An Efficient Two-Step Iterative Family Adaptive with Memory for Solving Nonlinear Equations and Their Applications
We propose a new iterative scheme without memory for solving nonlinear equations. The proposed scheme is based on a cubically convergent Hansen–Patrick-type method. The beauty of our techniques is that they work even though the derivative is very small in the vicinity of the required root or <inl...
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MDPI AG
2022-11-01
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| Series: | Mathematical and Computational Applications |
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| Online Access: | https://www.mdpi.com/2297-8747/27/6/97 |
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| author | Himani Sharma Munish Kansal Ramandeep Behl |
| author_facet | Himani Sharma Munish Kansal Ramandeep Behl |
| author_sort | Himani Sharma |
| collection | DOAJ |
| description | We propose a new iterative scheme without memory for solving nonlinear equations. The proposed scheme is based on a cubically convergent Hansen–Patrick-type method. The beauty of our techniques is that they work even though the derivative is very small in the vicinity of the required root or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>f</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>. On the contrary, the previous modifications either diverge or fail to work. In addition, we also extended the same idea for an iterative method with memory. Numerical examples and comparisons with some of the existing methods are included to confirm the theoretical results. Furthermore, basins of attraction are included to describe a clear picture of the convergence of the proposed method as well as that of some of the existing methods. Numerical experiments are performed on engineering problems, such as fractional conversion in a chemical reactor, Planck’s radiation law problem, Van der Waal’s problem, trajectory of an electron in between two parallel plates. The numerical results reveal that the proposed schemes are of utmost importance to be applied on various real–life problems. Basins of attraction also support this aspect. |
| first_indexed | 2024-03-09T16:08:36Z |
| format | Article |
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| institution | Directory Open Access Journal |
| issn | 1300-686X 2297-8747 |
| language | English |
| last_indexed | 2024-03-09T16:08:36Z |
| publishDate | 2022-11-01 |
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| spelling | doaj.art-f4266203b5c84f708cfa12babd86fd952023-11-24T16:30:49ZengMDPI AGMathematical and Computational Applications1300-686X2297-87472022-11-012769710.3390/mca27060097An Efficient Two-Step Iterative Family Adaptive with Memory for Solving Nonlinear Equations and Their ApplicationsHimani Sharma0Munish Kansal1Ramandeep Behl2School of Mathematics, Thapar Institute of Engineering and Technology, Patiala 147004, IndiaSchool of Mathematics, Thapar Institute of Engineering and Technology, Patiala 147004, IndiaDepartment of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi ArabiaWe propose a new iterative scheme without memory for solving nonlinear equations. The proposed scheme is based on a cubically convergent Hansen–Patrick-type method. The beauty of our techniques is that they work even though the derivative is very small in the vicinity of the required root or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>f</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>. On the contrary, the previous modifications either diverge or fail to work. In addition, we also extended the same idea for an iterative method with memory. Numerical examples and comparisons with some of the existing methods are included to confirm the theoretical results. Furthermore, basins of attraction are included to describe a clear picture of the convergence of the proposed method as well as that of some of the existing methods. Numerical experiments are performed on engineering problems, such as fractional conversion in a chemical reactor, Planck’s radiation law problem, Van der Waal’s problem, trajectory of an electron in between two parallel plates. The numerical results reveal that the proposed schemes are of utmost importance to be applied on various real–life problems. Basins of attraction also support this aspect.https://www.mdpi.com/2297-8747/27/6/97nonlinear equationiterative method with memory<i>R</i>-order of convergencebasin of attraction |
| spellingShingle | Himani Sharma Munish Kansal Ramandeep Behl An Efficient Two-Step Iterative Family Adaptive with Memory for Solving Nonlinear Equations and Their Applications Mathematical and Computational Applications nonlinear equation iterative method with memory <i>R</i>-order of convergence basin of attraction |
| title | An Efficient Two-Step Iterative Family Adaptive with Memory for Solving Nonlinear Equations and Their Applications |
| title_full | An Efficient Two-Step Iterative Family Adaptive with Memory for Solving Nonlinear Equations and Their Applications |
| title_fullStr | An Efficient Two-Step Iterative Family Adaptive with Memory for Solving Nonlinear Equations and Their Applications |
| title_full_unstemmed | An Efficient Two-Step Iterative Family Adaptive with Memory for Solving Nonlinear Equations and Their Applications |
| title_short | An Efficient Two-Step Iterative Family Adaptive with Memory for Solving Nonlinear Equations and Their Applications |
| title_sort | efficient two step iterative family adaptive with memory for solving nonlinear equations and their applications |
| topic | nonlinear equation iterative method with memory <i>R</i>-order of convergence basin of attraction |
| url | https://www.mdpi.com/2297-8747/27/6/97 |
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