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Axioms
Special Issues
The Numerical Analysis and Its Application
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The Numerical Analysis and Its Application

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: 26 February 2025 | Viewed by 5715

Special Issue Editor

Dr. Janis Bajars

E-Mail Website
Guest Editor
Faculty of Physics, Mathematics and Optometry, University of Latvia, Rīga, Latvia
Interests: numerical methods for differential and integral equations; dynamical systems; crystal lattice models; wave phenomena; data-driven algorithms

Special Issue Information

Dear Colleagues,

The value of applied numerical analysis and the power of computational modelling of real-life phenomena cannot be understated. The applications range from the computer, physical, life, and social sciences to engineering and emerging technologies. Further research in advancing applied and computational mathematics is indispensable for resolving real-life, engineering, technological, and scientific problems.

This Special Issue is dedicated to illustrating the numerical analysis and development of novel computational techniques and algorithms advancing diverse applications of mathematical modelling, theoretical physics, dynamical systems, differential and integral equations, optimization methods, structure-preserving methods, learning algorithms, and data-driven methods, but is not limited to these research fields only. With this Special Issue, we aim to appraise either the improvements of existing computational methods or demonstrate new algorithmic and computational methodologies, including learning and data-driven methods. For this Special Issue, theoretical and applied contributions with extensive numerical studies for different applications are welcome. Contributions demonstrating improved analytical results and computational efficiency of already existing or new algorithms as well as the discovery of new phenomena driven by the development of numerical algorithms are of particular interest.

Dr. Janis Bajars
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. Axioms is an international peer-reviewed open access monthly 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 2400 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

  • applied numerical analysis
  • computational mathematics
  • numerical methods for differential and integral equations
  • geometric numerical integration
  • data-driven dynamical systems
  • physics-informed learning
  • optimization techniques in mathematical modelling

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Published Papers (6 papers)

