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Mathematics
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Computational Methods in Analysis and Applications 2023
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Computational Methods in Analysis and Applications 2023

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: 31 October 2024 | Viewed by 8550

Special Issue Editor

Prof. Dr. Ioannis K. Argyros

E-Mail Website
Guest Editor
Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA
Interests: numerical analysis; numerical functional analysis; iterative methods for solving equations and systems of equations
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

A plethora of problems in mathematics, economics, physics, biology, chemistry, engineering, and other disciplines can be reduced to solving an equation or a system of equations in an abstract space. The solution of the equation can be found in closed form only in some special cases. That is why most researchers and practitioners introduce iterative methods to produce a sequence approximating the solution under certain conditions. The rapid development of digital computers, advanced computer arithmetic, and symbolic computation have made the implementation of high convergence order methods possible. Moreover, many methods that were previously only of academic interest have now become feasible. The main purpose of this Special Issue is to present new ideas in the field of iterative methods and their applications in the aforementioned disciplines.

This Special Issue provides an opportunity for researchers and practitioners to communicate their ideas. We are inviting contributions of original research papers to stimulate interest in nonlinear equations and related areas.

This issue is a continuation of the previous successful Special Issue “Computational Methods in Analysis and Applications 2020”.

Prof. Dr. Ioannis K. Argyros
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Newton-like methods
  • Steffensen-type methods
  • variational methods
  • iterative methods for image processing
  • methods for solving inverse problems
  • methods for generalized equilibrium problems
  • methods for optimization problems
  • methods in biology, chemistry, and medicine
  • methods in economics
  • methods in physics
  • methods in engineering

Published Papers (5 papers)

