High Convergence Order Iterative Procedures for Solving Equations Originating from Real Life Problems

Ramandeep Behl; Ioannis K. Argyros; Ali Saleh Alshomrani

doi:10.3390/math7090855

High Convergence Order Iterative Procedures for Solving Equations Originating from Real Life Problems

Mathematics ◽

10.3390/math7090855 ◽

2019 ◽

Vol 7 (9) ◽

pp. 855

Author(s):

Ramandeep Behl ◽

Ioannis K. Argyros ◽

Ali Saleh Alshomrani

Keyword(s):

Real Life ◽

Nonlinear Problems ◽

Local Study ◽

Precise Information ◽

Numerical Examples ◽

First Order ◽

Life Problems ◽

Solving Equations ◽

Iterative Procedures ◽

Good Number

The foremost aim of this paper is to suggest a local study for high order iterative procedures for solving nonlinear problems involving Banach space valued operators. We only deploy suppositions on the first-order derivative of the operator. Our conditions involve the Lipschitz or Hölder case as compared to the earlier ones. Moreover, when we specialize to these cases, they provide us: larger radius of convergence, higher bounds on the distances, more precise information on the solution and smaller Lipschitz or Hölder constants. Hence, we extend the suitability of them. Our new technique can also be used to broaden the usage of existing iterative procedures too. Finally, we check our results on a good number of numerical examples, which demonstrate that they are capable of solving such problems where earlier studies cannot apply.

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Some High-Order Convergent Iterative Procedures for Nonlinear Systems with Local Convergence

Mathematics ◽

10.3390/math9121375 ◽

2021 ◽

Vol 9 (12) ◽

pp. 1375

Author(s):

Ramandeep Behl ◽

Ioannis K. Argyros ◽

Fouad Othman Mallawi

Keyword(s):

Local Convergence ◽

Real Life ◽

High Order ◽

Systems Of Nonlinear Equations ◽

Convergence Criteria ◽

Restricted Area ◽

Iterative Schemes ◽

First Order ◽

Life Problems ◽

Iterative Procedures

In this study, we suggested the local convergence of three iterative schemes that works for systems of nonlinear equations. In earlier results, such as from Amiri et al. (see also the works by Behl et al., Argryos et al., Chicharro et al., Cordero et al., Geum et al., Guitiérrez, Sharma, Weerakoon and Fernando, Awadeh), authors have used hypotheses on high order derivatives not appearing on these iterative procedures. Therefore, these methods have a restricted area of applicability. The main difference of our study to earlier studies is that we adopt only the first order derivative in the convergence order (which only appears on the proposed iterative procedure). No work has been proposed on computable error distances and uniqueness in the aforementioned studies given on Rk. We also address these problems too. Moreover, by using Banach space, the applicability of iterative procedures is extended even further. We have examined the convergence criteria on several real life problems along with a counter problem that completes this study.

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

Mathematics ◽

10.3390/math9111242 ◽

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.

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Explicit Integrator of Runge-Kutta Type for Direct Solution of u(4)=f(x,u,u′,u′′)

Symmetry ◽

10.3390/sym11020246 ◽

2019 ◽

Vol 11 (2) ◽

pp. 246 ◽

Cited By ~ 2

Author(s):

Nizam Ghawadri ◽

Norazak Senu ◽

Firas Fawzi ◽

Fudziah Ismail ◽

Zarina Ibrahim

Keyword(s):

Real Life ◽

Order Ordinary Differential Equation ◽

New Techniques ◽

First Order ◽

New Methods ◽

Life Problems ◽

Algebraic Order ◽

Runge Kutta ◽

Order Four ◽

Primary Contribution

The primary contribution of this work is to develop direct processes of explicit Runge-Kutta type (RKT) as solutions for any fourth-order ordinary differential equation (ODEs) of the structure u ( 4 ) = f ( x , u , u ′ , u ′ ′ ) and denoted as RKTF method. We presented the associated B-series and quad-colored tree theory with the aim of deriving the prerequisites of the said order. Depending on the order conditions, the method with algebraic order four with a three-stage and order five with a four-stage denoted as RKTF4 and RKTF5 are discussed, respectively. Numerical outcomes are offered to interpret the accuracy and efficacy of the new techniques via comparisons with various currently available RK techniques after converting the problems into a system of first-order ODE systems. Application of the new methods in real-life problems in ship dynamics is discussed.

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An Optimal Reconstruction of Chebyshev–Halley-Type Methods with Local Convergence Analysis

International Journal of Computational Methods ◽

10.1142/s0219876219400176 ◽

2019 ◽

Vol 17 (05) ◽

pp. 1940017

Author(s):

Ali Saleh Alshomrani ◽

Ioannis K. Argyros ◽

Ramandeep Behl

Keyword(s):

Local Convergence ◽

Real Life ◽

Eighth Order ◽

Dynamical Study ◽

First Order ◽

Life Problems ◽

Local Convergence Analysis ◽

Lipschitz Conditions ◽

Scalar Equations ◽

Basins Of Attractions

Our principle aim in this paper is to present a new reconstruction of classical Chebyshev–Halley schemes having optimal fourth and eighth-order of convergence for all parameters [Formula: see text] unlike in the earlier studies. In addition, we analyze the local convergence of them by using hypotheses requiring the first-order derivative of the involved function [Formula: see text] and the Lipschitz conditions. In addition, we also formulate their theoretical radius of convergence. Several numerical examples originated from real life problems demonstrate that they are applicable to a broad range of scalar equations, where previous studies cannot be used. Finally, a dynamical study of them also demonstrates that bigger and more promising basins of attractions are obtained.