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Research

22 pages, 2921 KiB  
Article
A Robust and Optimal Iterative Algorithm Employing a Weight Function for Solving Nonlinear Equations with Dynamics and Applications
by Shahid Abdullah, Neha Choubey, Suresh Dara, Moin-ud-Din Junjua and Tawseef Abdullah
Axioms 2024, 13(10), 675; https://doi.org/10.3390/axioms13100675 - 30 Sep 2024
Viewed by 905
Abstract
This study introduces a novel, iterative algorithm that achieves fourth-order convergence for solving nonlinear equations. Satisfying the Kung–Traub conjecture, the proposed technique achieves an optimal order of four with an efficiency index (I) of 1.587, requiring three function evaluations. An [...] Read more.
This study introduces a novel, iterative algorithm that achieves fourth-order convergence for solving nonlinear equations. Satisfying the Kung–Traub conjecture, the proposed technique achieves an optimal order of four with an efficiency index (I) of 1.587, requiring three function evaluations. An analysis of convergence is presented to show the optimal fourth-order convergence. To verify the theoretical results, in-depth numerical comparisons are presented for both real and complex domains. The proposed algorithm is specifically examined on a variety of polynomial functions, and it is shown by the efficient and accurate results that it outperforms many existing algorithms in terms of speed and accuracy. The study not only explores the proposed method’s convergence properties, computational efficiency, and stability but also introduces a novel perspective by considering the count of black points as an indicator of a method’s divergence. By analyzing the mean number of iterations necessary for methods to converge within a cycle and measuring CPU time in seconds, this research provides a holistic assessment of both the efficiency and speed of iterative methods. Notably, the analysis of basins of attraction illustrates that our proposed method has larger sets of initial points that yield convergence. Full article
(This article belongs to the Special Issue The Numerical Analysis and Its Application)
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38 pages, 3307 KiB  
Article
A New Methodology for the Development of Efficient Multistep Methods for First–Order IVPs with Oscillating Solutions IV: The Case of the Backward Differentiation Formulae
by Theodore E. Simos
Axioms 2024, 13(9), 649; https://doi.org/10.3390/axioms13090649 - 22 Sep 2024
Viewed by 530
Abstract
A theory for the calculation of the phase–lag and amplification–factor for explicit and implicit multistep techniques for first–order differential equations was recently established by the author. His presentation also covered how the approaches’ efficacy is affected by the elimination of the phase–lag and [...] Read more.
A theory for the calculation of the phase–lag and amplification–factor for explicit and implicit multistep techniques for first–order differential equations was recently established by the author. His presentation also covered how the approaches’ efficacy is affected by the elimination of the phase–lag and amplification–factor derivatives. This paper will apply the theory for computing the phase–lag and amplification–factor, originally developed for implicit multistep methods, to a subset of implicit methods, called backward differentiation formulae (BDF), and will examine the impact of the phase–lag and amplification–factor derivatives on the efficiency of these strategies. Next, we will show you the stability zones of these brand-new approaches. Lastly, we will discuss the results of numerical experiments and draw some conclusions about the established approaches. Full article
(This article belongs to the Special Issue The Numerical Analysis and Its Application)
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14 pages, 654 KiB  
Article
Numerical Solution of Third-Order Rosenau–Hyman and Fornberg–Whitham Equations via B-Spline Interpolation Approach
by Tanveer Akbar, Sirajul Haq, Shams Ul Arifeen and Azhar Iqbal
Axioms 2024, 13(8), 501; https://doi.org/10.3390/axioms13080501 - 26 Jul 2024
Cited by 1 | Viewed by 638
Abstract
This study aims to find the numerical solution of the Rosenau–Hyman and Fornberg–Whitham equations via the quintic B-spline collocation method. Quintic B-spline, along with finite difference and theta-weighted schemes, is used for the discretization and approximation purposes. The effectiveness and robustness of the [...] Read more.
This study aims to find the numerical solution of the Rosenau–Hyman and Fornberg–Whitham equations via the quintic B-spline collocation method. Quintic B-spline, along with finite difference and theta-weighted schemes, is used for the discretization and approximation purposes. The effectiveness and robustness of the procedure is assessed by comparing the computed results with the exact and available results in the literature using absolute and relative error norms. The stability of the proposed scheme is studied using von Neumann stability analysis. Graphical representations are drawn to analyze the behavior of the solution. Full article
(This article belongs to the Special Issue The Numerical Analysis and Its Application)
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19 pages, 1790 KiB  
Article
Operator Smith Algorithm for Coupled Stein Equations from Jump Control Systems
by Bo Yu, Ning Dong and Baiquan Hu
Axioms 2024, 13(4), 249; https://doi.org/10.3390/axioms13040249 - 10 Apr 2024
Viewed by 795
Abstract
Consider a class of coupled Stein equations arising from jump control systems. An operator Smith algorithm is proposed for calculating the solution of the system. Convergence of the algorithm is established under certain conditions. For large-scale systems, the operator Smith algorithm is extended [...] Read more.
Consider a class of coupled Stein equations arising from jump control systems. An operator Smith algorithm is proposed for calculating the solution of the system. Convergence of the algorithm is established under certain conditions. For large-scale systems, the operator Smith algorithm is extended to a low-rank structured format, and the error of the algorithm is analyzed. Numerical experiments demonstrate that the operator Smith iteration outperforms existing linearly convergent iterative methods in terms of computation time and accuracy. The low-rank structured iterative format is highly effective in approximating the solutions of large-scale structured problems. Full article
(This article belongs to the Special Issue The Numerical Analysis and Its Application)
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14 pages, 1500 KiB  
Article
Computational Approach to Third-Order Nonlinear Boundary Value Problems via Efficient Decomposition Shooting Method
by K. A. Alzahrani, N. A. Alzaid, H. O. Bakodah and M. H. Almazmumy
Axioms 2024, 13(4), 248; https://doi.org/10.3390/axioms13040248 - 9 Apr 2024
Viewed by 1028
Abstract
The present manuscript proposes a computational approach to efficiently tackle a class of two-point boundary value problems that features third-order nonlinear ordinary differential equations. Specifically, this approach is based upon a combination of the shooting method with a modification of the renowned Adomian [...] Read more.
The present manuscript proposes a computational approach to efficiently tackle a class of two-point boundary value problems that features third-order nonlinear ordinary differential equations. Specifically, this approach is based upon a combination of the shooting method with a modification of the renowned Adomian decomposition method. The approach starts by transforming the governing BVP into two appropriate initial-value problems, and thereafter, solves the resulting IVPs recurrently. In addition, the application of this method to varied test models remains feasible—of course, this is supported by the competing Runge–Kutta method, among others, and reported through comparison plots and tables. Full article
(This article belongs to the Special Issue The Numerical Analysis and Its Application)
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16 pages, 437 KiB  
Article
A Reliable Combination of Double Laplace Transform and Homotopy Analysis Method for Solving a Singular Nonlocal Problem with Bessel Operator
by Said Mesloub, Hassan Eltayeb Gadain and Imed Bachar
Axioms 2023, 12(10), 933; https://doi.org/10.3390/axioms12100933 - 29 Sep 2023
Viewed by 795
Abstract
In this article, we present a numerical iterative scheme for solving a non-local singular initial-boundary value problem by combining two well known efficient methods. Namely, the homotopy analysis method and the double Laplace transform method. The resulting scheme is tested on a set [...] Read more.
In this article, we present a numerical iterative scheme for solving a non-local singular initial-boundary value problem by combining two well known efficient methods. Namely, the homotopy analysis method and the double Laplace transform method. The resulting scheme is tested on a set of test examples to illustrate its efficiency, it generates the exact analytical solution for each one of these examples. The convergence of the resulting numerical solutions of these examples is tested both graphically and numerically. Full article
(This article belongs to the Special Issue The Numerical Analysis and Its Application)
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