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Research

23 pages, 349 KiB  
Article
Extending the Domain with Application of Four-Step Nonlinear Scheme with Average Lipschitz Conditions
by Akanksha Saxena, Jai Prakash Jaiswal, Kamal Raj Pardasani and Ioannis K. Argyros
Mathematics 2023, 11(8), 1774; https://doi.org/10.3390/math11081774 - 07 Apr 2023
Viewed by 3033
Abstract
A novel local and semi-local convergence theorem for the four-step nonlinear scheme is presented. Earlier studies on local convergence were conducted without particular assumption on Lipschitz constant. In first part, the main local convergence theorems with a weak ϰ-average (assuming it as [...] Read more.
A novel local and semi-local convergence theorem for the four-step nonlinear scheme is presented. Earlier studies on local convergence were conducted without particular assumption on Lipschitz constant. In first part, the main local convergence theorems with a weak ϰ-average (assuming it as a positively integrable function and dropping the essential property of ND) are obtained. In comparison to previous research, in another part, we employ majorizing sequences that are more accurate in their precision along with the certain form of ϰ average Lipschitz criteria. A finer local and semi-local convergence criteria, boosting its utility, by relaxing the assumptions is derived. Applications in engineering to a variety of specific cases, such as object motion governed by a system of differential equations, are illustrated. Full article
(This article belongs to the Special Issue Computational Methods in Analysis and Applications 2023)
14 pages, 484 KiB  
Article
On the Order of Convergence of the Noor–Waseem Method
by Santhosh George, Ramya Sadananda, Jidesh Padikkal and Ioannis K. Argyros
Mathematics 2022, 10(23), 4544; https://doi.org/10.3390/math10234544 - 01 Dec 2022
Cited by 5 | Viewed by 962
Abstract
In 2009, Noor and Waseem studied an important third-order iterative method. The convergence order is obtained using Taylor expansion and assumptions on the derivatives of order up to four. In this paper, we have obtained convergence order three for this method using assumptions [...] Read more.
In 2009, Noor and Waseem studied an important third-order iterative method. The convergence order is obtained using Taylor expansion and assumptions on the derivatives of order up to four. In this paper, we have obtained convergence order three for this method using assumptions on the first and second derivatives of the involved operator. Further, we have extended the method to obtain a fifth- and a sixth-order methods. The dynamics of the methods are also provided in this study. Numerical examples are included. The same technique can be used to extend the utilization of other single or multistep methods. Full article
(This article belongs to the Special Issue Computational Methods in Analysis and Applications 2023)
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22 pages, 350 KiB  
Article
Convergence Criteria of a Three-Step Scheme under the Generalized Lipschitz Condition in Banach Spaces
by Akanksha Saxena, Jai Prakash Jaiswal, Kamal Raj Pardasani and Ioannis K. Argyros
Mathematics 2022, 10(21), 3946; https://doi.org/10.3390/math10213946 - 24 Oct 2022
Cited by 1 | Viewed by 863
Abstract
In the given study, we investigate the three-step NTS’s ball convergence for solving nonlinear operator equations with a convergence order of five in a Banach setting. A nonlinear operator’s first-order derivative is assumed to meet the generalized Lipschitz condition, also known as the [...] Read more.
In the given study, we investigate the three-step NTS’s ball convergence for solving nonlinear operator equations with a convergence order of five in a Banach setting. A nonlinear operator’s first-order derivative is assumed to meet the generalized Lipschitz condition, also known as the κ-average condition. Furthermore, several theorems on the convergence of the same method in Banach spaces are developed with the conditions that the derivative of the operators must satisfy the radius or center-Lipschitz condition with a weak κ-average and that κ is a positive integrable but not necessarily non-decreasing function. This novel approach allows for a more precise convergence analysis even without the requirement for new circumstances. As a result, we broaden the applicability of iterative approaches. The theoretical results are supported further by illuminating examples. The convergence theorem investigates the location of the solution ϵ* and the existence of it. In the end, we achieve weaker sufficient convergence criteria and more specific knowledge on the position of the ϵ* than previous efforts requiring the same computational effort. We obtain the convergence theorems as well as some novel results by applying the results to some specific functions for κ(u). Numerical tests are carried out to corroborate the hypotheses established in this work. Full article
(This article belongs to the Special Issue Computational Methods in Analysis and Applications 2023)
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14 pages, 1047 KiB  
Article
The Dynamics of a Continuous Newton-like Method
by Manoj K. Singh and Ioannis K. Argyros
Mathematics 2022, 10(19), 3602; https://doi.org/10.3390/math10193602 - 02 Oct 2022
Cited by 4 | Viewed by 1401
Abstract
The objective of the current work is to invent and introduce the continuous version of Newton’s method. This scheme is used to establish some interesting properties with examples. We have plotted the fractal pattern graphs for a Newton-like method and a Damped Newton-like [...] Read more.
The objective of the current work is to invent and introduce the continuous version of Newton’s method. This scheme is used to establish some interesting properties with examples. We have plotted the fractal pattern graphs for a Newton-like method and a Damped Newton-like method in the discrete case and hence we have introduced a new concept of streamline for the continuous version of the Newton-like method. The graph and streamlines of these patterns are in agreement with numerical results and describe the convergence and stability of the proposed method to different roots when the Newton method fails. Full article
(This article belongs to the Special Issue Computational Methods in Analysis and Applications 2023)
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24 pages, 569 KiB  
Article
A New Parameter Choice Strategy for Lavrentiev Regularization Method for Nonlinear Ill-Posed Equations
by Santhosh George, Jidesh Padikkal, Krishnendu Remesh and Ioannis K. Argyros
Mathematics 2022, 10(18), 3365; https://doi.org/10.3390/math10183365 - 16 Sep 2022
Cited by 3 | Viewed by 1063
Abstract
In this paper, we introduced a new source condition and a new parameter-choice strategy which also gives the known best error estimate. To obtain the results we used the assumptions used in earlier studies. Further, we studied the proposed new parameter-choice strategy and [...] Read more.
In this paper, we introduced a new source condition and a new parameter-choice strategy which also gives the known best error estimate. To obtain the results we used the assumptions used in earlier studies. Further, we studied the proposed new parameter-choice strategy and applied it to the method (in the finite-dimensional setting) considered in George and Nair (2017). Full article
(This article belongs to the Special Issue Computational Methods in Analysis and Applications 2023)
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