Efficient <i>k</i>-Step Linear Block Methods to Solve Second Order Initial Value Problems Directly
There are dozens of block methods in literature intended for solving second order initial-value problems. This article aimed at the analysis of the efficiency of <i>k</i>-step block methods for directly solving general second-order initial-value problems. Each of these methods consists o...
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MDPI AG
2020-10-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/10/1752 |
| _version_ | 1797551302250594304 |
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| author | Higinio Ramos Samuel N. Jator Mark I. Modebei |
| author_facet | Higinio Ramos Samuel N. Jator Mark I. Modebei |
| author_sort | Higinio Ramos |
| collection | DOAJ |
| description | There are dozens of block methods in literature intended for solving second order initial-value problems. This article aimed at the analysis of the efficiency of <i>k</i>-step block methods for directly solving general second-order initial-value problems. Each of these methods consists of a set of <inline-formula><math display="inline"><semantics><mrow><mn>2</mn><mi>k</mi></mrow></semantics></math></inline-formula> multi-step formulas (although we will see that this number can be reduced to <inline-formula><math display="inline"><semantics><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></semantics></math></inline-formula> in case of a special equation) that provides approximate solutions at <i>k</i> grid points at once. The usual way to obtain these formulas is by using collocation and interpolation at different points, which are not all necessarily in the mesh (it may also be considered intra-step or off-step points). An important issue is that for each <i>k</i>, all of them are essentially the same method, although they can adopt different formulations. Nevertheless, the performance of those formulations is not the same. The analysis of the methods presented give some clues as how to select the most appropriate ones in terms of computational efficiency. The numerical experiments show that using the proposed formulations, the computing time can be reduced to less than half. |
| first_indexed | 2024-03-10T15:42:37Z |
| format | Article |
| id | doaj.art-a5c9a9a79ae943719c13c7e437babc08 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T15:42:37Z |
| publishDate | 2020-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-a5c9a9a79ae943719c13c7e437babc082023-11-20T16:43:25ZengMDPI AGMathematics2227-73902020-10-01810175210.3390/math8101752Efficient <i>k</i>-Step Linear Block Methods to Solve Second Order Initial Value Problems DirectlyHiginio Ramos0Samuel N. Jator1Mark I. Modebei2Department of Applied Mathematics, University of Salamanca, 37008 Salamanca, SpainDepartment of Mathematics and Statistics, Austin Peay State University Clarksville, Clarksville, TN 37044, USADepartment of Mathematics Programme, National Mathematical Centre, Abuja 900211, NigeriaThere are dozens of block methods in literature intended for solving second order initial-value problems. This article aimed at the analysis of the efficiency of <i>k</i>-step block methods for directly solving general second-order initial-value problems. Each of these methods consists of a set of <inline-formula><math display="inline"><semantics><mrow><mn>2</mn><mi>k</mi></mrow></semantics></math></inline-formula> multi-step formulas (although we will see that this number can be reduced to <inline-formula><math display="inline"><semantics><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></semantics></math></inline-formula> in case of a special equation) that provides approximate solutions at <i>k</i> grid points at once. The usual way to obtain these formulas is by using collocation and interpolation at different points, which are not all necessarily in the mesh (it may also be considered intra-step or off-step points). An important issue is that for each <i>k</i>, all of them are essentially the same method, although they can adopt different formulations. Nevertheless, the performance of those formulations is not the same. The analysis of the methods presented give some clues as how to select the most appropriate ones in terms of computational efficiency. The numerical experiments show that using the proposed formulations, the computing time can be reduced to less than half.https://www.mdpi.com/2227-7390/8/10/1752ordinary differential equationssecond-order initial value problemsmultistep block methodscomputational efficiency |
| spellingShingle | Higinio Ramos Samuel N. Jator Mark I. Modebei Efficient <i>k</i>-Step Linear Block Methods to Solve Second Order Initial Value Problems Directly Mathematics ordinary differential equations second-order initial value problems multistep block methods computational efficiency |
| title | Efficient <i>k</i>-Step Linear Block Methods to Solve Second Order Initial Value Problems Directly |
| title_full | Efficient <i>k</i>-Step Linear Block Methods to Solve Second Order Initial Value Problems Directly |
| title_fullStr | Efficient <i>k</i>-Step Linear Block Methods to Solve Second Order Initial Value Problems Directly |
| title_full_unstemmed | Efficient <i>k</i>-Step Linear Block Methods to Solve Second Order Initial Value Problems Directly |
| title_short | Efficient <i>k</i>-Step Linear Block Methods to Solve Second Order Initial Value Problems Directly |
| title_sort | efficient i k i step linear block methods to solve second order initial value problems directly |
| topic | ordinary differential equations second-order initial value problems multistep block methods computational efficiency |
| url | https://www.mdpi.com/2227-7390/8/10/1752 |
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