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AppliedMath
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Contemporary Iterative Methods with Applications in Applied Sciences
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Contemporary Iterative Methods with Applications in Applied Sciences

A special issue of AppliedMath (ISSN 2673-9909).

Deadline for manuscript submissions: 30 April 2024 | Viewed by 2556

Special Issue Editors

Dr. Jai Prakash Jaiswal

E-Mail Website
Guest Editor
Department of Mathematics, Guru Ghasidas Vishwavidyalaya (A Central University), Bilaspur 495009, CG, India
Interests: numerical methods; numerical analysis; numerical functional analysis
Prof. Dr. Juan Ramón Torregrosa Sánchez

E-Mail Website
Guest Editor
Institute for Multidisciplinary Mathematics, Universitat Politècnica de València, 46022 València, Spain
Interests: iterative processes; matrix analysis; numerical analysis
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The unknowns of engineering equations can be functions (difference, differential and integral equations), vectors (systems of linear or non-linear algebraic equations), or real or complex numbers (single algebraic equations with single unknowns). Except for some special cases, the most commonly used solutions methods are iterative; when starting from one or several initial approximations, a sequence is constructed, which converges to a solution of the equation.

To complicate the matter further, many of these equations are non-linear. The local convergence of iterative methods (without and with memory IM for single or multivariate analysis) plays an important role in analyzing their rate of convergence and lowest requirement of presumption. The local convergence is also very important because it reveals the degree of difficulty in selecting initial points for the iterative method. The study of semilocal convergence for an iterative method I Banach spaces is very interesting because just by imposing conditions on the starting point, instead of on the solution, important results can be obtained, such as the existence and uniqueness of the solution, convergence order, a priori error bounds and convergence domains. These results can be applied to the solution of some practical problems arising from Mathematical Biology, Chemistry, Economics, Medicine, Physics, Engineering Science and Scientific Computing which are described by differential equations, partial differential equations and integral equations.

The papers are invited but not limited on the following topics:

  1. Iterative methods with and without memory and their applications.
  2. Derivative and derivative-free iterative techniques for non-linear systems and their applications.
  3. Local and semi-local convergence analysis of non-linear problems and their applications.

Dr. Jai Prakash Jaiswal
Prof. Dr. Juan Ramón Torregrosa Sánchez
Guest Editors

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. AppliedMath is an international peer-reviewed open access quarterly 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 1000 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

  • with and without memory iterative methods
  • with derivative & derivative-free iterative techniques for nonlinear systems
  • local and semi-local convergence of iterative methods

Published Papers (3 papers)

