Yazar "Mateen, Abdul" seçeneğine göre listele

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    Novel Fractional Boole's-Type Integral Inequalities via Caputo Fractional Operator and Their Implications in Numerical Analysis
    (Mdpi, 2025) Haider, Wali; Mateen, Abdul; Budak, Hueseyin; Shehzadi, Asia; Ciurdariu, Loredana
    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.
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    Numerical Approximations and Fractional Calculus: Extending Boole's Rule with Riemann-LiouvilleFractional Integral Inequalities
    (Mdpi, 2025) Mateen, Abdul; Haider, Wali; Shehzadi, Asia; Budak, Huseyin; Bin-Mohsin, Bandar
    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 q-calculus, symmetrized q-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.
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    Some novel inequalities of Weddle's formula type for Riemann-Liouville fractional integrals with their applications to numerical integration
    (Pergamon-Elsevier Science Ltd, 2025) Mateen, Abdul; Zhang, Zhiyue; Budak, Huseyin; Ozcan, Serap
    In numerical analysis, Weddle's formula is a pivotal tool for approximating definite integrals. The approximation of the definite integrals plays a significant role in numerical methods for differential equations, particularly in the finite volume method. We need to use the best approximation of the integrals to get better results. This paper thoroughly proves integral inequalities for first-time differentiable convex functions in fractional calculus. For this, first, we prove an integral identity involving Riemann-Liouville fractional integrals. Then, with the help of this identity, we prove fractional Weddle's formula-type inequalities for differentiable convex functions. Our approach involves significant functional classes, including convex, Lipschitzian and bounded functions. The primary motivation of this paper is that Weddle's formula should be employed when Simpson's 1/3 formula fails to yield the required precision. Simpson's formula is limited to third-order polynomial approximations, which may only sometimes capture the intricacies of more complex functions. On the other hand, Weddle's formula provides a higher degree of interpolation using sixth-order polynomials, offering a more refined approximation. Additionally, the paper highlights the significance of the Riemann- Liouville fractional operator in addressing problems involving non-integer-order differentiation, showcasing its critical role in enhancing classical inequalities. These new inequalities can help to find the error bounds for different numerical integration formulas in classical calculus. Moreover, we provide some applications to numerical quadrature formulas of these newly established inequalities. These approximations highlight their potential impact on computational mathematics and related fields. Furthermore, we give numerical examples, computational analysis, and graphical representations that show these newly established inequalities are numerically valid.