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Integer and Fractional Order q(Alpha)-Delta Integration, Sum and Its Fundamental Theorems.

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Title: Integer and Fractional Order q(Alpha)-Delta Integration, Sum and Its Fundamental Theorems.
Authors: D., Saraswathi1 dswathisaranya@gmail.com, G., Britto Antony Xavier2 brittoshc@gmail.com, S., Geethalakshmi1 geethathiru126@gmail.com, J., Joshvarajarathinam3 jjrr40731@gmail.com
Source: IAENG International Journal of Applied Mathematics. May2025, Vol. 55 Issue 5, p1448-1453. 6p.
Subjects: Integrable functions, Newton-Raphson method, Fractional integrals, Integers
Abstract: This research focuses on developing discrete fundamental theorems related to q(α)-delta anti-difference operator, applied to a class of q(α)-delta integrable functions. The fractional sum of a function f can be expressed in two ways: as a closed form and as a summation form. This dual representation motivates the creation of a new technique to derive several identities for q(α)-delta integrable functions, which possess both discrete anti-difference and integer summation referred to as discrete fundamental theorems. Additionally, by introducing the concept of 1-order q(α)-delta integrable functions, we derive the discrete integral related to the fractional sum of f using Newton's method. The theoretical results are illustrated and validated by numerical examples. [ABSTRACT FROM AUTHOR]
Copyright of IAENG International Journal of Applied Mathematics is the property of International Association of Engineers (IAENG) and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: <searchLink fieldCode="JN" term="%22IAENG+International+Journal+of+Applied+Mathematics%22">IAENG International Journal of Applied Mathematics</searchLink>. May2025, Vol. 55 Issue 5, p1448-1453. 6p.
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  Data: This research focuses on developing discrete fundamental theorems related to q(α)-delta anti-difference operator, applied to a class of q(α)-delta integrable functions. The fractional sum of a function f can be expressed in two ways: as a closed form and as a summation form. This dual representation motivates the creation of a new technique to derive several identities for q(α)-delta integrable functions, which possess both discrete anti-difference and integer summation referred to as discrete fundamental theorems. Additionally, by introducing the concept of 1-order q(α)-delta integrable functions, we derive the discrete integral related to the fractional sum of f using Newton's method. The theoretical results are illustrated and validated by numerical examples. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of IAENG International Journal of Applied Mathematics is the property of International Association of Engineers (IAENG) and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Type: general
      – SubjectFull: Newton-Raphson method
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      – SubjectFull: Fractional integrals
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              Text: May2025
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