Numerical Approximations and Fractional Calculus: Extending Boole’s Rule with Riemann–LiouvilleFractional Integral Inequalities
This paper develops integral inequalities for first-order differentiable convex functions within the framework of fractional calculus, extending Boole-type inequalities to this domain. An integral equality involving Riemann–Liouville fractional integrals is established, forming the foundation for de...
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2025-01-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/9/1/52 |
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| author | Abdul Mateen Wali Haider Asia Shehzadi Hüseyin Budak Bandar Bin-Mohsin |
| author_facet | Abdul Mateen Wali Haider Asia Shehzadi Hüseyin Budak Bandar Bin-Mohsin |
| author_sort | Abdul Mateen |
| collection | DOAJ |
| description | This paper develops integral inequalities for first-order differentiable convex functions within the framework of fractional calculus, extending Boole-type inequalities to this domain. An integral equality involving Riemann–Liouville fractional integrals is established, forming the foundation for deriving novel fractional Boole-type inequalities tailored to differentiable convex functions. The proposed framework encompasses a wide range of functional classes, including Lipschitzian functions, bounded functions, convex functions, and functions of bounded variation, thereby broadening the applicability of these inequalities to diverse mathematical settings. The research emphasizes the importance of the Riemann–Liouville fractional operator in solving problems related to non-integer-order differentiation, highlighting its pivotal role in enhancing classical inequalities. These newly established inequalities offer sharper error bounds for various numerical quadrature formulas in classical calculus, marking a significant advancement in computational mathematics. Numerical examples, computational analysis, applications to quadrature formulas and graphical illustrations substantiate the efficacy of the proposed inequalities in improving the accuracy of integral approximations, particularly within the context of fractional calculus. Future directions for this research include extending the framework to incorporate <i>q</i>-calculus, symmetrized <i>q</i>-calculus, alternative fractional operators, multiplicative calculus, and multidimensional spaces. These extensions would enable a comprehensive exploration of Boole’s formula and its associated error bounds, providing deeper insights into its performance across a broader range of mathematical and computational settings. |
| format | Article |
| id | doaj-art-7d58e0a735934e4a9a9ebacc791dffde |
| institution | Kabale University |
| issn | 2504-3110 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj-art-7d58e0a735934e4a9a9ebacc791dffde2025-01-24T13:33:30ZengMDPI AGFractal and Fractional2504-31102025-01-01915210.3390/fractalfract9010052Numerical Approximations and Fractional Calculus: Extending Boole’s Rule with Riemann–LiouvilleFractional Integral InequalitiesAbdul Mateen0Wali Haider1Asia Shehzadi2Hüseyin Budak3Bandar Bin-Mohsin4Ministry of Education Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, ChinaSchool of Mathematics and Statistics, Central South University, Changsha 410083, ChinaSchool of Mathematics and Statistics, Central South University, Changsha 410083, ChinaDepartment of Mathematics, Faculty of Science and Arts, Düzce University, Düzce 81620, TürkiyeDepartment of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi ArabiaThis paper develops integral inequalities for first-order differentiable convex functions within the framework of fractional calculus, extending Boole-type inequalities to this domain. An integral equality involving Riemann–Liouville fractional integrals is established, forming the foundation for deriving novel fractional Boole-type inequalities tailored to differentiable convex functions. The proposed framework encompasses a wide range of functional classes, including Lipschitzian functions, bounded functions, convex functions, and functions of bounded variation, thereby broadening the applicability of these inequalities to diverse mathematical settings. The research emphasizes the importance of the Riemann–Liouville fractional operator in solving problems related to non-integer-order differentiation, highlighting its pivotal role in enhancing classical inequalities. These newly established inequalities offer sharper error bounds for various numerical quadrature formulas in classical calculus, marking a significant advancement in computational mathematics. Numerical examples, computational analysis, applications to quadrature formulas and graphical illustrations substantiate the efficacy of the proposed inequalities in improving the accuracy of integral approximations, particularly within the context of fractional calculus. Future directions for this research include extending the framework to incorporate <i>q</i>-calculus, symmetrized <i>q</i>-calculus, alternative fractional operators, multiplicative calculus, and multidimensional spaces. These extensions would enable a comprehensive exploration of Boole’s formula and its associated error bounds, providing deeper insights into its performance across a broader range of mathematical and computational settings.https://www.mdpi.com/2504-3110/9/1/52inequalities of Boole’s typefractional calculusquadrature formulasRiemann–Liouville fractional integralserror boundsfunctions with convexity |
| spellingShingle | Abdul Mateen Wali Haider Asia Shehzadi Hüseyin Budak Bandar Bin-Mohsin Numerical Approximations and Fractional Calculus: Extending Boole’s Rule with Riemann–LiouvilleFractional Integral Inequalities Fractal and Fractional inequalities of Boole’s type fractional calculus quadrature formulas Riemann–Liouville fractional integrals error bounds functions with convexity |
| title | Numerical Approximations and Fractional Calculus: Extending Boole’s Rule with Riemann–LiouvilleFractional Integral Inequalities |
| title_full | Numerical Approximations and Fractional Calculus: Extending Boole’s Rule with Riemann–LiouvilleFractional Integral Inequalities |
| title_fullStr | Numerical Approximations and Fractional Calculus: Extending Boole’s Rule with Riemann–LiouvilleFractional Integral Inequalities |
| title_full_unstemmed | Numerical Approximations and Fractional Calculus: Extending Boole’s Rule with Riemann–LiouvilleFractional Integral Inequalities |
| title_short | Numerical Approximations and Fractional Calculus: Extending Boole’s Rule with Riemann–LiouvilleFractional Integral Inequalities |
| title_sort | numerical approximations and fractional calculus extending boole s rule with riemann liouvillefractional integral inequalities |
| topic | inequalities of Boole’s type fractional calculus quadrature formulas Riemann–Liouville fractional integrals error bounds functions with convexity |
| url | https://www.mdpi.com/2504-3110/9/1/52 |
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