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Characterizations of two different fractional operators without singular kernel

Mathematical Modelling of Natural Phenomena ◽

10.1051/mmnp/2018070 ◽

2019 ◽

Vol 14 (3) ◽

pp. 302 ◽

Cited By ~ 37

Author(s):

Mehmet Yavuz

Keyword(s):

Fractional Derivative ◽

Real Life ◽

Nonlinear Problems ◽

The Other ◽

Singular Kernel ◽

Sinc Function ◽

Life Problems ◽

The Laplace Transform ◽

Fractional Differential ◽

Fractional Derivative Operators

In this paper, we analyze the behaviours of two different fractional derivative operators defined in the last decade. One of them is defined with the normalized sinc function (NSF) and the other one is defined with the Mittag-Leffler function (MLF). Both of them have a non-singular kernel. The fractional derivative operator defined with the MLF is developed by Atangana and Baleanu (ABO) in 2016 and the other operator defined with the normalized sinc function (NSFDO) is created by Yang et al. in 2017. These mentioned operators have some advantages to model the real life problems and to solve them. On the other hand, since the Laplace transform (LT) of the ABO can be calculated more easily, it can be preferred to solve linear/nonlinear problems. In this study, we use the perturbation method with coupled the LTs of these operators to analyze their performance in solving some fractional differential equations. Furthermore, by constructing the error analysis, we test the practicability and usefulness of the method.

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Local Convergence Balls for Nonlinear Problems with Multiplicity and Their Extension to Eighth-Order Convergence

Mathematical Problems in Engineering ◽

10.1155/2019/1427809 ◽

2019 ◽

Vol 2019 ◽

pp. 1-17

Author(s):

Ramandeep Behl ◽

Eulalia Martínez ◽

Fabricio Cevallos ◽

Ali S. Alshomrani

Keyword(s):

Local Convergence ◽

Chemical Engineering ◽

Real Life ◽

Nonlinear Problems ◽

Engineering Problem ◽

Test Problems ◽

Order Convergence ◽

Eighth Order ◽

Life Problems ◽

Fourth Order Method

The main contribution of this study is to present a new optimal eighth-order scheme for locating zeros with multiplicity m≥1. An extensive convergence analysis is presented with the main theorem in order to demonstrate the optimal eighth-order convergence of the proposed scheme. Moreover, a local convergence study for the optimal fourth-order method defined by the first two steps of the new method is presented, allowing us to obtain the radius of the local convergence ball. Finally, numerical tests on some real-life problems, such as a Van der Waals equation of state, a conversion chemical engineering problem, and two standard academic test problems, are presented, which confirm the theoretical results established in this paper and the efficiency of this proposed iterative method. We observed from the numerical experiments that our proposed iterative methods have good values for convergence radii. Further, they not only have faster convergence towards the desired zero of the involved function but also have both smaller residual error and a smaller difference between two consecutive iterations than current existing techniques.

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A New Ranking Approach for Interval Valued Intuitionistic Fuzzy Sets and its Application in Decision Making

International Journal of Fuzzy System Applications ◽

10.4018/ijfsa.2019040106 ◽

2019 ◽

Vol 8 (2) ◽

pp. 110-125 ◽

Cited By ~ 2

Author(s):

Pranjal Talukdar ◽

Palash Dutta

Keyword(s):

Decision Making ◽

Fuzzy Sets ◽

Model Uncertainty ◽

Real Life ◽

Intuitionistic Fuzzy Sets ◽

Multi Criteria Decision Making ◽

Numerical Examples ◽

Intuitionistic Fuzzy ◽

Life Problems ◽

Interval Valued

Ranking of interval valued intuitionistic fuzzy sets (IVIFSs) plays an important role because of its attraction and applicability to model uncertainty in real life problems. In this article, an attempt has been made to devise a new method for ranking of IVIFSs based on exponential function. The significance of the method is illustrated with the help of some numerical examples and the results are compared with other existing methods. Furthermore, a multi criteria decision making method is presented here to evaluate the final ranking of the alternatives using the proposed ranking method and discussed the consistency of so obtained results.

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New derivative-free iterative family having optimal convergence order sixteen and its applications

Engineering Computations ◽

10.1108/ec-03-2021-0155 ◽

2021 ◽

Vol ahead-of-print (ahead-of-print) ◽

Author(s):

Manpreet Kaur ◽

Sanjeev Kumar ◽

Munish Kansal

Keyword(s):

Eigenvalue Problems ◽

Real Life ◽

Nonlinear Problems ◽

Chemical Reactor ◽

Basins Of Attraction ◽

Convergence Order ◽

Content Type ◽

New Class ◽

Life Problems ◽

Derivative Free

PurposeThe purpose of the article is to construct a new class of higher-order iterative techniques for solving scalar nonlinear problems.Design/methodology/approachThe scheme is generalized by using the power-mean notion. By applying Neville's interpolating technique, the methods are formulated into the derivative-free approaches. Further, to enhance the computational efficiency, the developed iterative methods have been extended to the methods with memory, with the aid of the self-accelerating parameter.FindingsIt is found that the presented family is optimal in terms of Kung and Traub conjecture as it evaluates only five functions in each iteration and attains convergence order sixteen. The proposed family is examined on some practical problems by modeling into nonlinear equations, such as chemical equilibrium problems, beam positioning problems, eigenvalue problems and fractional conversion in a chemical reactor. The obtained results confirm that the developed scheme works more adequately as compared to the existing methods from the literature. Furthermore, the basins of attraction of the different methods have been included to check the convergence in the complex plane.Originality/valueThe presented experiments show that the developed schemes are of great benefit to implement on real-life problems.

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