A Perturbed Milne’s Quadrature Rule for <i>n</i>-Times Differentiable Functions with <i>L<sup>p</sup></i>-Error Estimates
In this work, a perturbed Milne’s quadrature rule for <i>n</i>-times differentiable functions with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mi>p</mi>&l...
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MDPI AG
2023-08-01
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| Online Access: | https://www.mdpi.com/2075-1680/12/9/803 |
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| author | Ayman Hazaymeh Rania Saadeh Raed Hatamleh Mohammad W. Alomari Ahmad Qazza |
| author_facet | Ayman Hazaymeh Rania Saadeh Raed Hatamleh Mohammad W. Alomari Ahmad Qazza |
| author_sort | Ayman Hazaymeh |
| collection | DOAJ |
| description | In this work, a perturbed Milne’s quadrature rule for <i>n</i>-times differentiable functions with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mi>p</mi></msup></semantics></math></inline-formula>-error estimates is derived. One of the most important advantages of our result is that it is verified for <i>p</i>-variation and Lipschitz functions. Several error estimates involving <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mi>p</mi></msup></semantics></math></inline-formula>-bounds are proven. These estimates are useful if the fourth derivative is unbounded in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mo>∞</mo></msup></semantics></math></inline-formula>-norm or the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mi>p</mi></msup></semantics></math></inline-formula>-error estimate is less than the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mo>∞</mo></msup></semantics></math></inline-formula>-error estimate. Furthermore, since the classical Milne’s quadrature rule cannot be applied either when the fourth derivative is unbounded or does not exist, the proposed quadrature could be used alternatively. Numerical experiments showing that our proposed quadrature rule is better than the classical Milne rule for certain types of functions are also provided. The numerical experiments compare the accuracy of the proposed quadrature rule to the classical Milne rule when approximating different types of functions. The results show that, for certain types of functions, the proposed quadrature rule is more accurate than the classical Milne rule. |
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| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-10T23:04:03Z |
| publishDate | 2023-08-01 |
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| spelling | doaj.art-edce224e11dd4690aee977d510571cb02023-11-19T09:31:57ZengMDPI AGAxioms2075-16802023-08-0112980310.3390/axioms12090803A Perturbed Milne’s Quadrature Rule for <i>n</i>-Times Differentiable Functions with <i>L<sup>p</sup></i>-Error EstimatesAyman Hazaymeh0Rania Saadeh1Raed Hatamleh2Mohammad W. Alomari3Ahmad Qazza4Department of Mathematics, Faculty of Science and Information Technology, Jadara University, Irbid P.C. 21110, JordanDepartment of Mathematics, Faculty of Science, Zarqa University, Zarga 13110, JordanDepartment of Mathematics, Faculty of Science and Information Technology, Jadara University, Irbid P.C. 21110, JordanDepartment of Mathematics, Faculty of Science and Information Technology, Irbid National University, P.O. Box 2600, Irbid P.C. 21110, JordanDepartment of Mathematics, Faculty of Science, Zarqa University, Zarga 13110, JordanIn this work, a perturbed Milne’s quadrature rule for <i>n</i>-times differentiable functions with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mi>p</mi></msup></semantics></math></inline-formula>-error estimates is derived. One of the most important advantages of our result is that it is verified for <i>p</i>-variation and Lipschitz functions. Several error estimates involving <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mi>p</mi></msup></semantics></math></inline-formula>-bounds are proven. These estimates are useful if the fourth derivative is unbounded in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mo>∞</mo></msup></semantics></math></inline-formula>-norm or the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mi>p</mi></msup></semantics></math></inline-formula>-error estimate is less than the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mo>∞</mo></msup></semantics></math></inline-formula>-error estimate. Furthermore, since the classical Milne’s quadrature rule cannot be applied either when the fourth derivative is unbounded or does not exist, the proposed quadrature could be used alternatively. Numerical experiments showing that our proposed quadrature rule is better than the classical Milne rule for certain types of functions are also provided. The numerical experiments compare the accuracy of the proposed quadrature rule to the classical Milne rule when approximating different types of functions. The results show that, for certain types of functions, the proposed quadrature rule is more accurate than the classical Milne rule.https://www.mdpi.com/2075-1680/12/9/803Milne’s ruleSimpson’s rulequadrature ruleNewton–Cotes formulaenumerical integrationerror estimation |
| spellingShingle | Ayman Hazaymeh Rania Saadeh Raed Hatamleh Mohammad W. Alomari Ahmad Qazza A Perturbed Milne’s Quadrature Rule for <i>n</i>-Times Differentiable Functions with <i>L<sup>p</sup></i>-Error Estimates Axioms Milne’s rule Simpson’s rule quadrature rule Newton–Cotes formulae numerical integration error estimation |
| title | A Perturbed Milne’s Quadrature Rule for <i>n</i>-Times Differentiable Functions with <i>L<sup>p</sup></i>-Error Estimates |
| title_full | A Perturbed Milne’s Quadrature Rule for <i>n</i>-Times Differentiable Functions with <i>L<sup>p</sup></i>-Error Estimates |
| title_fullStr | A Perturbed Milne’s Quadrature Rule for <i>n</i>-Times Differentiable Functions with <i>L<sup>p</sup></i>-Error Estimates |
| title_full_unstemmed | A Perturbed Milne’s Quadrature Rule for <i>n</i>-Times Differentiable Functions with <i>L<sup>p</sup></i>-Error Estimates |
| title_short | A Perturbed Milne’s Quadrature Rule for <i>n</i>-Times Differentiable Functions with <i>L<sup>p</sup></i>-Error Estimates |
| title_sort | perturbed milne s quadrature rule for i n i times differentiable functions with i l sup p sup i error estimates |
| topic | Milne’s rule Simpson’s rule quadrature rule Newton–Cotes formulae numerical integration error estimation |
| url | https://www.mdpi.com/2075-1680/12/9/803 |
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