Novel Fractional Boole’s-Type Integral Inequalities via Caputo Fractional Operator and Their Implications in Numerical Analysis

Wali Haider, Abdul Mateen (), Hüseyin Budak, Asia Shehzadi and Loredana Ciurdariu ()
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Wali Haider: School of Mathematics and Statistics, Central South University, Changsha 410083, China
Abdul Mateen: Ministry of Education Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China
Hüseyin Budak: Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce 81620, Türkiye
Asia Shehzadi: School of Mathematics and Statistics, Central South University, Changsha 410083, China
Loredana Ciurdariu: Departament of Mathematics, Politehnica University of Timișoara, 300006 Timișoara, Romania

Mathematics, 2025, vol. 13, issue 4, 1-23

Abstract: The advancement of fractional calculus, particularly through the Caputo fractional derivative, has enabled more accurate modeling of processes with memory and hereditary effects, driving significant interest in this field. Fractional calculus also extends the concept of classical derivatives and integrals to noninteger (fractional) orders. This generalization allows for more flexible and accurate modeling of complex phenomena that cannot be adequately described using integer-order derivatives. Motivated by its applications in various scientific disciplines, this paper establishes novel n -times fractional Boole’s-type inequalities using the Caputo fractional derivative. For this, a fractional integral identity is first established. Using the newly derived identity, several novel Boole’s-type inequalities are subsequently obtained. The proposed inequalities generalize the classical Boole’s formula to the fractional domain. Further extensions are presented for bounded functions, Lipschitzian functions, and functions of bounded variation, providing sharper bounds compared to their classical counterparts. To demonstrate the precision and applicability of the obtained results, graphical illustrations and numerical examples are provided. These contributions offer valuable insights for applications in numerical analysis, optimization, and the theory of fractional integral equations.

Keywords: Caputo fractional operator; convex function; Lipschitzian function; Boole’s-type integral inequalities (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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