On the stability analysis of numerical schemes for solving non-linear polynomials arises in engineering problems
This study shows the link between computer science and applied mathematics. It conducts a dynamics investigation of new root solvers using computer tools and develops a new family of single-step simple root-finding methods. The convergence order of the proposed family of iterative methods is two, ac...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
AIMS Press
2024-03-01
|
| Series: | AIMS Mathematics |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2024433?viewType=HTML |
| _version_ | 1827317925538693120 |
|---|---|
| author | Mudassir Shams Nasreen Kausar Serkan Araci Liang Kong |
| author_facet | Mudassir Shams Nasreen Kausar Serkan Araci Liang Kong |
| author_sort | Mudassir Shams |
| collection | DOAJ |
| description | This study shows the link between computer science and applied mathematics. It conducts a dynamics investigation of new root solvers using computer tools and develops a new family of single-step simple root-finding methods. The convergence order of the proposed family of iterative methods is two, according to the convergence analysis carried out using symbolic computation in the computer algebra system CAS-Maple 18. Without further evaluations of a given nonlinear function and its derivatives, a very rapid convergence rate is achieved, demonstrating the remarkable computing efficiency of the novel technique. To determine the simple roots of nonlinear equations, this paper discusses the dynamic analysis of one-parameter families using symbolic computation, computer animation, and multi-precision arithmetic. To choose the best parametric value used in iterative schemes, it implements the parametric and dynamical plane technique using CAS-MATLAB$ ^{@}R2011b. $ The dynamic evaluation of the methods is also presented utilizing basins of attraction to analyze their convergence behavior. Aside from visualizing iterative processes, this method illustrates not only iterative processes but also gives useful information regarding the convergence of the numerical scheme based on initial guessed values. Some nonlinear problems that arise in science and engineering are used to demonstrate the performance and efficiency of the newly developed method compared to the existing method in the literature. |
| first_indexed | 2024-04-24T23:49:01Z |
| format | Article |
| id | doaj.art-bbfe3e0012084d3d8ba59a192a1d5cfd |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-04-24T23:49:01Z |
| publishDate | 2024-03-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-bbfe3e0012084d3d8ba59a192a1d5cfd2024-03-15T01:20:24ZengAIMS PressAIMS Mathematics2473-69882024-03-01948885890310.3934/math.2024433On the stability analysis of numerical schemes for solving non-linear polynomials arises in engineering problemsMudassir Shams 0Nasreen Kausar1Serkan Araci2Liang Kong31. Faculty of Engineering, Free University of Bozen-Bolzano (BZ), 39100, Italy 2. Department of Mathematics and Statistics, Riphah International University I-14, Islamabad 44000, Pakistan3. Department of Mathematics, Faculty of Arts and Science, Yildiz Technical University, Esenler 34220, Istanbul, Türkiye4. Department of Basic Sciences, Faculty of Engineering, Hasan Kalyoncu University, TR-27010 Gaziantep, Türkiye5. Department of Mathematical Sciences and Philosophy, University of Illinois Springfield, USAThis study shows the link between computer science and applied mathematics. It conducts a dynamics investigation of new root solvers using computer tools and develops a new family of single-step simple root-finding methods. The convergence order of the proposed family of iterative methods is two, according to the convergence analysis carried out using symbolic computation in the computer algebra system CAS-Maple 18. Without further evaluations of a given nonlinear function and its derivatives, a very rapid convergence rate is achieved, demonstrating the remarkable computing efficiency of the novel technique. To determine the simple roots of nonlinear equations, this paper discusses the dynamic analysis of one-parameter families using symbolic computation, computer animation, and multi-precision arithmetic. To choose the best parametric value used in iterative schemes, it implements the parametric and dynamical plane technique using CAS-MATLAB$ ^{@}R2011b. $ The dynamic evaluation of the methods is also presented utilizing basins of attraction to analyze their convergence behavior. Aside from visualizing iterative processes, this method illustrates not only iterative processes but also gives useful information regarding the convergence of the numerical scheme based on initial guessed values. Some nonlinear problems that arise in science and engineering are used to demonstrate the performance and efficiency of the newly developed method compared to the existing method in the literature.https://www.aimspress.com/article/doi/10.3934/math.2024433?viewType=HTMLcomplex dynamicsparametric planestability regioncpu timeengineering application |
| spellingShingle | Mudassir Shams Nasreen Kausar Serkan Araci Liang Kong On the stability analysis of numerical schemes for solving non-linear polynomials arises in engineering problems AIMS Mathematics complex dynamics parametric plane stability region cpu time engineering application |
| title | On the stability analysis of numerical schemes for solving non-linear polynomials arises in engineering problems |
| title_full | On the stability analysis of numerical schemes for solving non-linear polynomials arises in engineering problems |
| title_fullStr | On the stability analysis of numerical schemes for solving non-linear polynomials arises in engineering problems |
| title_full_unstemmed | On the stability analysis of numerical schemes for solving non-linear polynomials arises in engineering problems |
| title_short | On the stability analysis of numerical schemes for solving non-linear polynomials arises in engineering problems |
| title_sort | on the stability analysis of numerical schemes for solving non linear polynomials arises in engineering problems |
| topic | complex dynamics parametric plane stability region cpu time engineering application |
| url | https://www.aimspress.com/article/doi/10.3934/math.2024433?viewType=HTML |
| work_keys_str_mv | AT mudassirshams onthestabilityanalysisofnumericalschemesforsolvingnonlinearpolynomialsarisesinengineeringproblems AT nasreenkausar onthestabilityanalysisofnumericalschemesforsolvingnonlinearpolynomialsarisesinengineeringproblems AT serkanaraci onthestabilityanalysisofnumericalschemesforsolvingnonlinearpolynomialsarisesinengineeringproblems AT liangkong onthestabilityanalysisofnumericalschemesforsolvingnonlinearpolynomialsarisesinengineeringproblems |