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<p>This paper rigorously establishes integral inequalities for first-order differentiable convex functions in the context of fractional calculus, tailored to enhance the accuracy of Boole’s formula. Boole’s formula is a valuable tool for approximating definite integrals in numerical analysis, especially in methods involving numerical solutions of differential equations, such as the finite volume method. High-accuracy integral approximations are essential to improve computational precision in these applications. Initially, an integral identity involving Riemann–Liouville fractional integrals is derived, which serves as a foundation for proving fractional Boole’s formula-type inequalities for differentiable convex functions. This approach encompasses significant functional classes, including convex, Lipschitzian, functions of bounded variations, and bounded functions. The results are substantiated by rigorous proofs and accompanied by illustrative graphs, which provide visual insight into the bounds’ accuracy and robustness. These advancements contribute to numerical analysis by improving integral approximation accuracy, enabling higher precision in computational applications. This study thus provides valuable advancements in the theory of numerical integration and fractional calculus, extending the utility of Boole’s formula in high-precision applications and furthering the understanding of error behavior in integral approximations.</p>

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Error Bounds for Boole’s Formula Across Diverse Function Classes with Applications to Riemann–Liouville Fractional Integrals and Numerical Analysis

  • 期刊论文

  • 发表日期：

    2026年7月30日

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  • Wali Haider,
  • Hüseyin Budak,
  • Asia Shehzadi,
  • Abdul Mateen,
  • Haibo Chen

摘要

This paper rigorously establishes integral inequalities for first-order differentiable convex functions in the context of fractional calculus, tailored to enhance the accuracy of Boole’s formula. Boole’s formula is a valuable tool for approximating definite integrals in numerical analysis, especially in methods involving numerical solutions of differential equations, such as the finite volume method. High-accuracy integral approximations are essential to improve computational precision in these applications. Initially, an integral identity involving Riemann–Liouville fractional integrals is derived, which serves as a foundation for proving fractional Boole’s formula-type inequalities for differentiable convex functions. This approach encompasses significant functional classes, including convex, Lipschitzian, functions of bounded variations, and bounded functions. The results are substantiated by rigorous proofs and accompanied by illustrative graphs, which provide visual insight into the bounds’ accuracy and robustness. These advancements contribute to numerical analysis by improving integral approximation accuracy, enabling higher precision in computational applications. This study thus provides valuable advancements in the theory of numerical integration and fractional calculus, extending the utility of Boole’s formula in high-precision applications and furthering the understanding of error behavior in integral approximations.