Memory-Accelerating Methods for One-Step Iterative Schemes with Lie Symmetry Method Solving Nonlinear Boundary-Value Problem
In this paper, some one-step iterative schemes with memory-accelerating methods are proposed to update three critical values <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>f</mi&g...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2024-01-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/16/1/120 |
| _version_ | 1797342509392723968 |
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| author | Chein-Shan Liu Chih-Wen Chang Chung-Lun Kuo |
| author_facet | Chein-Shan Liu Chih-Wen Chang Chung-Lun Kuo |
| author_sort | Chein-Shan Liu |
| collection | DOAJ |
| description | In this paper, some one-step iterative schemes with memory-accelerating methods are proposed to update three critical values <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>r</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>f</mi><mrow><mo>″</mo></mrow></msup><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>f</mi><mrow><mo>‴</mo></mrow></msup><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of a nonlinear equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> with <i>r</i> being its simple root. We can achieve high values of the efficiency index (E.I.) over the bound <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mn>2</mn><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>=</mo><mn>1.587</mn></mrow></semantics></math></inline-formula> with three function evaluations and over the bound <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mn>2</mn><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>=</mo><mn>1.414</mn></mrow></semantics></math></inline-formula> with two function evaluations. The third-degree Newton interpolatory polynomial is derived to update these critical values per iteration. We introduce relaxation factors into the D<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi mathvariant="normal">z</mi><mo>ˇ</mo></mover></semantics></math></inline-formula>uni<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi mathvariant="normal">c</mi><mo>´</mo></mover></semantics></math></inline-formula> method and its variant, which are updated to render fourth-order convergence by the memory-accelerating technique. We developed six types optimal one-step iterative schemes with the memory-accelerating method, rendering a fourth-order convergence or even more, whose original ones are a second-order convergence without memory and without using specific optimal values of the parameters. We evaluated the performance of these one-step iterative schemes by the computed order of convergence (COC) and the E.I. with numerical tests. A Lie symmetry method to solve a second-order nonlinear boundary-value problem with high efficiency and high accuracy was developed. |
| first_indexed | 2024-03-08T10:34:36Z |
| format | Article |
| id | doaj.art-9cce3fc5769649d18472c040d5a66184 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-08T10:34:36Z |
| publishDate | 2024-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-9cce3fc5769649d18472c040d5a661842024-01-26T18:39:09ZengMDPI AGSymmetry2073-89942024-01-0116112010.3390/sym16010120Memory-Accelerating Methods for One-Step Iterative Schemes with Lie Symmetry Method Solving Nonlinear Boundary-Value ProblemChein-Shan Liu0Chih-Wen Chang1Chung-Lun Kuo2Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 202301, TaiwanDepartment of Mechanical Engineering, National United University, Miaoli 360302, TaiwanCenter of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 202301, TaiwanIn this paper, some one-step iterative schemes with memory-accelerating methods are proposed to update three critical values <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>r</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>f</mi><mrow><mo>″</mo></mrow></msup><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>f</mi><mrow><mo>‴</mo></mrow></msup><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of a nonlinear equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> with <i>r</i> being its simple root. We can achieve high values of the efficiency index (E.I.) over the bound <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mn>2</mn><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>=</mo><mn>1.587</mn></mrow></semantics></math></inline-formula> with three function evaluations and over the bound <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mn>2</mn><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>=</mo><mn>1.414</mn></mrow></semantics></math></inline-formula> with two function evaluations. The third-degree Newton interpolatory polynomial is derived to update these critical values per iteration. We introduce relaxation factors into the D<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi mathvariant="normal">z</mi><mo>ˇ</mo></mover></semantics></math></inline-formula>uni<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi mathvariant="normal">c</mi><mo>´</mo></mover></semantics></math></inline-formula> method and its variant, which are updated to render fourth-order convergence by the memory-accelerating technique. We developed six types optimal one-step iterative schemes with the memory-accelerating method, rendering a fourth-order convergence or even more, whose original ones are a second-order convergence without memory and without using specific optimal values of the parameters. We evaluated the performance of these one-step iterative schemes by the computed order of convergence (COC) and the E.I. with numerical tests. A Lie symmetry method to solve a second-order nonlinear boundary-value problem with high efficiency and high accuracy was developed.https://www.mdpi.com/2073-8994/16/1/120optimal fourth-order one-step iterative schemesmemory-accelerating methodoptimal combination functionoptimal relaxation factorLie symmetry method |
| spellingShingle | Chein-Shan Liu Chih-Wen Chang Chung-Lun Kuo Memory-Accelerating Methods for One-Step Iterative Schemes with Lie Symmetry Method Solving Nonlinear Boundary-Value Problem Symmetry optimal fourth-order one-step iterative schemes memory-accelerating method optimal combination function optimal relaxation factor Lie symmetry method |
| title | Memory-Accelerating Methods for One-Step Iterative Schemes with Lie Symmetry Method Solving Nonlinear Boundary-Value Problem |
| title_full | Memory-Accelerating Methods for One-Step Iterative Schemes with Lie Symmetry Method Solving Nonlinear Boundary-Value Problem |
| title_fullStr | Memory-Accelerating Methods for One-Step Iterative Schemes with Lie Symmetry Method Solving Nonlinear Boundary-Value Problem |
| title_full_unstemmed | Memory-Accelerating Methods for One-Step Iterative Schemes with Lie Symmetry Method Solving Nonlinear Boundary-Value Problem |
| title_short | Memory-Accelerating Methods for One-Step Iterative Schemes with Lie Symmetry Method Solving Nonlinear Boundary-Value Problem |
| title_sort | memory accelerating methods for one step iterative schemes with lie symmetry method solving nonlinear boundary value problem |
| topic | optimal fourth-order one-step iterative schemes memory-accelerating method optimal combination function optimal relaxation factor Lie symmetry method |
| url | https://www.mdpi.com/2073-8994/16/1/120 |
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