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Research

28 pages, 4033 KiB  
Article
A Block Hybrid Method with Equally Spaced Grid Points for Third-Order Initial Value Problems
by Salma A. A. Ahmedai Abd Allah, Precious Sibanda, Sicelo P. Goqo, Uthman O. Rufai, Hloniphile Sithole Mthethwa and Osman A. I. Noreldin
AppliedMath 2024, 4(1), 320-347; https://doi.org/10.3390/appliedmath4010017 - 01 Mar 2024
Viewed by 620
Abstract
In this paper, we extend the block hybrid method with equally spaced intra-step points to solve linear and nonlinear third-order initial value problems. The proposed block hybrid method uses a simple iteration scheme to linearize the equations. Numerical experimentation demonstrates that equally spaced [...] Read more.
In this paper, we extend the block hybrid method with equally spaced intra-step points to solve linear and nonlinear third-order initial value problems. The proposed block hybrid method uses a simple iteration scheme to linearize the equations. Numerical experimentation demonstrates that equally spaced grid points for the block hybrid method enhance its speed of convergence and accuracy compared to other conventional block hybrid methods in the literature. This improvement is attributed to the linearization process, which avoids the use of derivatives. Further, the block hybrid method is consistent, stable, and gives rapid convergence to the solutions. We show that the simple iteration method, when combined with the block hybrid method, exhibits impressive convergence characteristics while preserving computational efficiency. In this study, we also implement the proposed method to solve the nonlinear Jerk equation, producing comparable results with other methods used in the literature. Full article
(This article belongs to the Special Issue Contemporary Iterative Methods with Applications in Applied Sciences)
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39 pages, 5733 KiB  
Article
Convergence and Stability Improvement of Quasi-Newton Methods by Full-Rank Update of the Jacobian Approximates
by Peter Berzi
AppliedMath 2024, 4(1), 143-181; https://doi.org/10.3390/appliedmath4010008 - 26 Jan 2024
Viewed by 506
Abstract
A system of simultaneous multi-variable nonlinear equations can be solved by Newton’s method with local q-quadratic convergence if the Jacobian is analytically available. If this is not the case, then quasi-Newton methods with local q-superlinear convergence give solutions by approximating the Jacobian in [...] Read more.
A system of simultaneous multi-variable nonlinear equations can be solved by Newton’s method with local q-quadratic convergence if the Jacobian is analytically available. If this is not the case, then quasi-Newton methods with local q-superlinear convergence give solutions by approximating the Jacobian in some way. Unfortunately, the quasi-Newton condition (Secant equation) does not completely specify the Jacobian approximate in multi-dimensional cases, so its full-rank update is not possible with classic variants of the method. The suggested new iteration strategy (“T-Secant”) allows for a full-rank update of the Jacobian approximate in each iteration by determining two independent approximates for the solution. They are used to generate a set of new independent trial approximates; then, the Jacobian approximate can be fully updated. It is shown that the T-Secant approximate is in the vicinity of the classic quasi-Newton approximate, providing that the solution is evenly surrounded by the new trial approximates. The suggested procedure increases the superlinear convergence of the Secant method φS=1.618… to super-quadratic φT=φS+1=2.618… and the quadratic convergence of the Newton method φN=2 to cubic φT=φN+1=3 in one-dimensional cases. In multi-dimensional cases, the Broyden-type efficiency (mean convergence rate) of the suggested method is an order higher than the efficiency of other classic low-rank-update quasi-Newton methods, as shown by numerical examples on a Rosenbrock-type test function with up to 1000 variables. The geometrical representation (hyperbolic approximation) in single-variable cases helps explain the basic operations, and a vector-space description is also given in multi-variable cases. Full article
(This article belongs to the Special Issue Contemporary Iterative Methods with Applications in Applied Sciences)
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15 pages, 488 KiB  
Article
An Efficient Bi-Parametric With-Memory Iterative Method for Solving Nonlinear Equations
by Ekta Sharma, Shubham Kumar Mittal, J. P. Jaiswal and Sunil Panday
AppliedMath 2023, 3(4), 1019-1033; https://doi.org/10.3390/appliedmath3040051 - 11 Dec 2023
Viewed by 694
Abstract
New three-step with-memory iterative methods for solving nonlinear equations are presented. We have enhanced the convergence order of an existing eighth-order memory-less iterative method by transforming it into a with-memory method. Enhanced acceleration of the convergence order is achieved by introducing two self-accelerating [...] Read more.
New three-step with-memory iterative methods for solving nonlinear equations are presented. We have enhanced the convergence order of an existing eighth-order memory-less iterative method by transforming it into a with-memory method. Enhanced acceleration of the convergence order is achieved by introducing two self-accelerating parameters computed using the Hermite interpolating polynomial. The corresponding R-order of convergence of the proposed uni- and bi-parametric with-memory methods is increased from 8 to 9 and 10, respectively. This increase in convergence order is accomplished without requiring additional function evaluations, making the with-memory method computationally efficient. The efficiency of our with-memory methods NWM9 and NWM10 increases from 1.6818 to 1.7320 and 1.7783, respectively. Numeric testing confirms the theoretical findings and emphasizes the superior efficacy of suggested methods when compared to some well-known methods in the existing literature. Full article
(This article belongs to the Special Issue Contemporary Iterative Methods with Applications in Applied Sciences)
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