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Search Results (5)

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Keywords = polynomiographs

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15 pages, 1897 KiB  
Article
A Novel Three-Step Numerical Solver for Physical Models under Fractal Behavior
by Muath Awadalla, Sania Qureshi, Amanullah Soomro and Kinda Abuasbeh
Symmetry 2023, 15(2), 330; https://doi.org/10.3390/sym15020330 - 24 Jan 2023
Cited by 11 | Viewed by 1707
Abstract
In this paper, we suggest an iterative method for solving nonlinear equations that can be used in the physical sciences. This response is broken down into three parts. Our methodology is inspired by both the standard Taylor’s method and an earlier Halley’s method. [...] Read more.
In this paper, we suggest an iterative method for solving nonlinear equations that can be used in the physical sciences. This response is broken down into three parts. Our methodology is inspired by both the standard Taylor’s method and an earlier Halley’s method. Three evaluations of the given function and two evaluations of its first derivative are all that are needed for each iteration with this method. Because of this, the unique methodology can complete its goal far more quickly than many of the other methods currently in use. We looked at several additional practical research models, including population growth, blood rheology, and neurophysiology. Polynomiographs can be used to show the convergence zones of certain polynomials with complex values. Polynomiographs are produced as a byproduct, and these end up having an appealing look and being artistically engaging. The twisting of polynomiographs is symmetric when the parameters are all real and asymmetric when some of the parameters are imaginary. Full article
(This article belongs to the Special Issue Fractional-Order Systems and Its Applications in Engineering)
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14 pages, 7662 KiB  
Article
Visual Analysis of Mixed Algorithms with Newton and Abbasbandy Methods Using Periodic Parameters
by Safeer Hussain Khan, Lateef Olakunle Jolaoso and Maggie Aphane
Symmetry 2022, 14(12), 2484; https://doi.org/10.3390/sym14122484 - 23 Nov 2022
Cited by 1 | Viewed by 1302
Abstract
In this paper, we proposed two mixed algorithms of Newton’s and Abbasbandy’s methods using a known iteration scheme from fixed point theory in polynomiography. We numerically investigated some properties of the proposed algorithms using periodic sequence parameters instead of the constant parameters that [...] Read more.
In this paper, we proposed two mixed algorithms of Newton’s and Abbasbandy’s methods using a known iteration scheme from fixed point theory in polynomiography. We numerically investigated some properties of the proposed algorithms using periodic sequence parameters instead of the constant parameters that are mostly used by many authors. Two pseudo-Newton algorithms were introduced based on the mixed iterations for the purpose of generating polynomiographs. The properties of the obtained polynomiographs were studied with respect to their graphics, turning effects and computation time. Moreover, some of these polynomiographs exhibited symmetrical properties when the degree of the polynomial was even. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear and Convex Analysis)
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16 pages, 556 KiB  
Article
New Approach to Split Variational Inclusion Issues through a Three-Step Iterative Process
by Andreea Bejenaru and Mihai Postolache
Mathematics 2022, 10(19), 3617; https://doi.org/10.3390/math10193617 - 2 Oct 2022
Cited by 3 | Viewed by 1959
Abstract
Split variational inclusions are revealed as a large class of problems that includes several other pre-existing split-type issues: split feasibility, split zeroes problems, split variational inequalities and so on. This makes them not only a rich direction of theoretical study but also one [...] Read more.
Split variational inclusions are revealed as a large class of problems that includes several other pre-existing split-type issues: split feasibility, split zeroes problems, split variational inequalities and so on. This makes them not only a rich direction of theoretical study but also one with important and varied practical applications: large dimensional linear systems, optimization, signal reconstruction, boundary value problems and others. In this paper, the existing algorithmic tools are complemented by a new procedure based on a three-step iterative process. The resulting approximating sequence is proved to be weakly convergent toward a solution. The operation mode of the new algorithm is tracked in connection with mixed optimization–feasibility and mixed linear–feasibility systems. Standard polynomiographic techniques are applied for a comparative visual analysis of the convergence behavior. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications II)
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16 pages, 8253 KiB  
Article
Dynamics of RK Iteration and Basic Family of Iterations for Polynomiography
by Lateef Olakunle Jolaoso, Safeer Hussain Khan and Kazeem Olalekan Aremu
Mathematics 2022, 10(18), 3324; https://doi.org/10.3390/math10183324 - 13 Sep 2022
Cited by 6 | Viewed by 1743
Abstract
In this paper, we propose some modifications of the basic family of iterations with a new four-step iteration called RK iteration and its s-convexity. We present some graphical examples showing the dynamics of the new iteration in the colouring and shapes of [...] Read more.
In this paper, we propose some modifications of the basic family of iterations with a new four-step iteration called RK iteration and its s-convexity. We present some graphical examples showing the dynamics of the new iteration in the colouring and shapes of the obtained polynomiographs compared to the ones from the basic family only. Moreover, the computational results reveal that the value of s in the s-convex combination of the RK iteration has a significant impact on the time taken by the iteration process for approximating the roots of the polynomials. The obtained results are interesting from an artistic and computational point of view. Full article
(This article belongs to the Special Issue Advances in Fractals)
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27 pages, 3239 KiB  
Article
Visual Analysis of the Newton’s Method with Fractional Order Derivatives
by Krzysztof Gdawiec, Wiesław Kotarski and Agnieszka Lisowska
Symmetry 2019, 11(9), 1143; https://doi.org/10.3390/sym11091143 - 9 Sep 2019
Cited by 29 | Viewed by 4245
Abstract
The aim of this paper is to investigate experimentally and to present visually the dynamics of the processes in which in the standard Newton’s root-finding method the classic derivative is replaced by the fractional Riemann–Liouville or Caputo derivatives. These processes applied to polynomials [...] Read more.
The aim of this paper is to investigate experimentally and to present visually the dynamics of the processes in which in the standard Newton’s root-finding method the classic derivative is replaced by the fractional Riemann–Liouville or Caputo derivatives. These processes applied to polynomials on the complex plane produce images showing basins of attractions for polynomial zeros or images representing the number of iterations required to obtain polynomial roots. These latter images were called by Kalantari as polynomiographs. We use both: the colouring by roots to present basins of attractions, and the colouring by iterations that reveal the speed of convergence and dynamic properties of processes visualised by polynomiographs. Full article
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