth-order for a system of nonlinear equations. In addition, we present analyses of their convergences, as well as the computable radii for the guaranteed convergence of them for Banach space valued operators and error bounds based on the Lipschitz constants. Moreover, we show the applicability of them to some real-life problems, such as kinematic syntheses, Bratu’s, Fisher’s, boundary value, and Hammerstein integral problems. We finally wind up on the ground of achieved numerical experiments, where they perform better than other competing schemes.

scholarly journals An Optimal Fourth Order Derivative-Free Numerical Algorithm for Multiple Roots

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1038 ◽  
Author(s):  
Sunil Kumar ◽  
Deepak Kumar ◽  
Janak Raj Sharma ◽  
Clemente Cesarano ◽  
Praveen Agarwal ◽  
...  
Keyword(s):  
Iterative Methods ◽  
Numerical Algorithm ◽  
Fourth Order ◽  
Iterative Scheme ◽  
Higher Order ◽  
New Technique ◽  
Multiple Roots ◽  
Order Convergence ◽  
Higher Order Methods ◽  
Derivative Free

A plethora of higher order iterative methods, involving derivatives in algorithms, are available in the literature for finding multiple roots. Contrary to this fact, the higher order methods without derivatives in the iteration are difficult to construct, and hence, such methods are almost non-existent. This motivated us to explore a derivative-free iterative scheme with optimal fourth order convergence. The applicability of the new scheme is shown by testing on different functions, which illustrates the excellent convergence. Moreover, the comparison of the performance shows that the new technique is a good competitor to existing optimal fourth order Newton-like techniques.

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